Welcome to the Docs for the Forest SDK!¶
The Rigetti Forest Software Development Kit includes pyQuil, the Rigetti Quil Compiler (quilc), and the Quantum Virtual Machine (qvm).
Longtime users of Rigetti Forest will notice a few changes. First, the SDK now contains a downloadable compiler and a QVM. Second, the SDK contains pyQuil 2.0, with significant updates to previous versions. As a result, programs written using previous versions of the Forest toolkit will need to be updated to pyQuil 2.0 to be compatible with the QVM or compiler.
After installing the SDK and updating pyQuil in Installation and Getting Started, see the migration guide to get caught up on what’s new!
Quantum Cloud Services will provide users with a dedicated Quantum Machine Image, which will come prepackaged with the Forest SDK. We’re releasing a Preview to the Forest SDK now, so current users can begin migrating code (and share feedback with us early and often!). Longtime Forest users should start with the Migration Guide which outlines key changes in this SDK Preview release.
If you’re new to Forest, we hope this documentation will provide everything you need to get up and running with the toolkit. Once you’ve oriented yourself here, proceed to the section Installation and Getting Started to get started. If you’re new to quantum computing, you also go to our section on Introduction to Quantum Computing. There, you’ll learn the basic concepts needed to write quantum software. You can also work through an introduction to quantum computing in a jupyter notebook; launch the notebook from the source folder in pyquil’s docs:
A few terms to orient you as you get started with Forest:
 pyQuil: An open source Python library to help you write and run quantum programs. The source is hosted on github.
 Quil: The Quantum Instruction Language standard. Instructions written in Quil can be executed on any implementation of a quantum abstract machine, such as the quantum virtual machine (QVM), or on a real quantum processing unit (QPU). More details regarding Quil can be found in the whitepaper, A Practical Quantum Instruction Set Architecture.
 QVM: The Quantum Virtual Machine is an open source implementation of a quantum abstract machine on
 classical hardware. The QVM lets you use a regular computer to simulate a small quantum computer and execute Quil programs. Find QVM on GitHub.
 QPU: Quantum processing unit. This refers to the physical hardware chip which we run quantum programs on.
 Quil Compiler: The compiler,
quilc
, compiles Quil written for one quantum abstract machine (QAM) to another. Our open source compiler will take arbitrary Quil and compile it for the given QAM, according to its supported instruction set architecture. Find quilc on GitHub.  Forest SDK: Our software development kit, optimized for nearterm quantum computers that operate as coprocessors, working in concert with traditional processors to run hybrid quantumclassical algorithms. For references on problems addressable with nearterm quantum computers, see Quantum Computing in the NISQ era and beyond.
Our flagship product Quantum Cloud Services offers users an onpremise, dedicated access point to our quantum computers. This access point is a fullyconfigured VM, which we call a Quantum Machine Image. A QMI is bundled with the same downloadable SDK mentioned above, and a command line interface (CLI), which is used for scheduling compute time on our quantum computers. To sign up for our waitlist, please click the link above. If you’d like to access to our quantum computers for research, please email support@rigetti.com.
Note
To join our user community, connect to the Rigetti Slack workspace at https://rigettiforest.slack.com.
Contents¶
Installation and Getting Started¶
To make full use of the Rigetti Forest SDK, you will need pyQuil, the QVM, and the Quil Compiler. On this page, we will take you through the process of installing all three of these. We also step you through running a basic pyQuil program.
Note
If you’re running from a Quantum Machine Image, installation has been completed for you. Continue to Getting Started.
Upgrading or Installing pyQuil¶
PyQuil 2.0 is our library for generating and executing Quil programs on the Rigetti Forest platform.
Before you install, we recommend that you activate a Python 3.6+ virtual environment. Then, install pyQuil using pip:
pip install pyquil
For those of you that already have pyQuil, you can upgrade with:
pip install upgrade pyquil
If you would like to stay up to date with the latest changes and bug fixes, you can also opt to install pyQuil from the source here.
Note
PyQuil requires Python 3.6 or later.
Downloading the QVM and Compiler¶
The Forest 2.0 Downloadable SDK Preview currently contains:
 The Rigetti Quantum Virtual Machine (
qvm
) which allows highperformance simulation of Quil programs  The Rigetti Quil Compiler (
quilc
) which allows compilation and optimization of Quil programs to native gate sets
The QVM and the compiler are packed as program binaries that are accessed through the command line. Both of them provide support for direct commandline interaction, as well as a server mode. The server mode is required for use with pyQuil.
Request the Forest SDK here. You’ll receive an email right away with the download links for macOS, Linux (.deb), Linux (.rpm), and Linux (barebones).
All installation mechanisms, except the barebones package, require administrative privileges to install. To use the QVM and Quil Compiler from the barebones package, you will have to install the prerequisite dependencies on your own.
Note
You can also find the open source code for quilc and qvm on GitHub, where you can find instructions for compiling, installing, and contributing to the compiler and QVM.
Installing on macOS¶
Mount the file forestsdk.dmg
by double clicking on it in your email. From there, open forestsdk.pkg
by
doubleclicking on it. Follow the installation instructions.
Upon successful installation, one should be able to open a new terminal window and run the following two commands:
qvm version
quilc version
To uninstall, delete the following files:
/usr/local/bin/qvm
/usr/local/bin/quilc
/usr/local/share/man/man1/qvm.1
/usr/local/share/man/man1/quilc.1
Installing the QVM and Compiler on Linux (deb)¶
Download the Debian distribution by clicking on the link in your email. Unpack the tarball and change to that directory by doing:
tar xf forestsdklinuxdeb.tar.bz2
cd forestsdk2.0rc2linuxdeb
From here, run the following command:
sudo ./forestsdk2.0rc2linuxdeb.run
Upon successful installation, one should be able to run the following two commands:
qvm version
quilc version
To uninstall, type:
sudo apt remove forestsdk
Installing the QVM and Compiler on Linux (rpm)¶
Download the RPMbased distribution by clicking on the link in your email. Unpack the tarball and change to that directory by doing:
tar xf forestsdklinuxrpm.tar.bz2
cd forestsdk2.0rc2linuxrpm
From here, run the following command:
sudo ./forestsdk2.0rc2linuxrpm.run
Upon successful installation, one should be able to run the following two commands:
qvm version
quilc version
To uninstall, type:
sudo rpm e forestsdk
# or
sudo yum uninstall forestsdk
Installing the QVM and Compiler on Linux (barebones)¶
The barebones installation only contains the executable binaries and
manual pages, and doesn’t contain any of the requisite dynamic
libraries. As such, installation doesn’t require administrative or
sudo
privileges.
First, unpack the tarball and change to that directory by doing:
tar xf forestsdklinuxbarebones.tar.bz2
cd forestsdk2.1linuxbarebones
From here, run the following command:
./forestsdk2.1linuxbarebones.run
Upon successful installation, this will have created a new directory rigetti
in your home directory that contains all
of the binary and documentation artifacts.
This method of installation requires one, through whatever means, to install shared libraries for BLAS, LAPACK, and libffi. On a Debianderivative system, this could be accomplished with
sudo aptget install liblapackdev libblasdev libffidev libzmq3dev
Or on any rhelderivative systems (e.g. Amazon Linux) with
sudo yum install y lapackdevel blasdevel epelrelease
sudo yum install y zeromq3devel
To uninstall, remove the directory ~/rigetti
.
Getting Started¶
To get started using the SDK, you can either interact with the QVM and the compiler directly from the command line, or you can run them in server mode and use them with pyQuil. In this section, we’re going to explain how to do the latter.
For more information about directly interacting with the QVM and the compiler, refer to their respective manual pages.
After installation, you can read the manual pages by opening a new terminal window and typing man qvm
(for the QVM)
or man quilc
(for the compiler). Quit out of the manual page by typing q
.
Setting Up Server Mode for PyQuil¶
Note
This set up is only necessary to run pyQuil locally. If you’re running in a QMI, this has already been done for you.
It’s easy to start up local servers for the QVM and quilc on your laptop. You should have two terminal windows open
to run in the background. We recommend using a resource such as tmux
for running and managing multiple programs in one
terminal.
### CONSOLE 1
$ qvm S
Welcome to the Rigetti QVM
(Configured with 10240 MiB of workspace and 8 workers.)
[20180920 15:39:50] Starting server on port 5000.
### CONSOLE 2
$ quilc S
...  Launching quilc.
...  Spawning server at (tcp://*:5555) .
That’s it! You’re all set up to run pyQuil locally. Your programs will make requests to these server endpoints to compile your Quil programs to native Quil, and to simulate those programs on the QVM.
NOTE: We are transitioning from using an HTTP quilc
server to an RPCQ one.
In the near term, S
will spawn an HTTP server at port 6000 and an RPCQ server
at port 5555 (accessible via tcp://localhost:5555
).
Run Your First Program¶
Now that our local endpoints are up and running, we can start running pyQuil programs! We will run a simple program on the Quantum Virtual Machine (QVM).
The program we will create prepares a fully entangled state between two qubits, called a Bell State. This state is in an equal superposition between 00⟩ and 11⟩, meaning that it is equally likely that a measurement will result in measuring both qubits in the ground state or both qubits in the excited state. For more details about the physics behind these concepts, see Introduction to Quantum Computing.
To begin, start up python however you like. You can open a jupyter notebook (type jupyter notebook
in your terminal),
open an interactive python notebook in your terminal (with ipython3
), or simply launch python in your terminal
(type python3
). Recall that you need Python 3.6+ to use pyQuil.
Import a few things from pyQuil:
from pyquil import Program, get_qc
from pyquil.gates import *
The Program
object allows us to build up a Quil program. get_qc()
connects us to a
QuantumComputer
object, which specifies what our program should run on (see: The Quantum Computer). We’ve also imported all (*
)
gates from the pyquil.gates
module, which allows us to add operations to our program (Programs and Gates).
Note
PyQuil also provides a handy function for you to ensure that a local qvm and quilc are currently running in your environment. To make sure both are available you import from pyquil.api import local_qvm and then run local_qvm(). This will start a qvm and quilc instances using subprocesses if they have not already been started. You can also use it as a context manager as in the following example:
from pyquil import get_qc, Program
from pyquil.gates import CNOT, Z
from pyquil.api import local_qvm
qvm = get_qc('9qsquareqvm')
prog = Program(Z(0), CNOT(0, 1))
with local_qvm():
results = qvm.run_and_measure(prog, trials=10)
Next, let’s construct our Bell State.
# construct a Bell State program
p = Program(H(0), CNOT(0, 1))
We’ve accomplished this by driving qubit 0 into a superposition state (that’s what the “H” gate does), and then creating an entangled state between qubits 0 and 1 (that’s what the “CNOT” gate does). Finally, we’ll want to run our program:
# run the program on a QVM
qc = get_qc('9qsquareqvm')
result = qc.run_and_measure(p, trials=10)
print(result[0])
print(result[1])
Compare the two arrays of measurement results. The results will be correlated between the qubits and random from shot to shot.
The qc
is a simulated quantum computer. By specifying we want to .run_and_measure
, we’ve told our QVM to run
the program specified above, collapse the state with a measurement, and return the results to us. trials
refers to
the number of times we run the whole program.
The call to run_and_measure
will make a request to the two servers we
started up in the previous section: first, to the quilc
server
instance to compile the Quil program into native Quil, and then to the qvm
server
instance to simulate and return measurement results of the program 10 times. If you open up the terminal windows where your servers
are running, you should see output printed to the console regarding the requests you just made.
In the following sections, we’ll cover gates, program construction & execution, and go into detail about our Quantum Virtual Machine, our QPUs, noise models and more. If you’ve used pyQuil before, continue on to our New in Forest 2  Other. Once you’re set with that, jump to Programs and Gates to continue.
New in Forest 2  Running on the QVM¶
PyQuil is for constructing and running quantum programs on real quantum computers. With the release of pyQuil 2, we have changed parts of the API (compared to pyQuil 1.x) to better reflect that focus.
Program construction is largely the same¶
To construct runandmeasurestyle programs, there are no changes in program construction. When using explicit MEASURE
instructions or using parametric programs for massive speed increases, please read about the new quil memory model.
Below, we build a program that constructs a Bell state.
[1]:
from pyquil import Program
from pyquil.gates import *
program = Program(
H(0),
CNOT(0, 1),
)
print(program)
H 0
CNOT 0 1
Unphysical methods deprecated in QVMConnection
¶
In pyQuil 1.x, you would use the QVMConnection
to run quantum programs on a webhosted quantum simulator (the “Quantum Virtual Machine”). To run on a real quantum processor (QPU), you would switch all instances of QVMConnection
to QPUConnection
. However, QVMConnection
let you do many unphysical operations that are unsupported on a real QPU. These methods are detailed below and have been deprecated in favor of a new API that clearly delineats the physical from the unphysical.
Wavefunction¶
When simulating a quantum program, we can inspect its wavefunction. This is very useful! It is–however–impossible to inspect the wavefunction of a real quantum processor. This is one of the central tenets of quantum information! Attempting to switch instances of QVMConnection
to QPUConnection
results in an error. Additionally, using QVMConnection
with a noise model will cause the wavefunction
call to behave badly.
The old way¶
[2]:
from pyquil.api import QVMConnection
wfn_old = QVMConnection().wavefunction(program)
print(wfn_old)
(0.7071067812+0j)00> + (0.7071067812+0j)11>
The new way¶
WavefunctionSimulator
encapsulates all functionality that requires peering into a wavefunction. This also opens the door for different types of simulators other than those backed by a wavefunction. For example, you can simulate a quantum circuit with a density matrix simulation or a path integral simulation.
[3]:
from pyquil.api import WavefunctionSimulator
wfn = WavefunctionSimulator().wavefunction(program)
print(wfn)
(0.7071067812+0j)00> + (0.7071067812+0j)11>
Expectation¶
Many quantum algorithms involve calculating expectation values of quantum observables. We can represent quantum observables with PauliSum
s. When using a simulator, it is possible to exactly calculate expectation values (a consequence of having access to the full wavefunction) whereas when running on a QPU you must estimate expectation values by sampling from many runs of a program.
Let’s use our program defined above to construct a bell state.
[4]:
print(program)
H 0
CNOT 0 1
And we wish to measure the following quantum observables \(O\)  the expected value of the 0th bit \(O=(1Z_0)/2\)  the expected value of the 1st bit \(O=(1Z_1)/2\)  the expected value of the exclusive or (XOR) between the two qubits \(O=(1ZZ)/2\)
Exercise for the reader: convince yourself that \((1ZZ)/2\) is the XOR function
[5]:
from pyquil.paulis import sZ
z0 = (1sZ(0))*0.5
z1 = (1sZ(1))*0.5
xor = (1sZ(0)*sZ(1))*0.5
A Bell state is entangled, so each of the 1qubit operators should give an expectation value of 0.5 (as each qubit taken individually has a 50% change of being measured in the 0 or 1 state) whereas the twoqubit operator should give an expectation value of 0 (as the two qubits considered together will always have even parity)
The old way¶
Because of technical details, QVMConnection.expectation
requires transforming your observable represented as a PauliSum
into a program, and then remembering to multiply back in the coefficients. A more usable API was introduced in Pyquil 1.9 and was given the name pauli_expectation
.
[6]:
for observable in [z0, z1, xor]:
expectation = QVMConnection().pauli_expectation(prep_prog=program, pauli_terms=observable)
print(observable, '\t', expectation)
(0.5+0j)*I + (0.5+0j)*Z0 (0.4999999999999999+0j)
(0.5+0j)*I + (0.5+0j)*Z1 (0.4999999999999999+0j)
(0.5+0j)*I + (0.5+0j)*Z0*Z1 0j
The new way¶
If you want analytical expectation values, the solution is to use WavefunctionSimulator.expectation
Note that the method is not named pauli_expectation
as we do not support the pyQuil<1.9 way of using Program
s to represent PauliSum
s
[7]:
for observable in [z0, z1, xor]:
expectation = WavefunctionSimulator().expectation(prep_prog=program, pauli_terms=observable)
print(observable, '\t', expectation)
(0.5+0j)*I + (0.5+0j)*Z0 (0.4999999999999999+0j)
(0.5+0j)*I + (0.5+0j)*Z1 (0.4999999999999999+0j)
(0.5+0j)*I + (0.5+0j)*Z0*Z1 0j
Sampling expectations on a QPU.¶
Estimating expectation values via sampling from a QPU is often something we would like to do! Please look forward for helper functions for sampling observables in the near future.
Async methods removed¶
pyQuil 2 is tailored for Rigetti’s Quantum Cloud Services (QCS). In prior releases, executing programs on either the QVM or QPU involved communicating with a cloudhosted endpoint. Now, you are empowered with
 A preconfigured quantum machine image (QMI) with your own dedicated qvm and quilc instance.
 lowlatency QPU access all to yourself during a scheduled time window
 A downloadable local version of the qvm and quilc
Taken together, there is no longer any motivation for supporting asynchronous access to either the QVM or QPU.
When running many programs it was often possible to reduce runtime by batching jobs and exploiting the async queue. The following example does not work in pyQuil 2 but gives a sketch about how this would have worked.
import numpy as np
cxn = QVMConnection()
thetas = np.linspace(0, 2*np.pi, 20)
jobs = [Program(RY(theta, 0)) for theta in thetas]
job_ids = [cxn.run_and_measure_async(job, qubits=[0], trials=1000) for job in jobs]
bitstrings = [np.asarray(cxn.wait_for_job(job)) for job in jobs]
Since this is such an important use case, there have been many changes to support running many programs as quickly as possible. We demonstrate an equivalent, synchronous version of the example given above. To idiomatically run this set of jobs, there are additional features you should use that are not covered in this document. Please continue reading the documentation, especially the page covering parametric programs.
[8]:
import numpy as np
cxn = QVMConnection()
thetas = np.linspace(0, np.pi, 20)
bitstrings = [np.asarray(cxn.run_and_measure(Program(RY(theta, 0)), qubits=[0], trials=1000))
for theta in thetas]
[9]:
%matplotlib inline
from matplotlib import pyplot as plt
averages = [np.mean(bs[:,0]) for bs in bitstrings]
_ = plt.plot(thetas, averages, 'o')
New in Forest 2  QuantumComputer¶
PyQuil is for constructing and running quantum programs on real quantum computers. With the release of pyQuil 2, we have changed parts of the API to better reflect that focus. Instead of swapping between a QVMConnection
and a QPUConnection
, you will primarily deal with a QuantumComputer
with consistent API and behavior regardless of
 QVM / QPU
 Presence of noise model
 Device topology
Running a program¶
Let’s show how you can run a simple program on a QuantumComputer
first we start with the relevant imports.
[1]:
from pyquil import Program
from pyquil.gates import *
We’ll write a function that takes a list of qubits and returns a pyQuil Program
that constructs an entangled “GHZ” state. This is a generalization of the twoqubit Bell state.
[2]:
def ghz_state(qubits):
"""Create a GHZ state on the given list of qubits by applying
a Hadamard gate to the first qubit followed by a chain of CNOTs
"""
program = Program()
program += H(qubits[0])
for q1, q2 in zip(qubits, qubits[1:]):
program += CNOT(q1, q2)
return program
For example, creating a GHZ state on qubits 1, 2, and 3 would look like:
[3]:
program = ghz_state(qubits=[0, 1, 2])
print(program)
H 0
CNOT 0 1
CNOT 1 2
Debugging with WavefunctionSimulator
¶
We can check that this program gives us the desired wavefunction by using WavefunctionSimulator.wavefunction()
[4]:
from pyquil.api import WavefunctionSimulator
wfn = WavefunctionSimulator().wavefunction(program)
print(wfn)
(0.7071067812+0j)000> + (0.7071067812+0j)111>
We can’t get the wavefunction from a real quantum computer though, so instead we’ll sample bitstrings. We expect to always measure the bitstring 000 or the bitstring 111 based on the definition of a GHZ state and confirmed by our wavefunction simulation.
get_qc
¶
We’ll construct a QuantumComputer
via the helper method get_qc. You may be tempted to use the QuantumComputer
constructor directly. Please refer to the advanced documentation to see how to do that. Our program uses 3 qubits, so we’ll ask for a 3qubit QVM.
[5]:
from pyquil import get_qc
qc = get_qc('3qqvm')
qc
[5]:
QuantumComputer[name="3qqvm"]
We can do a quick check to make sure it has 3 qubits
[6]:
qc.qubits()
[6]:
[0, 1, 2]
Sampling with run_and_measure
¶
QuantumComputer.run_and_measure
will run a given program (that does not have explicit MEASURE
instructions) and then measure all qubits present in the quantum computer.
[7]:
bitstrings = qc.run_and_measure(program, trials=10)
bitstrings
[7]:
{0: array([1, 0, 0, 1, 1, 1, 1, 0, 1, 0]),
1: array([1, 0, 0, 1, 1, 1, 1, 0, 1, 0]),
2: array([1, 0, 0, 1, 1, 1, 1, 0, 1, 0])}
Let’s programatically verify that we always measure 000 or 111 by “summing” each bitstring and checking if it’s eather 0 (for 000) or 3 (for 111)
[8]:
import numpy as np
bitstring_array = np.vstack(bitstrings[q] for q in qc.qubits()).T
sums = np.sum(bitstring_array, axis=1)
sums
[8]:
array([3, 0, 0, 3, 3, 3, 3, 0, 3, 0])
[9]:
sample_is_ghz = np.logical_or(sums == 0, sums == 3)
sample_is_ghz
[9]:
array([ True, True, True, True, True, True, True, True, True,
True])
[10]:
np.all(sample_is_ghz)
[10]:
True
Change alert: run_and_measure
will return a dictionary of 1d bitstrings.¶
Not a 2d array. To demonstrate why, consider a lattice whose qubits are not contiguously indexed from 0.
[11]:
# TODO: we need a lattice that is not zeroindexed
# qc = get_qc('Aspen03QB')
# qc.run_and_measure(ghz_state(qubits=[1,2,3]))
Change alert: All qubits are measured¶
PyQuil 1.x’s run_and_measure
would only measure qubits used in the given program. Now all qubits (per qc.qubits()
) are measured. This is easier to reason about and reflects the reality of running on a QPU. When accounting for noise or when running QCVV tasks, you may be interested in the measurement results of qubits that weren’t even used in your program!
[12]:
qc = get_qc('4qqvm')
bitstrings = qc.run_and_measure(Program(X(0), X(1), X(2)), trials=10)
bitstrings
[12]:
{0: array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1]),
1: array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1]),
2: array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1]),
3: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])}
You can drop qubits you’re not interested in by indexing into the returned dictionary
[13]:
# Stacking everything
np.vstack(bitstrings[q] for q in qc.qubits()).T
[13]:
array([[1, 1, 1, 0],
[1, 1, 1, 0],
[1, 1, 1, 0],
[1, 1, 1, 0],
[1, 1, 1, 0],
[1, 1, 1, 0],
[1, 1, 1, 0],
[1, 1, 1, 0],
[1, 1, 1, 0],
[1, 1, 1, 0]])
[14]:
# Stacking what you want (contrast with above)
qubits = [0, 1, 2]
np.vstack(bitstrings[q] for q in qubits).T
[14]:
array([[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]])
Change alert: run_and_measure
works with noise models now.¶
In pyQuil 1.x, run_and_measure
would not work with noise models. Now noise models are supported. Preconfigured noise models can be used via get_qc('xxxxnoisyqvm')
.
As a consequence, run_and_measure
for large numbers of trials will be slower in Pyquil 2.
[15]:
qc = get_qc('3qnoisyqvm')
bitstrings = qc.run_and_measure(program, trials=10)
bitstrings
[15]:
{0: array([0, 1, 1, 0, 0, 1, 1, 0, 1, 1]),
1: array([0, 1, 1, 0, 0, 1, 1, 0, 1, 1]),
2: array([0, 1, 1, 0, 0, 1, 1, 0, 1, 1])}
[16]:
bitstring_array = np.vstack(bitstrings[q] for q in qc.qubits()).T
sums = np.sum(bitstring_array, axis=1)
sums
[16]:
array([0, 3, 3, 0, 0, 3, 3, 0, 3, 3])
[17]:
# Noise means now we measure things other than 000 or 111
np.all(np.logical_or(sums == 0, sums == 3))
[17]:
True
list_quantum_computers
¶
You can find all possible arguments to get_qc
with list_quantum_computers
[18]:
from pyquil import list_quantum_computers
# TODO: unauthenticated endpoint
# list_quantum_computers()
QuantumComputer
s have a topology¶
An important restriction when running on a real quantum computer is the mapping of qubits to the supported twoqubit gates. The QVM is designed to provide increasing levels of “realism” to guarantee that if your program executes successfully on get_qc("Aspenxxxnoisyqvm")
then it will execute successfully on get_qc("Aspenxxx")
*
* guarantee not currently guaranteed. This is a work in progress.
Inspecting the topology¶
You can access a topology by qc.qubit_topology()
, which will return a NetworkX representation of qubit connectivity. You can access the full set of supported instructions by qc.get_isa()
. For example, we include a generic QVM named "9qsquareqvm"
that has a square topology.
[19]:
qc = get_qc('9qsquareqvm')
%matplotlib inline
import networkx as nx
nx.draw(qc.qubit_topology())
from matplotlib import pyplot as plt
_ = plt.title('9qsquareqvm', fontsize=18)
What If I don’t want a topology?¶
WavefunctionSimulator
still has no notion of qubit connectivity, so feel free to use that for simulating quantum algorithms that you aren’t concerned about running on an actual QPU.
Above we used get_qc("3qqvm")
, "4qqvm"
, and indeed you can do any "{n}qqvm"
(subject to computational resource constraints). These QVM’s are constructed with a topology! It just happens to be fully connected
[20]:
nx.draw(get_qc('5qqvm').qubit_topology())
_ = plt.title('5qqvm is fully connected', fontsize=16)
Heirarchy of realism¶
WavefunctionSimulator
to debug algorithmget_qc("5qqvm")
to debug samplingget_qc("9qsquareqvm")
to debug mapping to a latticeget_qc("9qsquarenoisyqvm"
) to debug generic noise characteristicsget_qc("Aspen016QAqvm")
to debug mapping to a real latticeget_qc("Aspen016QAnoisyqvm")
to debug noise characteristics of a real deviceget_qc("Aspen016QA")
to run on a real device
“What is a QuantumComputer
?” Advanced Edition¶
A QuantumComputer
is a wrapper around three constituent parts, each of which has a programatic interface that must be respected by all classes that implement the interface. By having clear interfaces we can write backendagnostic methods on QuantumComputer
and mixandmatch backing objects.
The following diagram shows the three objects that must be provided when constructing a QuantumComputer
“by hand”. The abstract classes are backed in grey with example implementing classes listed below. Please consult the api reference for details on each interface.
As an example, let’s construct a 5qubit QVM with one central node and only even numbered qubits.
[21]:
topology = nx.from_edgelist([
(10, 2),
(10, 4),
(10, 6),
(10, 8),
])
from pyquil.device import NxDevice
device = NxDevice(topology)
from pyquil.api._qac import AbstractCompiler
class MyLazyCompiler(AbstractCompiler):
def quil_to_native_quil(self, program):
return program
def native_quil_to_executable(self, nq_program):
return nq_program
from pyquil.api import QuantumComputer, QVM, ForestConnection
my_qc = QuantumComputer(
name='myqvm',
qam=QVM(connection=ForestConnection()),
device=device,
compiler=MyLazyCompiler(),
)
nx.draw(my_qc.qubit_topology())
[22]:
my_qc.run_and_measure(Program(X(10)), trials=5)
[22]:
{2: array([0, 0, 0, 0, 0]),
4: array([0, 0, 0, 0, 0]),
6: array([0, 0, 0, 0, 0]),
8: array([0, 0, 0, 0, 0]),
10: array([1, 1, 1, 1, 1])}
New in Forest 2  Parametric Programs¶
pyQuil is for constructing and running hybrid quantum/classical algorithms on real quantum computers. With the release of pyQuil 2, we have changed parts of the API to take advantage of some exciting new features available on QCS.
A hybrid algorithm involves using the quantum computer to create a quantum state that would be difficult to prepare classically; measure it in a way particular to your problem; and then update your procedure for creating the state so that the measurements are closer to the correct answer. A real hybrid algorithm involves structured ansatze like QAOA for optimization or a UCC ansatz for chemistry. Here, we’ll do a much simpler parameterized program
[1]:
from pyquil import Program, get_qc
from pyquil.gates import *
def ansatz(theta):
program = Program()
program += RY(theta, 0)
return program
print(ansatz(theta=0.2))
RY(0.2) 0
Scan over the parameter (the old way)¶
For this extrordinarily simple ansatz, we can discretize the parameter theta and try all possible values. As the number of parameters increases, the number of combinations increases exponentially so doing a full grid search will become intractable for anything more than ~two parameters.
[2]:
import numpy as np
qc = get_qc("9qsquareqvm")
thetas = np.linspace(0, 2*np.pi, 21)
results = []
for theta in thetas:
program = ansatz(theta)
bitstrings = qc.run_and_measure(program, trials=1000)
results.append(np.mean(bitstrings[0]))
[3]:
%matplotlib inline
from matplotlib import pyplot as plt
plt.plot(thetas, results, 'o')
plt.xlabel(r'$\theta$', fontsize=18)
_ = plt.ylabel(r'$\langle \Psi(\theta)  \frac{1  Z}{2}  \Psi(\theta) \rangle$',
fontsize=18)
Do an optimization (the old way)¶
Instead of doing a full grid search, we will employ a classical optimizer to find the best parameter values. Here we use scipy to find the theta that results in sampling the most 1
s in our resultant bitstrings.
[4]:
def objective_function(theta):
program = ansatz(theta[0])
bitstrings = qc.run_and_measure(program, trials=1000)
result = np.mean(bitstrings[0])
return result
import scipy.optimize
res = scipy.optimize.minimize(objective_function, x0=[0.1], method='COBYLA')
res
[4]:
fun: 1.0
maxcv: 0.0
message: 'Optimization terminated successfully.'
nfev: 13
status: 1
success: True
x: array([3.1])
[5]:
plt.plot(thetas, results, label='scan')
plt.plot([res.x], [res.fun], '*', ms=20, label='optimization result')
plt.legend()
[5]:
<matplotlib.legend.Legend at 0x1015dedf28>
Compilation¶
Prior to QCS, a QPU job would be routed via a series of cloudbased queues and compilation steps. With Forest 2, you are in control of the two stages of compilation so you can amortize the cost of compiling. Your QMI and all classical infrastructure is hosted on the Rigetti premises, so network latency is minimal.
Quil to native quil¶
The first step of compilation converts gates to their hardwaresupported equivalent. For example, our parametric RY is converted into RX’s and RZ’s because these are physically realizable on a Rigetti QPU
[6]:
nq_program = qc.compiler.quil_to_native_quil(ansatz(theta=0.5))
print(nq_program)
PRAGMA EXPECTED_REWIRING "#(0 1 2 3 4 5 6 7 8)"
RX(pi/2) 0
RZ(0.5) 0
RX(pi/2) 0
PRAGMA CURRENT_REWIRING "#(0 1 2 3 4 5 6 7 8)"
PRAGMA EXPECTED_REWIRING "#(0 1 2 3 4 5 6 7 8)"
PRAGMA CURRENT_REWIRING "#(0 1 2 3 4 5 6 7 8)"
Native quil to executable¶
The second step of compilation will turn named gates into calibrated pulses stored in a binary format suitable for consumption by the control electronics. This means that you can fully compile a given program and run it many times with minimal classical overhead.
Note: since we’re using a QVM, for which there is no binary format, this stage is mocked out and you can see the original Quil inside the PyQuilExecutableResponse
that is returned. When running on the QPU, this will return a BinaryExecutableResponse
whose contents are opaque.
TODO: obscure the contents of PyQuilExecutableResponse
: https://github.com/rigetti/pyquil/issues/700
[7]:
qc.compiler.native_quil_to_executable(nq_program)
[7]:
PyQuilExecutableResponse(attributes={'native_quil_metadata': {'finalrewiring': [0, 1, 2, 3, 4, 5, 6, 7, 8], 'topological_swaps': 0, 'gate_depth': 3, 'gate_volume': 3, 'program_duration': 18.01, 'program_fidelity': 1.0, 'multiqubit_gate_depth': 0}, 'num_shots': 1}, program='PRAGMA EXPECTED_REWIRING "#(0 1 2 3 4 5 6 7 8)"\nRX(pi/2) 0\nRZ(0.5) 0\nRX(pi/2) 0\nPRAGMA CURRENT_REWIRING "#(0 1 2 3 4 5 6 7 8)"\nPRAGMA EXPECTED_REWIRING "#(0 1 2 3 4 5 6 7 8)"\nPRAGMA CURRENT_REWIRING "#(0 1 2 3 4 5 6 7 8)"\n')
Parametric compilation¶
This doesn’t buy us much if we have to know exactly what circuit we want to run before compiling it and amortizing the compilation cost. Maybe you could get away with it when you’re doing a parameter scan, but for hybrid algorithms, the circuit parameter (here: theta
) depends on the results of a circuit before. This is the essence of hybrid programming! Therefore, all compilation steps have been upgraded to support named, symbolic parameters that will be updated at runtime with minimal
overhead.
With this feature, you can compile a parametric program once and run it many times will different parameter values and you need not know the parameter values at compilation time.
There are a couple of prerequisites to use this feature effectively from PyQuil, which we address in this document.
First, you must declare a parameter when constructing your quil program. When declaring a named classical variable, you must specify at least a name and a type. It is conventional to make sure the Python variable name of the reference memory matches the Quil variable name. In our case, we name both things theta
. Our circuit above would be modified in this way:
[8]:
program = Program()
theta = program.declare('theta', memory_type='REAL')
program += RY(theta, 0)
print(program)
DECLARE theta REAL[1]
RY(theta) 0
Measuring¶
In the documentation so far, we’ve been using the run_and_measure
functionality of QuantumComputer
. It’s time to get our hands dirty and introduce explicit measure instructions.
Above, we declared a classical piece of memory, we’ve given it a name (theta
), and we’ve given it a type (REAL
). The bits that we measure (or “read out” – ro
for short) must now also be declared, given a name, and a type. Additionally, we’ll usually be measuring more than one qubit so we can give this register a size.
The index of the readout register need not match the qubit index. For example below, we create a bell state on qubits 5 and 6 and measure the readout results into ro[0]
and ro[1]
.
Note: The readout register must be named “ro” (for now)
[9]:
program = Program()
ro = program.declare('ro', memory_type='BIT', memory_size=2)
program += H(5)
program += CNOT(5, 6)
program += MEASURE(5, ro[0])
program += MEASURE(6, ro[1])
print(program)
DECLARE ro BIT[2]
H 5
CNOT 5 6
MEASURE 5 ro[0]
MEASURE 6 ro[1]
Our very simple ansatz only has one qubit, so the measurement is quite simple.
[10]:
program = Program()
theta = program.declare('theta', memory_type='REAL')
ro = program.declare('ro', memory_type='BIT', memory_size=1)
program += RY(theta, 0)
program += MEASURE(0, ro[0])
print(program)
DECLARE theta REAL[1]
DECLARE ro BIT[1]
RY(theta) 0
MEASURE 0 ro[0]
Number of shots¶
The number of trials is compiled into the executable binary, so we must specify this number prior to compilation.
TODO: add to str / repr https://github.com/rigetti/pyquil/issues/701
[11]:
program = Program()
theta = program.declare('theta', memory_type='REAL')
ro = program.declare('ro', memory_type='BIT', memory_size=1)
program += RY(theta, 0)
program += MEASURE(0, ro[0])
program.wrap_in_numshots_loop(shots=1000)
print(program)
DECLARE theta REAL[1]
DECLARE ro BIT[1]
RY(theta) 0
MEASURE 0 ro[0]
Using qc.run()
¶
To use the lowerlevel but more powerful qc.run
interface, we have had to take control of these three things
 We decalred a readout register named
ro
of typeBIT
and included explicitMEASURE
instructions. Since this sets up a (potentially sprase) mapping from qubits to classical addresses, we can expectqc.run()
to return the classic 2d ndarray of yore instead of the dictionary returned byrun_and_measure
 We have called
program.wrap_in_numshots_loop()
prior to compilation so the number of shots can be encoded in an efficient binary representation of the program  We have taken control of compilation; either by calling
qc.compile(program)
or by using the lowerlevel functions:nq_program = qc.compiler.quil_to_native_quil(program) executable = qc.compiler.native_quil_to_executable(nq_program)
[12]:
def ansatz(theta):
program = Program()
ro = program.declare('ro', memory_type='BIT', memory_size=1)
program += RY(theta, 0)
program += MEASURE(0, ro[0])
return program
print(ansatz(theta=np.pi))
DECLARE ro BIT[1]
RY(pi) 0
MEASURE 0 ro[0]
We can run the program with a preset angle (here, theta = np.pi
).
[13]:
program = ansatz(theta=np.pi)
program.wrap_in_numshots_loop(shots=5)
executable = qc.compile(program)
bitstrings = qc.run(executable)
print(bitstrings.shape)
bitstrings
(5, 1)
[13]:
array([[1],
[1],
[1],
[1],
[1]])
Scan over the parameter (the new way)¶
Finally, all the pieces are in place to compile and run parameterized executable binaries. We declare parameters that will be compiled symbolically into the binary allowing us to amortize the cost of compilation when running hybrid algorithms.
[14]:
def ansatz():
program = Program()
theta = program.declare('theta', memory_type='REAL')
ro = program.declare('ro', memory_type='BIT', memory_size=1)
program += RY(theta, 0)
program += MEASURE(0, ro[0])
return program
print(ansatz())
DECLARE theta REAL[1]
DECLARE ro BIT[1]
RY(theta) 0
MEASURE 0 ro[0]
Using memory_map
¶
Now, when we call qc.run
we provide a memory_map
argument which will substitute in values for previouslydeclared Quil variables in a precompiled executable.
[15]:
program = ansatz() # look ma, no arguments!
program.wrap_in_numshots_loop(shots=1000)
executable = qc.compile(program)
thetas = np.linspace(0, 2*np.pi, 21)
results = []
for theta in thetas:
bitstrings = qc.run(executable, memory_map={'theta': [theta]})
results.append(np.mean(bitstrings[:, 0]))
%matplotlib inline
from matplotlib import pyplot as plt
plt.plot(thetas, results, 'o')
plt.xlabel(r'$\theta$', fontsize=18)
_ = plt.ylabel(r'$\langle \Psi(\theta)  \frac{1  Z}{2}  \Psi(\theta) \rangle$', fontsize=18)
Do an optimization (the new way)¶
Since parameters are compiled symbolically, we can do hybrid algorithms just as fast as parameter scans.
[16]:
program = ansatz() # look ma, no arguments!
program.wrap_in_numshots_loop(shots=1000)
executable = qc.compile(program)
def objective_function(thetas):
bitstrings = qc.run(executable, memory_map={'theta': thetas})
result = np.mean(bitstrings[:, 0])
return result
res = scipy.optimize.minimize(objective_function, x0=[0.1], method='COBYLA')
res
[16]:
fun: 1.0
maxcv: 0.0
message: 'Optimization terminated successfully.'
nfev: 12
status: 1
success: True
x: array([3.1])
[17]:
plt.plot(thetas, results, label='scan')
plt.plot([res.x], [res.fun], '*', ms=20, label='optimization result')
plt.legend()
[17]:
<matplotlib.legend.Legend at 0x1015f13898>
New in Forest 2  Other¶
There are many other changes to the Forest SDK (comprising pyQuil, Quil, the Quil Compiler, and the QVM).
Note
For installation & setup, follow the download instructions in the section Installation and Getting Started at the top of the page.
Updates to the Quil language¶
The primary differences in the programming language Quil 1.0 (as appearing in pyQuil 1.x) and Quil 2 (as appearing in pyQuil 2) amount to an enhanced memory model. Whereas the classical memory model in Quil 1.0 amounted to an flat bit array of indefinite size, the memory model in Quil 2 is segmented into typed, sized, named regions.
In terms of compatibility with Quil 1.0, this primarily changes how MEASURE
instructions are formulated, since their
classical address targets must be modified to fit the new framework. In terms of new functionality, this allows angle
values to be read in from classical memory.
Quil 2 also introduces easier ways to manipulate gates by using gate modifiers. Two gate modifiers are supported currently, DAGGER and CONTROLLED.
DAGGER can be written before a gate to refer to its inverse. For instance:
DAGGER RX(pi/3) 0
would have the same effect as:
RX(pi/3) 0
DAGGER can be applied to any gate, but also circuits defined with DEFCIRCUIT. This allows for easy reversal of unitary circuits:
DEFCIRCUIT BELL:
H 0
CNOT 0 1
# construct a Bell state
BELL
# disentangle, bringing us back to identity
DAGGER BELL
Classical memory regions must be explicitly requested and named by a Quil program using DECLARE
directive. A generic
DECLARE
directive has the following syntax:
DECLARE regionname type([count])? (SHARING parentregionname (OFFSET (offsetcount offsettype)+))?
The nonkeyword items have the following allowable values:
regionname
: any nonkeyword formal name.type
: one ofREAL
,BIT
,OCTET
, orINTEGER
parentregionname
: any nonkeyword formal name previously used asregionname
in a differentDECLARE
statement.offsetcount
: a nonnegative integer.offsettype
: the same allowable values astype
.
Here are some examples:
DECLARE beta REAL[32]
DECLARE ro BIT[128]
DECLARE betabits BIT[1436] SHARING beta
DECLARE fourthbitinbeta1 BIT SHARING beta OFFSET 1 REAL 4 BIT
In order, the intention of these DECLARE
statements is:
 Allocate an array called
beta
of length 32, each entry of which is aREAL
number.  Allocate an array called
ro
of length 128, each entry of which is aBIT
.  Name an array called
betabits
, which is an overlay onto the existing arraybeta
, so that the bit representations of elements ofbeta
can be directly examined and manipulated.  Name a single
BIT
calledfourthbitinbeta1
which overlays the fourth bit of the bit representation of theREAL
valuebeta[1]
.
Backwards compatibility¶
Quil 1.0 is not compatible with Quil 2 in the following ways:
 The unnamed memory references
[n]
and[nm]
have no direct equivalent in Quil 2 and must be replaced by named memory references. (This primarily affectsMEASURE
instructions.)  The classical memory manipulation instructions have been modified: the operands of
AND
have been reversed (so that in Quil 2, the left operand is the target address) andOR
has been replaced byIOR
and its operands reversed (so that, again, in Quil 2 the left operand is the target address).
In all other instances, Quil 1.0 will operate identically with Quil 2.
When confronted with program text conforming to Quil 1.0, pyQuil 2 will automatically rewrite MEASURE q [n]
to
MEASURE q ro[n]
and insert a DECLARE
statement which allocates a BIT
array of the appropriate size named
ro
.
Updates to Forest¶
 In Forest 1.3, job submission to the QPU was done from your workstation and the ability was gated by on user ID. In Forest 2, job submission to the QPU must be done from your remote virtual machine, called a QMI (Quantum Machine Image).
 In Forest 1.3, user data persisted indefinitely in cloud storage and could be accessed using the assigned job ID. In Forest 2, user data is stored only transiently, and it is the user’s responsibility to handle longterm data storage on their QMI.
 Forest 1.3 refered to the software developer kit (pyQuil, QVM, Quilc) and the cloud platform for submitting jobs. Forest 2 is the SDK which you can install on your own computer or use preinstalled on a QMI. The entire platform is called Quantum Cloud Services (QCS).
Example: Computing the bond energy of molecular hydrogen, pyQuil 1.9 vs 2.0¶
By way of example, let’s consider the following pyQuil 1.9 program, which computes the natural bond distance in molecular hydrogen using a VQEtype algorithm:
from pyquil.api import QVMConnection
from pyquil.quil import Program
def setup_forest_objects():
qvm = QVMConnection()
return qvm
def build_wf_ansatz_prep(theta):
program = Program(f"""
# set up initial state
X 0
X 1
# build the exponentiated operator
RX(pi/2) 0
H 1
H 2
H 3
CNOT 0 1
CNOT 1 2
CNOT 2 3
RZ({theta}) 3
CNOT 2 3
CNOT 1 2
CNOT 0 1
RX(pi/2) 0
H 1
H 2
H 3
# measure out the results
MEASURE 0 [0]
MEASURE 1 [1]
MEASURE 2 [2]
MEASURE 3 [3]""")
return program
# some constants
bond_step, bond_min, bond_max = 0.05, 0, 200
angle_step, angle_min, angle_max = 0.1, 0, 63
convolution_coefficients = [0.1698845197777728, 0.16988451977777283, 0.2188630663199042,
0.2188630663199042]
shots = 1000
# set up the Forest object
qvm = setup_forest_objects()
# get all the unweighted expectations for all the sample wavefunctions
occupations = list(range(angle_min, angle_max))
indices = list(range(4))
for offset in occupations:
# set up the Program object, each time we have a new parameter
program = build_wf_ansatz_prep(angle_min + offset * angle_step)
bitstrings = qvm.run(program, indices, trials=shots)
totals = [0, 0, 0, 0]
for bitstring in bitstrings:
for index in indices:
totals[index] += bitstring[index]
occupations[offset] = [t / shots for t in totals]
# compute minimum energy as a function of bond length
min_energies = list(range(bond_min, bond_max))
for bond_length in min_energies:
energies = []
for offset in range(angle_min, angle_max):
energy = 0
for j in range(4):
energy += occupations[offset][j] * convolution_coefficients[j]
energies.append(energy)
min_energies[bond_length] = min(energies)
min_index = min_energies.index(min(min_energies))
min_energy, relaxed_length = min_energies[min_index], min_index * bond_step
In order to port this code to pyQuil 2.0, we need change only one thing: the part referencing QVMConnection
should be replaced by an equivalent part referencing a QuantumComputer
connected to a QVM
. Specifically, the following
snippet
from pyquil.api import QVMConnection
def setup_forest_objects():
qvm = QVMConnection()
return qvm
can be changed to
from pyquil.api import get_qc
def setup_forest_objects():
qc = get_qc("9qsquareqvm")
return qc
and the references to qvm
in the main body are changed to qc
instead. Since the QuantumComputer
object also
exposes a run
routine and pyQuil itself automatically rewrites 1.9style MEASURE
instructions into 2.0style
instructions, this is all we need to do.
If we are willing to be more intrusive, we can also take advantage of pyQuil 2.0’s classical memory and parametric
programs. The first piece to change is the Quil program itself: we remove the argument theta
from the Python
function build_wf_ansatz_prep
, with the intention of letting the QPU fill it in later. In turn, we modify the Quil
program itself to have a REAL
memory parameter named theta
. We also declare a few BIT
s for our MEASURE
instructions to target.
def build_wf_ansatz_prep():
program = Program("""
# set up memory
DECLARE ro BIT[4]
DECLARE theta REAL
# set up initial state
X 0
X 1
# build the exponentiated operator
RX(pi/2) 0
H 1
H 2
H 3
CNOT 0 1
CNOT 1 2
CNOT 2 3
RZ(theta) 3
CNOT 2 3
CNOT 1 2
CNOT 0 1
RX(pi/2) 0
H 1
H 2
H 3
# measure out the results
MEASURE 0 ro[0]
MEASURE 1 ro[1]
MEASURE 2 ro[2]
MEASURE 3 ro[3]""")
return program
Next, we modify the execution loop. Rather than reformulating the Program
object each time, we build and compile it
once, then use the .load()
method to transfer the parametric program to the (simulated) quantum device. We then set
only the angle value within the inner loop, and we change to using .run()
and .wait()
methods to manage control
between us and the quantum device.
More specifically, the old execution loop
# get all the unweighted expectations for all the sample wavefunctions
occupations = list(range(angle_min, angle_max))
indices = list(range(4))
for offset in occupations:
# set up the Program object, each time we have a new parameter
program = build_wf_ansatz_prep(angle_min + offset * angle_step)
bitstrings = qvm.run(program, indices, trials=shots)
totals = [0, 0, 0, 0]
for bitstring in bitstrings:
for index in indices:
totals[index] += bitstring[index]
occupations[offset] = [t / shots for t in totals]
becomes
# set up the Program object, ONLY ONCE
program = build_wf_ansatz_prep().wrap_in_numshots_loop(shots=shots)
binary = qc.compile(program)
# get all the unweighted expectations for all the sample wavefunctions
occupations = list(range(angle_min, angle_max))
indices = list(range(4))
for offset in occupations:
bitstrings = qc.run(binary, {'theta': [angle_min + offset * angle_step]})
totals = [0, 0, 0, 0]
for bitstring in bitstrings:
for index in indices:
totals[index] += bitstring[index]
occupations[offset] = [t / shots for t in totals]
Overall, the resulting program looks like this:
from pyquil.api import get_qc
from pyquil.quil import Program
def setup_forest_objects():
qc = get_qc("9qsquareqvm")
return qc
def build_wf_ansatz_prep():
program = Program("""
# set up memory
DECLARE ro BIT[4]
DECLARE theta REAL
# set up initial state
X 0
X 1
# build the exponentiated operator
RX(pi/2) 0
H 1
H 2
H 3
CNOT 0 1
CNOT 1 2
CNOT 2 3
RZ(theta) 3
CNOT 2 3
CNOT 1 2
CNOT 0 1
RX(pi/2) 0
H 1
H 2
H 3
# measure out the results
MEASURE 0 ro[0]
MEASURE 1 ro[1]
MEASURE 2 ro[2]
MEASURE 3 ro[3]""")
return program
# some constants
bond_step, bond_min, bond_max = 0.05, 0, 200
angle_step, angle_min, angle_max = 0.1, 0, 63
convolution_coefficients = [0.1698845197777728, 0.16988451977777283, 0.2188630663199042,
0.2188630663199042]
shots = 1000
# set up the Forest object
qc = setup_forest_objects()
# set up the Program object, ONLY ONCE
program = build_wf_ansatz_prep().wrap_in_numshots_loop(shots=shots)
binary = qc.compile(program)
# get all the unweighted expectations for all the sample wavefunctions
occupations = list(range(angle_min, angle_max))
indices = list(range(4))
for offset in occupations:
bitstrings = qc.run(binary, {'theta': [angle_min + offset * angle_step]})
totals = [0, 0, 0, 0]
for bitstring in bitstrings:
for index in indices:
totals[index] += bitstring[index]
occupations[offset] = [t / shots for t in totals]
# compute minimum energy as a function of bond length
min_energies = list(range(bond_min, bond_max))
for bond_length in min_energies:
energies = []
for offset in range(angle_min, angle_max):
energy = 0
for j in range(4):
energy += occupations[offset][j] * convolution_coefficients[j]
energies.append(energy)
min_energies[bond_length] = min(energies)
min_index = min_energies.index(min(min_energies))
min_energy, relaxed_length = min_energies[min_index], min_index * bond_step
Miscellanea¶
Quil promises that a BIT is 1 bit and that an OCTET is 8 bits. Quil does not promise, however, the size or layout of INTEGER or REAL. These are implementationdependent.
On the QPU, INTEGER
refers to an unsigned integer stored in a 48bit wide littleendian word, and REAL
refers to
a 48bit wide littleendian fixed point number of type <0.48>. In general, these datatypes are implementationdependent.
OCTET
always refers to an 8bit wide unsigned integer and is independent of implementation.
Memory regions are all “global”: DECLARE
directives cannot appear in the body of a DEFCIRCUIT
.
On the QVM, INTEGER is a two’s complement signed 64bit integer. REAL is an IEEE754 doubleprecision floatingpoint number.
Error reporting¶
Because the Forest 2.0 execution model is no longer asynchronous, our error reporting model has also changed. Rather than writing to technical support with a job ID, users will need to provide all pertinent details to how they produced an error.
PyQuil 2 makes this task easy with the function decorator @pyquil_protect
, found in the module
pyquil.api
. By decorating a failing function (or a function that has the potential to fail), any
unhandled exceptions will cause an error log to be written to disk (at a userspecifiable location). For example, the
nonsense code block
from pyquil.api import pyquil_protect
...
@pyquil_protect
def my_function():
...
qc.qam.load(qc)
...
my_function()
causes the following error to be printed:
>>> PYQUIL_PROTECT <<<
An uncaught exception was raised in a function wrapped in pyquil_protect. We are writing out a
log file to "/Users/your_name/Documents/pyquil/pyquil_error.log".
Along with a description of what you were doing when the error occurred, send this file to Rigetti Computing
support by email at support@rigetti.com for assistance.
>>> PYQUIL_PROTECT <<<
as well as the following log file to be written to disk at the indicated location:
{
"stack_trace": [
{
"name": "pyquil_protect_wrapper",
"filename": "/Users/your_name/Documents/pyquil/pyquil/error_reporting.py",
"line_number": 197,
"locals": {
"e": "TypeError('quil_binary argument must be a QVMExecutableResponse. This error is typically triggered by
forgetting to pass (nativized) Quil to native_quil_to_executable or by using a compiler meant to be used
for jobs bound for a QPU.',)",
"old_filename": "'pyquil_error.log'",
"kwargs": "{}",
"args": "()",
"log_filename": "'pyquil_error.log'",
"func": "<function my_function at 0x106dc4510>"
}
},
{
"name": "my_function",
"filename": "<stdin>",
"line_number": 10,
"locals": {
"offset": "0",
"occupations": "[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62]"
}
},
{
"name": "wrapper",
"filename": "/Users/your_name/Documents/pyquil/pyquil/error_reporting.py",
"line_number": 228,
"locals": {
"pre_entry": "CallLogValue(timestamp_in=datetime.datetime(2018, 9, 11, 18, 40, 19, 65538),
timestamp_out=None, return_value=None)",
"key": "run('<pyquil.api._qvm.QVM object at 0x1027e3940>', )",
"kwargs": "{}",
"args": "(<pyquil.api._qvm.QVM object at 0x1027e3940>,)",
"func": "<function QVM.run at 0x106db4e18>"
}
},
{
"name": "run",
"filename": "/Users/your_name/Documents/pyquil/pyquil/api/_qvm.py",
"line_number": 376,
"locals": {
"self": "<pyquil.api._qvm.QVM object at 0x1027e3940>",
"__class__": "<class 'pyquil.api._qvm.QVM'>"
}
}
],
"timestamp": "20180911T18:40:19.253286",
"call_log": {
"__init__('<pyquil.api._qvm.QVM object at 0x1027e3940>', '<pyquil.api._base_connection.ForestConnection object at
0x1027e3588>', )": {
"timestamp_in": "20180911T18:40:18.967750",
"timestamp_out": "20180911T18:40:18.968170",
"return_value": "None"
},
"run('<pyquil.api._qvm.QVM object at 0x1027e3940>', )": {
"timestamp_in": "20180911T18:40:19.065538",
"timestamp_out": null,
"return_value": null
}
},
"exception": "TypeError('quil_binary argument must be a QVMExecutableResponse. This error is typically triggered
by forgetting to pass (nativized) Quil to native_quil_to_executable or by using a compiler meant to be used for
jobs bound for a QPU.',)",
"system_info": {
"python_version": "3.6.3 (default, Jan 25 2018, 13:55:02) \n[GCC 4.2.1 Compatible Apple LLVM 9.0.0
(clang900.0.39.2)]",
"pyquil_version": "2.0.0internal.1"
}
}
Please attach such a logfile to any request for support.
Parametric Programs¶
In PyQuil 1.x, there was an object named ParametricProgram
:
# This function returns a quantum circuit with different rotation angles on a gate on qubit 0
def rotator(angle):
return Program(RX(angle, 0))
from pyquil.parametric import ParametricProgram
par_p = ParametricProgram(rotator) # This produces a new type of parameterized program object
This object has been removed from PyQuil 2. Please consider simply using a Python function for the above functionality:
par_p = rotator
Or using declared classical memory:
p = Program()
angle = p.declare('angle', 'REAL')
p += RX(angle, 0)
Programs and Gates¶
Note
If you’re running locally, remember set up the QVM and quilc in server mode before trying to use them: Setting Up Server Mode for PyQuil.
Introduction¶
Quantum programs are written in Forest using the Program
object. This Program
abstraction will help us
compose Quil programs.
from pyquil import Program
Programs are constructed by adding quantum gates to it, which are defined in the gates
module. We can import all
standard gates with the following:
from pyquil.gates import *
Let’s instantiate a Program
and add an operation to it. We will act an X
gate on qubit 0.
p = Program()
p += X(0)
All qubits begin in the ground state. This means that if we measure a qubit without applying operations on it, we expect to receive
a 0 result. The X
gate will rotate qubit 0 from the ground state to the excited state, so a measurement immediately
after should return a 1 result. More details about gate operations are explained in Introduction to Quantum Computing.
We can print our pyQuil program (print(p)
) to see the equivalent Quil representation:
X 0
This isn’t going to be very useful to us without measurements. In pyQuil 2.0, we have to DECLARE
a memory space
to read measurement results, which we call “readout results” and abbreviate as ro
. With measurement, our whole program
looks like this:
from pyquil import Program
from pyquil.gates import *
p = Program()
ro = p.declare('ro', 'BIT', 1)
p += X(0)
p += MEASURE(0, ro[0])
print(p)
DECLARE ro BIT[1]
X 0
MEASURE 0 ro[0]
We’ve instantiated a program, declared a memory space named ro
with one single bit of memory, applied
an X
gate on qubit 0, and finally measured qubit 0 into the zeroth index of the memory space named ro
.
Awesome! That’s all we need to get results back. Now we can actually see what happens if we run this program on the Quantum Virtual Machine (QVM). We just have to add a few lines to do this.
from pyquil import get_qc
...
qc = get_qc('1qqvm') # You can make any 'nqqvm' this way for any reasonable 'n'
executable = qc.compile(p)
result = qc.run(executable)
print(result)
Congratulations! You just ran your program on the QVM. The returned value should be:
[[1]]
For more information on what the above result means, and on executing quantum programs on the QVM in general, see The Quantum Computer. The remainder of this section of the docs will be dedicated to constructing programs in detail, an essential part of becoming fluent in quantum programming.
The Standard Gate Set¶
The following gates methods come standard with Quil and gates.py
:
 Pauli gates
I
,X
,Y
,Z
 Hadamard gate:
H
 Phase gates:
PHASE(theta)
,S
,T
 Controlled phase gates:
CZ
,CPHASE00(alpha)
,CPHASE01(alpha)
,CPHASE10(alpha)
,CPHASE(alpha)
 Cartesian rotation gates:
RX(theta)
,RY(theta)
,RZ(theta)
 Controlled \(X\) gates:
CNOT
,CCNOT
 Swap gates:
SWAP
,CSWAP
,ISWAP
,PSWAP(alpha)
The parameterized gates take a real or complex floating point number as an argument.
Declaring Memory¶
Classical memory regions must be explicitly requested and named by a Quil program using the DECLARE
directive.
Details about the Quil directive can be found in declare.
In pyQuil, we declare memory with the .declare
method on a Program
. Let’s inspect the function signature
# pyquil.quil.Program
def declare(self, name, memory_type='BIT', memory_size=1, shared_region=None, offsets=None):
and break down each argument:
name
is any name you want to give this memory region.memory_type
is one of'REAL'
,'BIT'
,'OCTET'
, or'INTEGER'
(given as a string). OnlyBIT
andOCTET
always have a determined size, which is 1 bit and 8 bits respectively.memory_size
is the number of elements of that type to reserve.shared_region
andoffsets
allow you to alias memory regions. For example, you might want to name the third bit in your readout array asq3_ro
.SHARING
is currently disallowed for our QPUs, so we won’t focus on this here.
Now we can get into an example.
from pyquil import Program
p = Program()
ro = p.declare('ro', 'BIT', 16)
theta = p.declare('theta', 'REAL')
Warning
.declare
cannot be chained, since it doesn’t return a modified Program
object.
Notice that the .declare
method returns a reference to the memory we’ve just declared. We will need this reference
to make use of these memory spaces again. Let’s see how the Quil is looking so far:
DECLARE ro BIT[16]
DECLARE theta REAL[1]
That’s all we have to do to declare the memory. Continue to the next section on Measurement to learn more about
using ro
to store measured readout results. Check out Parametric Compilation to see how you might use
theta
to compile gate parameters dynamically.
Measurement¶
There are several ways you can handle measurements in your program. We will start with the simplest method – letting
the QuantumComputer
abstraction do it for us.
from pyquil import Program, get_qc
from pyquil.gates import H, CNOT
# Get our QuantumComputer instance, with a Quantum Virutal Machine (QVM) backend
qc = get_qc("8qqvm")
# Construct a simple Bell State
p = Program(H(0), CNOT(0, 1))
results = qc.run_and_measure(p, trials=10)
print(results)
{0: array([1, 1, 0, 1, 0, 0, 1, 1, 0, 1]),
1: array([1, 1, 0, 1, 0, 0, 1, 1, 0, 1]),
2: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]),
3: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]),
4: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]),
5: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]),
6: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]),
7: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])}
The method .run_and_measure
will handle declaring memory for readout results, adding MEASURE
instructions for each
qubit in the QVM, telling the QVM how many trials to run, running and returning the measurement results.
You might sometimes want finer grained control. In this case, we’re probably only interested in the results on
qubits 0 and 1, but .run_and_measure
returns the results for all eight qubits in the QVM. We can change our program
to be more particular about what we want.
from pyquil import Program
from pyquil.gates import *
p = Program()
ro = p.declare('ro', 'BIT', 2)
p += H(0)
p += CNOT(0, 1)
p += MEASURE(0, ro[0])
p += MEASURE(1, ro[1])
In the last two lines, we’ve added our MEASURE
instructions, saying that we want to store the result of qubit 0
into the 0th bit of ro
, and the result of qubit 1 into the 1st bit of ro
. The following snippet could be a
useful way to measure many qubits, in particular, on a lattice that doesn’t start at qubit 0 (although you can
use the compiler to reindex your qubits):
qubits = [5, 6, 7]
# ...
for i, q in enumerate(qubits):
p += MEASURE(q, ro[i])
Note
The QPU can only handle MEASURE
final programs. You can’t operate gates after measurements.
Specifying the number of trials¶
Quantum computing is inherently probabilistic. We often have to repeat the same experiment many times to get the
results we need. Sometimes we expect the results to all be the same, such as when we apply no gates, or only an X
gate. When we prepare a superposition state, we expect probabilistic outcomes, such as a 50% probability measuring 0 or 1.
The number of shots (also called trials) is the number of times to execute a program at once. This determines the length of the results that are returned.
As we saw above, the .run_and_measure
method of the QuantumComputer
object can handle multiple executions of a program.
If you would like more explicit control for representing multishot execution, another way to do this is
with .wrap_in_numshots_loop
. This puts the number of shots to be run in the representation of the program itself,
as opposed to in the arguments list of the execution method itself. Below, we specify that our program should
be executed 1000 times.
p = Program()
... # build up your program here...
p.wrap_in_numshots_loop(1000)
Note
Did You Know?
The word “shot” comes from experimental physics where an experiment is performed many times, and each result is called a shot.
Parametric Compilation¶
Modern quantum algorithms are often parametric, following a hybrid model. In this hybrid model, the program ansatz (template of gates) is fixed, and iteratively updated with new parameters. These new parameters are often determined by an update given by a classical optimizer. Depending on the complexity of the algorithm, problem of interest, and capabilities of the classical optimizer, this loop may need to run many times. In order to efficiently operate within this hybrid model, parametric compilation can be used.
Parametric compilation allows one to compile the program ansatz just once. Making use of declared memory regions, we can load values to the parametric gates at execution time, after compilation. Taking the compiler out of the execution loop for programs like this offers a huge performance improvement compared to compiling the program each time a parameter update is required. (Some more details about this and an example are found here.)
The first step is to build our parametric program, which functions like a template for all the precise programs we will run. Below we create a simple example program to illustrate, which puts the qubit onto the equator of the Bloch Sphere and then rotates it around the Z axis for some variable angle theta before applying another X pulse and measuring.
import numpy as np
from pyquil import Program
from pyquil.gates import RX, RZ, MEASURE
qubit = 0
p = Program()
ro = p.declare("ro", "BIT", 1)
theta_ref = p.declare("theta", "REAL")
p += RX(np.pi / 2, qubit)
p += RZ(theta_ref, qubit)
p += RX(np.pi / 2, qubit)
p += MEASURE(qubit, ro[0])
Note
The example program, although simple, is actually more than just a toy example. It is similar to an experiment which measures the qubit frequency.
Notice how theta
hasn’t been specified yet. The next steps will have to involve a QuantumComputer
or a compiler
implementation. For simplicity, we will demonstrate with a QuantumComputer
instance.
from pyquil import get_qc
# Get a Quantum Virtual Machine to simulate execution
qc = get_qc("1qqvm")
executable = qc.compile(p)
We are able to compile our program, even with theta
still not specified. Now we want to run our program with theta
filled in for, say, 200 values between \(0\) and \(2\pi\). We demonstrate this below.
# Somewhere to store each list of results
parametric_measurements = []
for theta in np.linspace(0, 2 * np.pi, 200):
# Get the results of the run with the value we want to execute with
bitstrings = qc.run(executable, {'theta': [theta]})
# Store our results
parametric_measurements.append(bitstrings)
In the example here, if you called qc.run(executable)
and didn’t specify 'theta'
, the program would apply
RZ(0, qubit)
for every execution.
Note
Classical memory defaults to zero. If you don’t specify a value for a declared memory region, it will be zero.
Gate Modifiers¶
Gate applications in Quil can be preceded by a gate modifier. There are two supported modifiers:
DAGGER
and CONTROLLED
. The DAGGER
modifier represents the dagger of the gate. For instance,
DAGGER RX(pi/3) 0
would have an equivalent effect to RX(pi/3) 0
.
The CONTROLLED
modifier takes a gate and makes it a controlled gate. For instance, one could write the Toffoli gate in any of the three following ways:
CCNOT 0 1 2
CONTROLLED CNOT 0 1 2
CONTROLLED CONTROLLED X 0 1 2
Note
The letter C
in the gate name has no semantic significance in Quil. To make a controlled Y
gate, one cannot write CY
, but rather one has to write CONTROLLED Y
.
All gates (objects deriving from the Gate
class) provide the
methods Gate.controlled(control_qubit)
and Gate.dagger()
that
can be used to programmatically apply the CONTROLLED
and
DAGGER
modifiers.
For example, to produce the controlledNOT gate (CNOT
) with
control qubit 0
and target qubit 1
prog = Program(X(1).controlled(0))
You can achieve the oftused controloff gate (flip the target qubit
1
if the control qubit 0
is zero) with
prog = Program(X(0), X(1).controlled(0), X(0))
Defining New Gates¶
New gates can be easily added inline to Quil programs. All you need is a matrix representation of the gate. For example, below we define a \(\sqrt{X}\) gate.
import numpy as np
from pyquil import Program
from pyquil.quil import DefGate
# First we define the new gate from a matrix
sqrt_x = np.array([[ 0.5+0.5j, 0.50.5j],
[ 0.50.5j, 0.5+0.5j]])
# Get the Quil definition for the new gate
sqrt_x_definition = DefGate("SQRTX", sqrt_x)
# Get the gate constructor
SQRT_X = sqrt_x_definition.get_constructor()
# Then we can use the new gate
p = Program()
p += sqrt_x_definition
p += SQRT_X(0)
print(p)
DEFGATE SQRTX:
0.5+0.5i, 0.50.5i
0.50.5i, 0.5+0.5i
SQRTX 0
Below we show how we can define \(X_0\otimes \sqrt{X_1}\) as a single gate.
# A multiqubit defgate example
x_gate_matrix = np.array(([0.0, 1.0], [1.0, 0.0]))
sqrt_x = np.array([[ 0.5+0.5j, 0.50.5j],
[ 0.50.5j, 0.5+0.5j]])
x_sqrt_x = np.kron(x_gate_matrix, sqrt_x)
Now we can use this gate in the same way that we used SQRT_X
, but we will pass it two arguments
rather than one, since it operates on two qubits.
x_sqrt_x_definition = DefGate("XSQRTX", x_sqrt_x)
X_SQRT_X = x_sqrt_x_definition.get_constructor()
# Then we can use the new gate
p = Program(x_sqrt_x_definition, X_SQRT_X(0, 1))
Tip
To inspect the wavefunction that will result from applying your new gate, you can use
the Wavefunction Simulator
(e.g. print(WavefunctionSimulator().wavefunction(p))
).
Defining Parametric Gates¶
Let’s say we want to have a controlled RX gate. Since RX is a parametric gate, we need a slightly different way of defining it than in the previous section.
from pyquil import Program, WavefunctionSimulator
from pyquil.parameters import Parameter, quil_sin, quil_cos
from pyquil.quilbase import DefGate
import numpy as np
# Define the new gate from a matrix
theta = Parameter('theta')
crx = np.array([
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, quil_cos(theta / 2), 1j * quil_sin(theta / 2)],
[0, 0, 1j * quil_sin(theta / 2), quil_cos(theta / 2)]
])
gate_definition = DefGate('CRX', crx, [theta])
CRX = gate_definition.get_constructor()
# Create our program and use the new parametric gate
p = Program()
p += gate_definition
p += H(0)
p += CRX(np.pi/2)(0, 1)
quil_sin
and quil_cos
work as the regular sines and cosines, but they support the parametrization. Parametrized
functions you can use with pyQuil are: quil_sin
, quil_cos
, quil_sqrt
, quil_exp
, and quil_cis
.
Tip
To inspect the wavefunction that will result from applying your new gate, you can use
the Wavefunction Simulator
(e.g. print(WavefunctionSimulator().wavefunction(p))
).
Defining Permutation Gates¶
Note
quilc
supports permutation gate syntax since version 1.8.0
. pyQuil introduced support in version 2.8.0
.
Some gates can be compactly represented as a permutation. For example, CCNOT
gate can be represented by the matrix
import numpy as np
from pyquil.quilbase import DefGate
ccnot_matrix = np.array([
[1, 0, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1],
[1, 0, 0, 0, 0, 0, 1, 0]
])
ccnot_gate = DefGate("CCNOT", ccnot_matrix)
# etc
It can equivalently be defined by the permutation
import numpy as np
from pyquil.quilbase import DefPermutationGate
ccnot_gate = DefPermutationGate("CCNOT", [0, 1, 2, 3, 4, 5, 7, 6])
# etc
Pragmas¶
PRAGMA
directives give users more control over how Quil programs are processed or simulated but generally do not
change the semantics of the Quil program itself. As a general rule of thumb, deleting all PRAGMA
directives in a Quil
program should leave a valid and semantically equivalent program.
In pyQuil, PRAGMA
directives play many roles, such as controlling the behavior of gates in noisy simulations,
or commanding the Quil compiler to perform actions in a certain way. Here, we will cover the basics of two very
common use cases for including a PRAGMA
in your program: qubit rewiring and delays. For a more comprehensive
review of what pragmas are and what the compiler supports, check out The Quil Compiler. For more information about
PRAGMA
in Quil, see
A Practical Quantum ISA, and
Simulating Quantum Processor Errors.
Specifying A Qubit Rewiring Scheme¶
Qubit rewiring is one of the most powerful features of the Quil compiler. We are able to write Quil programs which are agnostic to the topology of the chip, and the compiler will intelligently relabel our qubits to give better performance.
When we intend to run a program on the QPU, sometimes we write programs which use specific qubits targeting a specific device topology, perhaps to achieve a highperformance program. Other times, we write programs that are agnostic to the underlying topology, thereby making the programs more portable. Qubit rewiring accommodates both use cases in an automatic way.
Consider the following program.
from pyquil import Program
from pyquil.gates import *
p = Program(X(3))
We’ve tested this on the QVM, and we’ve reserved a lattice on the QPU which has qubits 4, 5, and 6, but not qubit 3. Rather than rewrite our program for each reservation, we modify our program to tell the compiler to do this for us.
from pyquil.quil import Pragma
p = Program(Pragma('INITIAL_REWIRING', ['"GREEDY"']))
p += X(3)
Now, when we pass our program through the compiler (such as with QuantumComputer.compile()
) we will get native Quil
with the qubit reindexed to one of 4, 5, or 6. If qubit 3 is available, and we don’t want that pulse to be applied to
any other qubit, we would instead use Pragma('INITIAL_REWIRING', ['"NAIVE"']]
. Detailed information about the
available options is here.
Note
In general, we assume that the qubits you’re supplying as input are also the ones which you prefer to operate on, and so NAIVE rewiring is the default.
Asking for a Delay¶
At times, we may want to add a delay in our program. Usually this is associated with qubit characterization. Delays
are not regular gate operations, and they do not affect the abstract semantics of the Quil program, so they’re implemented with a PRAGMA
directive.
# ...
# qubit index and time in seconds must be defined and provided
# the time argument accepts exponential notation e.g. 200e9
p += Pragma('DELAY', [qubit], str(time))
Warning
These delays currently have effects on the real QPU. They have no effect on QVM’s even when those QVM’s have noise models applied.
Warning
Keep in mind, the program duration is currently capped at 15 seconds, and the length of the program is multiplied by the number of shots. If you have a 1000 shot program, where each shot contains a 100ms delay, you won’t be able to execute it.
Ways to Construct Programs¶
PyQuil supports a variety of methods for constructing programs however you prefer.
Multiple instructions can be applied at once, and programs can be added together. PyQuil can also produce a
Program
by interpreting raw Quil text. You can still use the more pyQuil 1.X style of using
the .inst
method to add instruction gates. Thus, the following are all valid programs:
# Preferred method
p = Program()
p += X(0)
p += Y(1)
print(p)
# Multiple instructions in declaration
print(Program(X(0), Y(1)))
# A composition of two programs
print(Program(X(0)) + Program(Y(1)))
# Raw Quil with newlines
print(Program("X 0\nY 1"))
# Raw Quil comma separated
print(Program("X 0", "Y 1"))
# Chained inst; less preferred
print(Program().inst(X(0)).inst(Y(1)))
All of the above methods will produce the same output:
X 0
Y 1
The pyquil.parser
submodule provides a frontend to other similar parser
functionality.
Fixing a Mistaken Instruction¶
If an instruction was appended to a program incorrectly, you can pop it off.
p = Program(X(0), Y(1))
print(p)
print("We can fix by popping:")
p.pop()
print(p)
X 0
Y 1
We can fix by popping:
X 0
QPUallowable Quil¶
Apart from DECLARE
and PRAGMA
directives, a program must break into the following three regions, each optional:
 A
RESET
command.  A sequence of quantum gate applications.
 A sequence of
MEASURE
commands.
The only memory that is writeable is the region named ro
, and only through MEASURE
instructions. All other
memory is readonly.
The keyword SHARING
is disallowed.
Compilation is unavailable for invocations of DEFGATE
s with parameters read from classical memory.
The Quantum Computer¶
PyQuil is used to build Quil (Quantum Instruction Language) programs and execute them on simulated or real quantum devices. Quil is an opinionated quantum instruction language: its basic belief is that in the near term quantum computers will operate as coprocessors, working in concert with traditional CPUs. This means that Quil is designed to execute on a Quantum Abstract Machine (QAM) that has a shared classical/quantum architecture at its core.
A QAM must, therefore, implement certain abstract methods to manipulate classical and quantum states, such as loading programs, writing to shared classical memory, and executing programs.
The program execution itself is sent from pyQuil to quantum computer endpoints, which will be one of two options:
 A Rigetti Quantum Virtual Machine (QVM)
 A Rigetti Quantum Processing Unit (QPU)
Within pyQuil, there is a QVM
object and a QPU
object which use
the exposed APIs of the QVM and QPU servers, respectively.
On this page, we’ll learn a bit about the QVM and QPU. Then we will show you how to use them from pyQuil with a The QuantumComputer.
For information on constructing quantum programs, please refer back to Programs and Gates.
The Quantum Virtual Machine (QVM)¶
The Rigetti Quantum Virtual Machine is an implementation of the Quantum Abstract Machine from A Practical Quantum Instruction Set Architecture. [1] It is implemented in ANSI Common LISP and executes programs specified in Quil.
The QVM simulates the unitary evolution of a wavefunction with classical control. The QVM has a plethora of other features, including:
 Stochastic purestate evolution, density matrix evolution, and Pauli noise channels;
 Shared memory access to the quantum state, allowing direct NumPy access to the state without copying or transmission delay; and
 A fast justintime compilation mode for rapid simulation of large programs with many qubits.
The QVM is part of the Forest SDK, and it’s available for you to use on your local machine.
After downloading and installing the SDK, you can initialize a local
QVM server by typing qvm S
into your terminal. You should see the following message.
$ qvm S
******************************
* Welcome to the Rigetti QVM *
******************************
Copyright (c) 2018 Rigetti Computing.
(Configured with 2048 MiB of workspace and 8 workers.)
[20181106 18:18:18] Starting server on port 5000.
By default, the server is started on port 5000 on your local machine. Consequently, the endpoint which
the pyQuil QVM
will default to for the QVM address is http://127.0.0.1:5000
. When you
run your program, a pyQuil client will send a Quil program to the QVM server and wait for a response back.
It’s also possible to use the QVM from the command line. You can write a Quil program in its own file:
# example.quil
DECLARE ro BIT[1]
RX(pi/2) 0
CZ 0 1
and then execute it with the QVM directly from the command line:
$ qvm < example.quil
[20181130 11:13:58] Reading program.
[20181130 11:13:58] Allocating memory for QVM of 2 qubits.
[20181130 11:13:58] Allocation completed in 4 ms.
[20181130 11:13:58] Loading quantum program.
[20181130 11:13:58] Executing quantum program.
[20181130 11:13:58] Execution completed in 6 ms.
[20181130 11:13:58] Printing 2qubit state.
[20181130 11:13:58] Amplitudes:
[20181130 11:13:58] 00>: 0.0, P= 0.0%
[20181130 11:13:58] 01>: 0.01.0i, P=100.0%
[20181130 11:13:58] 10>: 0.0, P= 0.0%
[20181130 11:13:58] 11>: 0.0, P= 0.0%
[20181130 11:13:58] Classical memory (low > high indexes):
[20181130 11:13:58] ro: 1 0
The QVM offers a simple benchmarking mode with qvm verbose
benchmark
. Example output looks like this:
$ ./qvm verbose benchmark
******************************
* Welcome to the Rigetti QVM *
******************************
Copyright (c) 20162019 Rigetti Computing.
(Configured with 8192 MiB of workspace and 8 workers.)
<135>1 20190501T18:26:14Z workstation.local qvm 96177   Selected simulation method: purestate
<135>1 20190501T18:26:15Z workstation.local qvm 96177   Computing baseline serial norm timing...
<135>1 20190501T18:26:15Z workstation.local qvm 96177   Baseline serial norm timing: 96 ms
<135>1 20190501T18:26:15Z workstation.local qvm 96177   Starting "bell" benchmark with 26 qubits...
; Transition H 0 took 686 ms (gc: 0 ms; alloc: 65536 bytes)
; Transition CNOT 0 1 took 651 ms (gc: 0 ms; alloc: 0 bytes)
; Transition CNOT 1 2 took 658 ms (gc: 0 ms; alloc: 32656 bytes)
; Transition CNOT 2 3 took 661 ms (gc: 0 ms; alloc: 0 bytes)
; Transition CNOT 3 4 took 650 ms (gc: 0 ms; alloc: 0 bytes)
; Transition CNOT 4 5 took 662 ms (gc: 0 ms; alloc: 0 bytes)
; Transition CNOT 5 6 took 673 ms (gc: 0 ms; alloc: 0 bytes)
[...]
<135>1 20190501T18:30:13Z workstation.local qvm 96288   Total time for program run: 24385 ms
The QVM also has mode for faster execution of long quantum programs
operating on a large number of qubits, called compiled
mode. Compiled mode can be enabled by adding c
to the command
line options. Observe the speedup in the benchmark:
$ ./qvm verbose benchmark c
******************************
* Welcome to the Rigetti QVM *
******************************
Copyright (c) 20162019 Rigetti Computing.
(Configured with 8192 MiB of workspace and 8 workers.)
<135>1 20190501T18:28:07Z workstation.local qvm 96285   Selected simulation method: purestate
<135>1 20190501T18:28:08Z workstation.local qvm 96285   Computing baseline serial norm timing...
<135>1 20190501T18:28:08Z workstation.local qvm 96285   Baseline serial norm timing: 95 ms
<135>1 20190501T18:28:08Z workstation.local qvm 96285   Starting "bell" benchmark with 26 qubits...
; Compiling program loaded into QVM...
; Compiled in 87 ms.
; Optimization eliminated 26 instructions ( 50.0%).
; Transition compiled{ FUSEDGATE0 1 0 } took 138 ms (gc: 0 ms; alloc: 0 bytes)
; Transition compiled{ CNOT 1 2 } took 144 ms (gc: 0 ms; alloc: 0 bytes)
; Transition compiled{ CNOT 2 3 } took 137 ms (gc: 0 ms; alloc: 0 bytes)
; Transition compiled{ CNOT 3 4 } took 143 ms (gc: 0 ms; alloc: 0 bytes)
; Transition compiled{ CNOT 4 5 } took 95 ms (gc: 0 ms; alloc: 0 bytes)
; Transition compiled{ CNOT 5 6 } took 75 ms (gc: 0 ms; alloc: 0 bytes)
[...]
<135>1 20190501T18:29:12Z workstation.local qvm 96287   Total time for program run: 2416 ms
The runtime reduced to 2.4 seconds from 24 seconds, a 10x speedup.
Note
Compiled mode speeds up the execution of a program at the cost of an initial compilation. Note in the above example that compilation took 87 ms. If you are running small programs with low qubit counts, this cost may be significant, and it may be worth executing in the usual (“interpreted”) mode. However, if your programs contain a large number of qubits or a large number of instructions, the initial cost is far outweighed by the benefits.
For a detailed description of how to use the qvm
from the command line, see the QVM README or type man qvm
in your terminal.
We also offer a Wavefunction Simulator (formerly a part of the QVM
object),
which allows users to contruct and inspect wavefunctions of quantum programs. Learn more
about the Wavefunction Simulator here.
The Quantum Processing Unit (QPU)¶
To access a QPU endpoint, you will have to sign up for Quantum Cloud Services (QCS).
Documentation for getting started with your Quantum Machine Image (QMI) is found
here. Using QCS, you will ssh
into your QMI, and reserve a
QPU lattice for a particular time block.
When your reservation begins, you will be authorized to access the QPU. A configuration file will be
automatically populated for you with the proper QPU endpoint for your reservation. Both your QMI and the QPU
are located on premises, giving you low latency access to the QPU server. That server accepts jobs in the form
of a BinaryExecutableRequest
object, which is precisely what you get back when you compile your program in
pyQuil and target the QPU (more on this soon). This request contains all the information necessary to run
your program on the control rack which sends and receives waveforms from the QPU, so that you can receive
classical binary readout results.
For information on available lattices, you can check out your dashboard at https://qcs.rigetti.com/dashboard after you’ve been invited to QCS.
The QuantumComputer
¶
The QuantumComputer
abstraction offered by pyQuil provides an easy access point to the most
critical objects used in pyQuil for building and executing your quantum programs.
We will cover the main methods and attributes on this page.
The QuantumComputer API Reference provides a reference for all of its methods and
options.
At a high level, the QuantumComputer
wraps around our favorite quantum computing tools:
 A quantum abstract machine
.qam
: this is our general purpose quantum computing device, which implements the required abstract methods described above. It is implemented as aQVM
orQPU
object in pyQuil. A compiler
.compiler
: this determines how we manipulate the Quil input to something more efficient when possible, and then into a form which our QAM can accept as input. A device
.device
: this specifies the topology and Instruction Set Architecture (ISA) of the targeted device by listing the supported 1Q and 2Q gates.
When you instantiate a QuantumComputer
instance, these subcomponents will be compatible with
each other. So, if you get a QPU
implementation for the .qam
, you will have a QPUCompiler
for the
.compiler
, and your .device
will match the device used by the .compiler.
The QuantumComputer
instance makes methods available which are built on the above objects. If
you need more fine grained controls for your work, you might try exploring what is offered by these objects.
For more information on each of the above, check out the following pages:
Instantiation¶
A decent amount of information needs to be provided to initialize the compiler
, device
, and qam
attributes,
much of which is already in your config files (or provided reasonable defaults when running locally).
Typically, you will want a QuantumComputer
which either:
 pertains to a real, available QPU device
 is a QVM but mimics the topology of a QPU
 is some generic QVM
All of this can be accomplished with get_qc()
.
def get_qc(name: str, *, as_qvm: bool = None, noisy: bool = None,
connection: ForestConnection = None) > QuantumComputer:
from pyquil import get_qc
# Get a QPU
qc = get_qc(QPU_LATTICE_NAME) # QPU_LATTICE_NAME is just a string naming the device
# Get a QVM with the same topology as the QPU lattice
qc = get_qc(QPU_LATTICE_NAME, as_qvm=True)
# or, equivalently
qc = get_qc(f"{QPU_LATTICE_NAME}qvm")
# A fully connected QVM
number_of_qubits = 10
qc = get_qc(f"{number_of_qubits}qqvm")
For now, you will have to join QCS to get QPU_LATTICE_NAME
by running the
qcs lattices
command from your QMI. Access to the QPU is only possible from a QMI, during a booked reservation.
If this sounds unfamiliar, check out our documentation for QCS
and join the waitlist.
For more information about creating and adding your own noise models, check out Noise and Quantum Computation.
Note
When connecting to a QVM locally (such as with get_qc(..., as_qvm=True)
) you’ll have to set up the QVM
in server mode.
Methods¶
Now that you have your qc
, there’s a lot you can do with it. Most users will want to use compile
, run
or
run_and_measure
, and qubits
very regularly. The general flow of use would look like this:
from pyquil import get_qc, Program
from pyquil.gates import *
qc = get_qc('9qsquareqvm') # not general to any number of qubits, 9qsquareqvm is special
qubits = qc.qubits() # this information comes from qc.device
p = Program()
# ... build program, potentially making use of the qubits list
compiled_program = qc.compile(p) # this makes multiple calls to qc.compiler
results = qc.run(compiled_program) # this makes multiple calls to qc.qam
Note
In addition to a running QVM server, you will need a running quilc
server to compile your program. Setting
up both of these is very easy, as explained here.
The .run_and_measure(...)
method¶
This is the most high level way to run your program. With this method, you are not responsible for compiling your program
before running it, nor do you have to specify any MEASURE
instructions; all qubits will get measured.
from pyquil import Program, get_qc
from pyquil.gates import X
qc = get_qc("8qqvm")
p = Program(X(0))
results = qc.run_and_measure(p, trials=5)
print(results)
trials
specifies how many times to run this program. Let’s see our results:
{0: array([1, 1, 1, 1, 1]),
1: array([0, 0, 0, 0, 0]),
2: array([0, 0, 0, 0, 0]),
3: array([0, 0, 0, 0, 0]),
4: array([0, 0, 0, 0, 0]),
5: array([0, 0, 0, 0, 0]),
6: array([0, 0, 0, 0, 0]),
7: array([0, 0, 0, 0, 0])}
The return value is a dictionary from qubit index to results for all trials. Every qubit in the lattice is measured for you, and as expected, qubit 0 has been flipped to the excited state for each trial.
The .run(...)
method¶
The lowerlevel .run(...)
method gives you more control over how you want to build and compile your program than
.run_and_measure
does. You are responsible for compiling your program before running it.
The above program would be written in this way to execute with run
:
from pyquil import Program, get_qc
from pyquil.gates import X, MEASURE
qc = get_qc("8qqvm")
p = Program()
ro = p.declare('ro', 'BIT', 1)
p += X(0)
p += MEASURE(0, ro[0])
p += MEASURE(1, ro[1])
p.wrap_in_numshots_loop(5)
executable = qc.compile(p)
bitstrings = qc.run(executable) # .run takes in a compiled program, unlike .run_and_measure
print(bitstrings)
By specifying MEASURE
ourselves, we will only get the results that we are interested in. To be completely equivalent
to the previous example, we would have to measure all eight qubits.
The results returned is a list of lists of integers. In the above case, that’s
[[1, 0], [1, 0], [1, 0], [1, 0], [1, 0]]
Let’s unpack this. The outer list is an enumeration over the trials; the argument given to
wrap_in_numshots_loop
will match the length of results
.
The inner list, on the other hand, is an enumeration over the results stored in the memory region named ro
, which
we use as our readout register. We see that the result of this program is that the memory region ro[0]
now stores
the state of qubit 0, which should be 1
after an \(X\)gate. See Declaring Memory and Measurement
for more details about declaring and accessing classical memory regions.
Tip
Get the results for qubit 0 with numpy.array(bitstrings)[:,0]
.
Providing Your Own Device Topology¶
It is simple to provide your own device topology as long as you can give your qubits each a number, and specify which edges exist. Here is an example, using the topology of our 16Q chip (two octagons connected by a square):
import networkx as nx
from pyquil.device import NxDevice, gates_in_isa
from pyquil.noise import decoherence_noise_with_asymmetric_ro
qubits = [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17] # qubits are numbered by octagon
edges = [(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 0), # first octagon
(1, 16), (2, 15), # connections across the square
(10, 11), (11, 12), (13, 14), (14, 15), (15, 16), (16, 17), (10, 17)] # second octagon
# Build the NX graph
topo = nx.from_edgelist(edges)
# You would uncomment the next line if you have disconnected qubits
# topo.add_nodes_from(qubits)
device = NxDevice(topo)
device.noise_model = decoherence_noise_with_asymmetric_ro(gates_in_isa(device.get_isa())) # Optional
Now that you have your device, you could set qc.device
and qc.compiler.device
to point to your new device,
or use it to make new objects.
Simulating the QPU using the QVM¶
The QAM
methods are intended to be used in the same way, whether a QVM or QPU is being targeted.
Everywhere on this page,
you can swap out the type of the QAM (QVM <=> QPU) and you will still
get reasonable results back. As long as the topology of the devices are the same, programs compiled and ran on the QVM
will be able to run on the QPU and visaversa. Since QuantumComputer
is built on the QAM
abstract class, its methods will also work for both QAM implementations.
This makes the QVM a powerful tool for testing quantum programs before executing them on the QPU.
qpu = get_qc(QPU_LATTICE_NAME)
qvm = get_qc(QPU_LATTICE_NAME, as_qvm=True)
By simply providing as_qvm=True
, we get a QVM which will have the same topology as
the named QPU. It’s a good idea to run your programs against the QVM before booking QPU time to iron out
bugs. To learn more about how to add noise models to your virtual QuantumComputer
instance, check out
Noise and Quantum Computation.
In the next section, we will see how to use the Wavefunction Simulator aspect of the Rigetti QVM to inspect the full wavefunction set up by a Quil program.
[1]  https://arxiv.org/abs/1608.03355 
The Wavefunction Simulator¶
Formerly a part of the QVM object in pyQuil, the Wavefunction Simulator allows you to directly inspect the wavefunction of a quantum state prepared by your program. Because of the probabilistic nature of quantum information, the programs you’ll be running on the QPU can give a distribution of outputs. When running on the QPU or QVM, you would aggregate results (anywhere from tens of trials to 100k+!) that you can sample to get back a distribution.
With the Wavefunction Simulator, you can look at the distribution without having to collect samples from your program. This can save a lot of time for small programs. Let’s walk through a basic example of using WavefunctionSimulator:
from pyquil import Program
from pyquil.gates import *
from pyquil.api import WavefunctionSimulator
wf_sim = WavefunctionSimulator()
coin_flip = Program(H(0))
wf_sim.wavefunction(coin_flip)
<pyquil.wavefunction.Wavefunction at 0x1088a2c10>
The return value is a Wavefunction object that stores the amplitudes of the quantum state. We can print this object
coin_flip = Program(H(0))
wavefunction = wf_sim.wavefunction(coin_flip)
print(wavefunction)
(0.7071067812+0j)0> + (0.7071067812+0j)1>
to see the amplitudes listed as a sum of computational basis states. We can index into those amplitudes directly or look at a dictionary of associated outcome probabilities.
assert wavefunction[0] == 1 / np.sqrt(2)
# The amplitudes are stored as a numpy array on the Wavefunction object
print(wavefunction.amplitudes)
prob_dict = wavefunction.get_outcome_probs() # extracts the probabilities of outcomes as a dict
print(prob_dict)
prob_dict.keys() # these store the bitstring outcomes
assert len(wavefunction) == 1 # gives the number of qubits
[ 0.70710678+0.j 0.70710678+0.j]
{'1': 0.49999999999999989, '0': 0.49999999999999989}
It is important to remember that this wavefunction
method is a useful debugging tool for small quantum systems, and
cannot be feasibly obtained on a quantum processor.
MultiQubit Basis Enumeration¶
The WavefunctionSimulator enumerates bitstrings such that qubit 0
is the least significant bit (LSB)
and therefore on the right end of a bitstring as shown in the table below which contains some
examples.
bitstring  qubit_(n1)  …  qubit_2  qubit_1  qubit_0 

1…101  1  …  1  0  1 
0…110  0  …  1  1  0 
This convention is counter to that often found in the quantum computing literature where
bitstrings are often ordered such that the lowestindex qubit is on the left.
The vector representation of a wavefunction assumes the “canonical” ordering of basis elements.
I.e., for two qubits this order is 00, 01, 10, 11
.
In the typical Dirac notation for quantum states, the tensor product of two different degrees of
freedom is not always explicitly understood as having a fixed order of those degrees of freedom.
This is in contrast to the kronecker product between matrices which uses the same mathematical
symbol and is clearly not commutative.
This, however, becomes important when writing things down as coefficient vectors or matrices:
As a consequence there arise some subtle but important differences in the ordering of wavefunction and multiqubit gate matrix coefficients. According to our conventions the matrix
corresponds to the Quil instruction CNOT(1, 0)
which is counter to how most other people in the
field order their tensor product factors (or more specifically their kronecker products).
In this convention CNOT(0, 1)
is given by
For additional information why we decided on this basis ordering check out our note Someone shouts, “01000>!” Who is Excited?.
The Quil Compiler¶
Expectations for Program Contents¶
The QPUs have much more limited natural gate sets than the standard gate set offered by pyQuil: on Rigetti QPUs, the
gate operators are constrained to lie in RZ(θ)
, RX(k*π/2)
, and CZ
; and the
gates are required to act on physically available hardware (for singlequbit gates, this means
acting only on live qubits, and for qubitpair gates, this means acting on neighboring qubits). However, as a programmer, it is often (though not always) desirable to to be able to write programs which don’t take these details into account. These generally leads to more portable code if one isn’t tied to a specific set of gates or QPU architecture.
To ameliorate these limitations, the Rigetti software toolkit contains an optimizing compiler that
translates arbitrary Quil to native Quil and native Quil to executables suitable for Rigetti
hardware.
Interacting with the Compiler¶
After downloading the SDK, the Quil Compiler, quilc
is available on your local machine.
You can initialize a local quilc
server by typing quilc S
into your terminal. You should see the following message.
$ quilc S
++
 W E L C O M E 
 T O T H E 
 R I G E T T I 
 Q U I L 
 C O M P I L E R 
++
Copyright (c) 2018 Rigetti Computing.
...  Launching quilc.
...  Spawning server at (tcp://*:5555) .
To get a description of quilc
, and options and examples of its command line use, see quilc_man.
A QuantumComputer
object supplied by the function pyquil.api.get_qc()
comes equipped with a
connection to your local Rigetti Quil compiler. This can be accessed using the instance method .compile()
,
as in the following:
from pyquil.quil import Pragma, Program
from pyquil.api import get_qc
from pyquil.gates import CNOT, H
qc = get_qc("9qsquareqvm")
ep = qc.compile(Program(H(0), CNOT(0,1), CNOT(1,2)))
print(ep.program) # here ep is of type PyquilExecutableResponse, which is not always inspectable
with output
RZ(pi/2) 0
RX(pi/2) 0
RZ(pi/2) 1
RX(pi/2) 1
CZ 1 0
RX(pi/2) 1
RZ(pi/2) 2
RX(pi/2) 2
CZ 2 1
RZ(pi/2) 0
RZ(pi/2) 1
RX(pi/2) 2
RZ(pi/2) 2
The compiler connection is also available directly via the property qc.compiler
. The
precise class of this object changes based on context (e.g., QPUCompiler
,
QVMCompiler
), but it always conforms to the interface laid out by pyquil.api._qac
:
compiler.quil_to_native_quil(program)
: This method converts a Quil program into native Quil, according to the ISA that the compiler is initialized with. The input parameter is specified as aProgram
object, and the output is given as a newProgram
object, equipped with a.metadata
property that gives extraneous information about the compilation output (e.g., gate depth, as well as many others). This call blocks until Quil compilation finishes.compiler.native_quil_to_executable(nq_program)
: This method converts a native Quil program, which is promised to consist only of native gates for a given ISA, into an executable suitable for submission to one of a QVM or a QPU. This call blocks until the executable is generated.
The instance method qc.compile
described above is a combination of these two methods: first the
incoming Quil is nativized, and then that is immediately turned into an executable. Accordingly,
the previous example snippet is identical to the following:
from pyquil.quil import Pragma, Program
from pyquil.api import get_qc
from pyquil.gates import CNOT, H
qc = get_qc("9qsquareqvm")
p = Program(H(0), CNOT(0,1), CNOT(1,2))
np = qc.compiler.quil_to_native_quil(p)
ep = qc.compiler.native_quil_to_executable(np)
print(ep.program) # here ep is of type PyquilExecutableResponse, which is not always inspectable
Legal compiler input¶
The QPU is not able to execute all possible Quil programs. At present, a Quil program qualifies for execution if has the following form:
 The program may or may not begin with a
RESET
instruction. (If provided, the QPU will actively reset the state of the quantum device to the ground state before program execution. If omitted, the QPU will wait for a relaxation period to pass before program execution instead.)  This is then followed by a block of native quantum gates. A gate is native if it is of the form
RZ(θ)
for any valueθ
,RX(k*π/2)
for an integerk
, orCZ q0 q1
forq0
,q1
a pair of qubits participating in a qubitqubit interaction.  This is then followed by a block of
MEASURE
instructions.
Regionspecific compiler features through PRAGMA¶
The Quil compiler can also be communicated with through PRAGMA
commands embedded in the Quil
program.
Note
The interface to the Quil compiler from pyQuil is under construction, and some of the PRAGMA
directives will soon be replaced by finergrained method calls.
Preserved regions¶
The compiler can be circumvented in userspecified regions. The start of such a region is denoted by
PRAGMA PRESERVE_BLOCK
, and the end is denoted by PRAGMA END_PRESERVE_BLOCK
. The Quil
compiler promises not to modify any instructions contained in such a region.
Warning
If a preserved block is not legal QPU input, then it is not guaranteed to execute or it may produced unexpected results.
The following is an example of a program that prepares a Bell state on qubits 0 and 1, then performs
a time delay to invite noisy system interaction before measuring the qubits. The time delay region
is marked by PRAGMA PRESERVE_BLOCK
and PRAGMA END_PRESERVE_BLOCK
; without these delimiters,
the compiler will remove the identity gates that serve to provide the time delay. However, the
regions outside of the PRAGMA
region will still be compiled, converting the Bell state preparation
to the native gate set.
DECLARE ro BIT[2]
# prepare a Bell state
H 0
CNOT 0 1
# wait a while
PRAGMA PRESERVE_BLOCK
I 0
I 1
I 0
I 1
# ...
I 0
I 1
PRAGMA END_PRESERVE_BLOCK
# and read out the results
MEASURE 0 ro[0]
MEASURE 1 ro[1]
Parallelizable regions¶
The compiler can sometimes arrange gate sequences more cleverly if the user gives it hints about
sequences of gates that commute. A region containing commuting sequences is bookended by
PRAGMA COMMUTING_BLOCKS
and PRAGMA END_COMMUTING_BLOCKS
; within such a region, a given
commuting sequence is bookended by PRAGMA BLOCK
and PRAGMA END_BLOCK
.
Warning
Lying to the compiler about what blocks can commute can cause incorrect results.
The following snippet demonstrates this hinting syntax in a context typical of VQEtype algorithms: after a first stage of performing some state preparation on individual qubits, there is a second stage of “mixing operations” that both reuse qubit resources and mutually commute, followed by a final rotation and measurement. The following program is naturally laid out on a ring with vertices (read either clockwise or counterclockwise) as 0, 1, 2, 3. After scheduling the first round of preparation gates, the compiler will use the hinting to schedule the first and third blocks (which utilize qubit pairs 01 and 23) before the second and fourth blocks (which utilize qubit pairs 12 and 03), resulting in a reduction in circuit depth by one half. Without hinting, the compiler will instead execute the blocks in their written order.
DECLARE ro BIT[4]
# Stage one
H 0
H 1
H 2
H 3
# Stage two
PRAGMA COMMUTING_BLOCKS
PRAGMA BLOCK
CNOT 0 1
RZ(0.4) 1
CNOT 0 1
PRAGMA END_BLOCK
PRAGMA BLOCK
CNOT 1 2
RZ(0.6) 2
CNOT 1 2
PRAGMA END_BLOCK
PRAGMA BLOCK
CNOT 2 3
RZ(0.8) 3
CNOT 2 3
PRAGMA END_BLOCK
PRAGMA BLOCK
CNOT 0 3
RZ(0.9) 3
CNOT 0 3
PRAGMA END_BLOCK
PRAGMA END_COMMUTING_BLOCKS
# Stage three
H 0
H 1
H 2
H 3
MEASURE 0 ro[0]
MEASURE 1 ro[1]
MEASURE 2 ro[2]
MEASURE 3 ro[3]
Rewirings¶
When a Quil program contains multiqubit instructions that do not name qubitqubit links present on a
target device, the compiler will rearrange the qubits so that execution becomes possible. In order to
help the user understand what rearrangement may have been done, the compiler emits comments at various
points in the raw Quil code (which is not currently visible from a pyQuil Program
object’s .out()
method): # Entering rewiring
and # Exiting rewiring
. From the perspective of the user, both
comments serve the same purpose: # Entering rewiring: #(n0 n1 ... nk)
indicates that the logical
qubit labeled j
in the program has been assigned to lie on the physical qubit labeled nj
on
the device. This is strictly for humanreadability: these comments are discarded and have no effect.
In addition, you have some control over how the compiler constructs its
rewiring, which is controlled by PRAGMA INITIAL_REWIRING
. The syntax is as follows.
# <type> can be NAIVE, RANDOM, PARTIAL, or GREEDY
#
# The double quotes are required.
PRAGMA INITIAL_REWIRING "<type>"
Including this before any nonpragmas will allow the compiler to alter its rewiring behavior. The possible options are:
NAIVE
(default): The compiler will start with an identity mapping as the initial rewiring. In particular, qubits will not be rewired unless the program requests a qubitqubit interaction not natively available on the QPU.PARTIAL
: The compiler will start with nothing assigned to each physical qubit. Then, it will fill in the logicaltophysical mapping as it encounters new qubits in the program, making its best guess for where they should be placed.RANDOM
: the compiler will start with a random permutation.GREEDY
: the compiler will make a guess for the initial rewiring based on a quick initial scan of the entire program.
Note
NAIVE
rewiring is the default, and for the most part, it
follows the “Do What I Mean” (DWIM) principle. It is the least
sophisticated, but attempts to follow what the user has constructed
with their program. Choosing another rewiring, such as PARTIAL
,
may lead to higherperforming programs because the compiler has
more freedom to optimize the layout of the gates on the qubits.
Common Error Messages¶
The compiler itself is subject to some limitations, and some of the more commonly observed errors follow:
! ! ! Error: Matrices do not lie in the same projective class.
The compiler attempted to decompose an operator as native Quil instructions, and the resulting instructions do not match the original operator. This can happen when the original operator is not a unitary matrix, and could indicate an invalidDEFGATE
block. In some rare circumstances, it can also happen due to floating point precision issues.
Noise and Quantum Computation¶
Modeling Noisy Quantum Gates¶
Pure States vs. Mixed States¶
Errors in quantum computing can introduce classical uncertainty in what the underlying state is. When this happens we sometimes need to consider not only wavefunctions but also probabilistic sums of wavefunctions when we are uncertain as to which one we have. For example, if we think that an X gate was accidentally applied to a qubit with a 5050 chance then we would say that there is a 50% chance we have the \(\ket{0}\) state and a 50% chance that we have a \(\ket{1}\) state. This is called an “impure” or “mixed”state in that it isn’t just a wavefunction (which is pure) but instead a distribution over wavefunctions. We describe this with something called a density matrix, which is generally an operator. Pure states have very simple density matrices that we can write as an outer product of a ket vector \(\ket{\psi}\) with its own bra version \(\bra{\psi}=\ket{\psi}^\dagger\). For a pure state the density matrix is simply
The expectation value of an operator for a mixed state is given by
where \(\tr{\cdot}\) is the trace of an operator, which is the sum of its diagonal elements, which is independent of choice of basis. Pure state density matrices satisfy
which you can easily verify for \(\rho_\psi\) assuming that the state is normalized. If we want to describe a situation with classical uncertainty between states \(\rho_1\) and \(\rho_2\), then we can take their weighted sum
where \(p\in [0,1]\) gives the classical probability that the state is \(\rho_1\).
Note that classical uncertainty in the wavefunction is markedly different from superpositions. We can represent superpositions using wavefunctions, but use density matrices to describe distributions over wavefunctions. You can read more about density matrices here [DensityMatrix].
[DensityMatrix]  https://en.wikipedia.org/wiki/Density_matrix 
Quantum Gate Errors¶
For a quantum gate given by its unitary operator \(U\), a “quantum gate error” describes the scenario in which the actually induces transformation deviates from \(\ket{\psi} \mapsto U\ket{\psi}\). There are two basic types of quantum gate errors:
coherent errors are those that preserve the purity of the input state, i.e., instead of the above mapping we carry out a perturbed, but unitary operation \(\ket{\psi} \mapsto \tilde{U}\ket{\psi}\), where \(\tilde{U} \neq U\).
incoherent errors are those that do not preserve the purity of the input state, in this case we must actually represent the evolution in terms of density matrices. The state \(\rho := \ket{\psi}\bra{\psi}\) is then mapped as
\[\rho \mapsto \sum_{j=1}^m K_j\rho K_j^\dagger,\]where the operators \(\{K_1, K_2, \dots, K_m\}\) are called Kraus operators and must obey \(\sum_{j=1}^m K_j^\dagger K_j = I\) to conserve the trace of \(\rho\). Maps expressed in the above form are called Kraus maps. It can be shown that every physical map on a finite dimensional quantum system can be represented as a Kraus map, though this representation is not generally unique. You can find more information about quantum operations here
In a way, coherent errors are in principle amendable by more precisely calibrated control. Incoherent errors are more tricky.
Why Do Incoherent Errors Happen?¶
When a quantum system (e.g., the qubits on a quantum processor) is not perfectly isolated from its environment it generally coevolves with the degrees of freedom it couples to. The implication is that while the total time evolution of system and environment can be assumed to be unitary, restriction to the system state generally is not.
Let’s throw some math at this for clarity: Let our total Hilbert space be given by the tensor product of system and environment Hilbert spaces: \(\mathcal{H} = \mathcal{H}_S \otimes \mathcal{H}_E\). Our system “not being perfectly isolated” must be translated to the statement that the global Hamiltonian contains a contribution that couples the system and environment:
where \(V\) nontrivally acts on both the system and the environment. Consequently, even if we started in an initial state that factorized over system and environment \(\ket{\psi}_{S,0}\otimes \ket{\psi}_{E,0}\) if everything evolves by the Schrödinger equation
the final state will generally not admit such a factorization.
A Toy Model¶
In this (somewhat technical) section we show how environment interaction can corrupt an identity gate and derive its Kraus map. For simplicity, let us assume that we are in a reference frame in which both the system and environment Hamiltonian’s vanish \(H_S = 0, H_E = 0\) and where the crosscoupling is small even when multiplied by the duration of the time evolution \(\\frac{tV}{\hbar}\^2 \sim \epsilon \ll 1\) (any operator norm \(\\cdot\\) will do here). Let us further assume that \(V = \sqrt{\epsilon} V_S \otimes V_E\) (the more general case is given by a sum of such terms) and that the initial environment state satisfies \(\bra{\psi}_{E,0} V_E\ket{\psi}_{E,0} = 0\). This turns out to be a very reasonable assumption in practice but a more thorough discussion exceeds our scope.
Then the joint system + environment state \(\rho = \rho_{S,0} \otimes \rho_{E,0}\) (now written as a density matrix) evolves as
Using the BakerCampbellHausdorff theorem we can expand this to second order in \(\epsilon\)
We can insert the initially factorizable state \(\rho = \rho_{S,0} \otimes \rho_{E,0}\) and trace over the environmental degrees of freedom to obtain
where the coefficient in front of the second part is by our initial assumption very small \(\gamma := \frac{\epsilon t^2}{2\hbar^2}\tr{V_E^2 \rho_{E,0}} \ll 1\). This evolution happens to be approximately equal to a Kraus map with operators \(K_1 := I  \frac{\gamma}{2} V_S^2, K_2:= \sqrt{\gamma} V_S\):
This agrees to \(O(\epsilon^{3/2})\) with the result of our derivation above. This type of derivation can be extended to many other cases with little complication and a very similar argument is used to derive the Lindblad master equation.
Noisy Gates on the Rigetti QVM¶
As of today, users of our Forest SDK can annotate their QUIL programs by certain pragma statements that inform the QVM that a particular gate on specific target qubits should be replaced by an imperfect realization given by a Kraus map.
The QVM propagates pure states — so how does it simulate noisy gates? It does so by yielding the correct outcomes in the average over many executions of the QUIL program: When the noisy version of a gate should be applied the QVM makes a random choice which Kraus operator is applied to the current state with a probability that ensures that the average over many executions is equivalent to the Kraus map. In particular, a particular Kraus operator \(K_j\) is applied to \(\ket{\psi}_S\)
with probability \(p_j:= \bra{\psi}_S K_j^\dagger K_j \ket{\psi}_S\). In the average over many execution \(N \gg 1\) we therefore find that
where \(j_n\) is the chosen Kraus operator label in the \(n\)th trial. This is clearly a Kraus map itself! And we can group identical terms and rewrite it as
where \(N_{\ell}\) is the number of times that Kraus operator label \(\ell\) was selected. For large enough \(N\) we know that \(N_{\ell} \approx N p_\ell\) and therefore
which proves our claim. The consequence is that noisy gate simulations must generally be repeated many times to obtain representative results.
Getting Started¶
Come up with a good model for your noise. We will provide some examples below and may add more such examples to our public repositories over time. Alternatively, you can characterize the gate under consideration using Quantum Process Tomography or Gate Set Tomography and use the resulting process matrices to obtain a very accurate noise model for a particular QPU.
Define your Kraus operators as a list of numpy arrays
kraus_ops = [K1, K2, ..., Km]
.For your QUIL program
p
, call:p.define_noisy_gate("MY_NOISY_GATE", [q1, q2], kraus_ops)
where you should replace
MY_NOISY_GATE
with the gate of interest andq1, q2
the indices of the qubits.
Scroll down for some examples!
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom
import matplotlib.colors as colors
%matplotlib inline
from pyquil import Program, get_qc
from pyquil.gates import CZ, H, I, X, MEASURE
from scipy.linalg import expm
# We could ask for "2qnoisyqvm" but we will be specifying
# our noise model as PRAGMAs on the Program itself.
qc = get_qc('2qqvm')
Example 1: Amplitude Damping¶
Amplitude damping channels are imperfect identity maps with Kraus operators
where \(p\) is the probability that a qubit in the \(\ket{1}\) state decays to the \(\ket{0}\) state.
def damping_channel(damp_prob=.1):
"""
Generate the Kraus operators corresponding to an amplitude damping
noise channel.
:params float damp_prob: The onestep damping probability.
:return: A list [k1, k2] of the Kraus operators that parametrize the map.
:rtype: list
"""
damping_op = np.sqrt(damp_prob) * np.array([[0, 1],
[0, 0]])
residual_kraus = np.diag([1, np.sqrt(1damp_prob)])
return [residual_kraus, damping_op]
def append_kraus_to_gate(kraus_ops, g):
"""
Follow a gate `g` by a Kraus map described by `kraus_ops`.
:param list kraus_ops: The Kraus operators.
:param numpy.ndarray g: The unitary gate.
:return: A list of transformed Kraus operators.
"""
return [kj.dot(g) for kj in kraus_ops]
def append_damping_to_gate(gate, damp_prob=.1):
"""
Generate the Kraus operators corresponding to a given unitary
single qubit gate followed by an amplitude damping noise channel.
:params np.ndarraylist gate: The 2x2 unitary gate matrix.
:params float damp_prob: The onestep damping probability.
:return: A list [k1, k2] of the Kraus operators that parametrize the map.
:rtype: list
"""
return append_kraus_to_gate(damping_channel(damp_prob), gate)
%%time
# single step damping probability
damping_per_I = 0.02
# number of program executions
trials = 200
results_damping = []
lengths = np.arange(0, 201, 10, dtype=int)
for jj, num_I in enumerate(lengths):
print("\r{}/{}, ".format(jj, len(lengths)), end="")
p = Program(X(0))
# want increasing number of Igates
p.inst([I(0) for _ in range(num_I)])
p.inst(MEASURE(0, 0))
# overload identity I on qc 0
p.define_noisy_gate("I", [0], append_damping_to_gate(np.eye(2), damping_per_I))
p.wrap_in_numshots_loop(trials)
qc.qam.random_seed = int(num_I)
res = qc.run(p)
results_damping.append([np.mean(res), np.std(res) / np.sqrt(trials)])
results_damping = np.array(results_damping)
dense_lengths = np.arange(0, lengths.max()+1, .2)
survival_probs = (1damping_per_I)**dense_lengths
logpmf = binom.logpmf(np.arange(trials+1)[np.newaxis, :], trials, survival_probs[:, np.newaxis])/np.log(10)
DARK_TEAL = '#48737F'
FUSCHIA = "#D6619E"
BEIGE = '#EAE8C6'
cm = colors.LinearSegmentedColormap.from_list('anglemap', ["white", FUSCHIA, BEIGE], N=256, gamma=1.5)
plt.figure(figsize=(14, 6))
plt.pcolor(dense_lengths, np.arange(trials+1)/trials, logpmf.T, cmap=cm, vmin=4, vmax=logpmf.max())
plt.plot(dense_lengths, survival_probs, c=BEIGE, label="Expected mean")
plt.errorbar(lengths, results_damping[:,0], yerr=2*results_damping[:,1], c=DARK_TEAL,
label=r"noisy qvm, errorbars $ = \pm 2\hat{\sigma}$", marker="o")
cb = plt.colorbar()
cb.set_label(r"$\log_{10} \mathrm{Pr}(n_1; n_{\rm trials}, p_{\rm survival}(t))$", size=20)
plt.title("Amplitude damping model of a single qubit", size=20)
plt.xlabel(r"Time $t$ [arb. units]", size=14)
plt.ylabel(r"$n_1/n_{\rm trials}$", size=14)
plt.legend(loc="best", fontsize=18)
plt.xlim(*lengths[[0, 1]])
plt.ylim(0, 1)
Example 2: Dephased CZgate¶
Dephasing is usually characterized through a qubit’s \(T_2\) time. For a single qubit the dephasing Kraus operators are
where \(p = 1  \exp(T_2/T_{\rm gate})\) is the probability that the qubit is dephased over the time interval of interest, \(I_2\) is the \(2\times 2\)identity matrix and \(\sigma_Z\) is the PauliZ operator.
For two qubits, we must construct a Kraus map that has four different outcomes:
 No dephasing
 Qubit 1 dephases
 Qubit 2 dephases
 Both dephase
The Kraus operators for this are given by
where we assumed a dephasing probability \(p\) for the first qubit and \(q\) for the second.
Dephasing is a diagonal error channel and the CZ gate is also diagonal, therefore we can get the combined map of dephasing and the CZ gate simply by composing \(U_{\rm CZ}\) the unitary representation of CZ with each Kraus operator
Note that this is not always accurate, because a CZ gate is often achieved through nondiagonal interaction Hamiltonians! However, for sufficiently small dephasing probabilities it should always provide a good starting point.
def dephasing_kraus_map(p=.1):
"""
Generate the Kraus operators corresponding to a dephasing channel.
:params float p: The onestep dephasing probability.
:return: A list [k1, k2] of the Kraus operators that parametrize the map.
:rtype: list
"""
return [np.sqrt(1p)*np.eye(2), np.sqrt(p)*np.diag([1, 1])]
def tensor_kraus_maps(k1, k2):
"""
Generate the Kraus map corresponding to the composition
of two maps on different qubits.
:param list k1: The Kraus operators for the first qubit.
:param list k2: The Kraus operators for the second qubit.
:return: A list of tensored Kraus operators.
"""
return [np.kron(k1j, k2l) for k1j in k1 for k2l in k2]
%%time
# single step damping probabilities
ps = np.linspace(.001, .5, 200)
# number of program executions
trials = 500
results = []
for jj, p in enumerate(ps):
corrupted_CZ = append_kraus_to_gate(
tensor_kraus_maps(
dephasing_kraus_map(p),
dephasing_kraus_map(p)
),
np.diag([1, 1, 1, 1]))
print("\r{}/{}, ".format(jj, len(ps)), end="")
# make Bellstate
p = Program(H(0), H(1), CZ(0,1), H(1))
p.inst(MEASURE(0, 0))
p.inst(MEASURE(1, 1))
# overload identity I on qc 0
p.define_noisy_gate("CZ", [0, 1], corrupted_CZ)
p.wrap_in_numshots_loop(trials)
qc.qam.random_seed = jj
res = qc.run(p)
results.append(res)
results = np.array(results)
Z1s = (2*results[:,:,0]1.)
Z2s = (2*results[:,:,1]1.)
Z1Z2s = Z1s * Z2s
Z1m = np.mean(Z1s, axis=1)
Z2m = np.mean(Z2s, axis=1)
Z1Z2m = np.mean(Z1Z2s, axis=1)
plt.figure(figsize=(14, 6))
plt.axhline(y=1.0, color=FUSCHIA, alpha=.5, label="Bell state")
plt.plot(ps, Z1Z2m, "x", c=FUSCHIA, label=r"$\overline{Z_1 Z_2}$")
plt.plot(ps, 12*ps, "", c=FUSCHIA, label=r"$\langle Z_1 Z_2\rangle_{\rm theory}$")
plt.plot(ps, Z1m, "o", c=DARK_TEAL, label=r"$\overline{Z}_1$")
plt.plot(ps, 0*ps, "", c=DARK_TEAL, label=r"$\langle Z_1\rangle_{\rm theory}$")
plt.plot(ps, Z2m, "d", c="k", label=r"$\overline{Z}_2$")
plt.plot(ps, 0*ps, "", c="k", label=r"$\langle Z_2\rangle_{\rm theory}$")
plt.xlabel(r"Dephasing probability $p$", size=18)
plt.ylabel(r"$Z$moment", size=18)
plt.title(r"$Z$moments for a Bellstate prepared with dephased CZ", size=18)
plt.xlim(0, .5)
plt.legend(fontsize=18)
Adding Decoherence Noise¶
In this example, we investigate how a program might behave on a
nearterm device that is subject to T1 and T2type noise using the convenience function
pyquil.noise.add_decoherence_noise()
. The same module also contains some other useful
functions to define your own types of noise models, e.g.,
pyquil.noise.tensor_kraus_maps()
for generating multiqubit noise processes,
pyquil.noise.combine_kraus_maps()
for describing the succession of two noise processes and
pyquil.noise.append_kraus_to_gate()
which allows appending a noise process to a unitary
gate.
from pyquil.quil import Program
from pyquil.paulis import PauliSum, PauliTerm, exponentiate, exponential_map, trotterize
from pyquil.gates import MEASURE, H, Z, RX, RZ, CZ
import numpy as np
The Task¶
We want to prepare \(e^{i \theta XY}\) and measure it in the \(Z\) basis.
from numpy import pi
theta = pi/3
xy = PauliTerm('X', 0) * PauliTerm('Y', 1)
The Idiomatic PyQuil Program¶
prog = exponential_map(xy)(theta)
print(prog)
H 0
RX(pi/2) 1
CNOT 0 1
RZ(2*pi/3) 1
CNOT 0 1
H 0
RX(pi/2) 1
The Compiled Program¶
To run on a real device, we must compile each program to the native gate set for the device. The highlevel noise model is similarly constrained to use a small, native gate set. In particular, we can use
 \(I\)
 \(RZ(\theta)\)
 \(RX(\pm \pi/2)\)
 \(CZ\)
For simplicity, the compiled program is given below but generally you will want to use a compiler to do this step for you.
def get_compiled_prog(theta):
return Program([
RZ(pi/2, 0),
RX(pi/2, 0),
RZ(pi/2, 1),
RX( pi/2, 1),
CZ(1, 0),
RZ(pi/2, 1),
RX(pi/2, 1),
RZ(theta, 1),
RX( pi/2, 1),
CZ(1, 0),
RX( pi/2, 0),
RZ( pi/2, 0),
RZ(pi/2, 1),
RX( pi/2, 1),
RZ(pi/2, 1),
])
Scan Over Noise Parameters¶
We perform a scan over three levels of noise each at 20 theta points.
Specifically, we investigate T1 values of 1, 3, and 10 us. By default, T2 = T1 / 2, 1 qubit gates take 50 ns, and 2 qubit gates take 150 ns.
In alignment with the device, \(I\) and parametric \(RZ\) are noiseless while \(RX\) and \(CZ\) gates experience 1q and 2q gate noise, respectively.
from pyquil.api import QVMConnection
cxn = QVMConnection()
t1s = np.logspace(6, 5, num=3)
thetas = np.linspace(pi, pi, num=20)
t1s * 1e6 # us
array([ 1. , 3.16227766, 10. ])
from pyquil.noise import add_decoherence_noise
records = []
for theta in thetas:
for t1 in t1s:
prog = get_compiled_prog(theta)
noisy = add_decoherence_noise(prog, T1=t1).inst([
MEASURE(0, 0),
MEASURE(1, 1),
])
bitstrings = np.array(cxn.run(noisy, [0,1], 1000))
# Expectation of Z0 and Z1
z0, z1 = 1  2*np.mean(bitstrings, axis=0)
# Expectation of ZZ by computing the parity of each pair
zz = 1  (np.sum(bitstrings, axis=1) % 2).mean() * 2
record = {
'z0': z0,
'z1': z1,
'zz': zz,
'theta': theta,
't1': t1,
}
records += [record]
Plot the Results¶
Note that to run the code below you will need to install the pandas and seaborn packages.
%matplotlib inline
from matplotlib import pyplot as plt
import seaborn as sns
sns.set(style='ticks', palette='colorblind')
import pandas as pd
df_all = pd.DataFrame(records)
fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(12,4))
for t1 in t1s:
df = df_all.query('t1 == @t1')
ax1.plot(df['theta'], df['z0'], 'o')
ax2.plot(df['theta'], df['z1'], 'o')
ax3.plot(df['theta'], df['zz'], 'o', label='T1 = {:.0f} us'.format(t1*1e6))
ax3.legend(loc='best')
ax1.set_ylabel('Z0')
ax2.set_ylabel('Z1')
ax3.set_ylabel('ZZ')
ax2.set_xlabel(r'$\theta$')
fig.tight_layout()
Modeling Readout Noise¶
QubitReadout can be corrupted in a variety of ways. The two most relevant error mechanisms on the Rigetti QPU right now are:
 Transmission line noise that makes a 0state look like a 1state or vice versa. We call this classical readout bitflip error. This type of readout noise can be reduced by tailoring optimal readout pulses and using superconducting, quantum limited amplifiers to amplify the readout signal before it is corrupted by classical noise at the higher temperature stages of our cryostats.
 T1 qubit decay during readout (our readout operations can take more than a µsecond unless they have been specially optimized), which leads to readout signals that initially behave like 1states but then collapse to something resembling a 0state. We will call this T1readout error. This type of readout error can be reduced by achieving shorter readout pulses relative to the T1 time, i.e., one can try to reduce the readout pulse length, or increase the T1 time or both.
Qubit Measurements¶
This section provides the necessary theoretical foundation for accurately modeling noisy quantum measurements on superconducting quantum processors. It relies on some of the abstractions (density matrices, Kraus maps) introduced in our notebook on gate noise models.
The most general type of measurement performed on a single qubit at a single time can be characterized by some set \(\mathcal{O}\) of measurement outcomes, e.g., in the simplest case \(\mathcal{O} = \{0, 1\}\), and some unnormalized quantum channels (see notebook on gate noise models) that encapsulate 1. the probability of that outcome 2. how the qubit state is affected conditional on the measurement outcome.
Here the outcome is understood as classical information that has been extracted from the quantum system.
Projective, Ideal Measurement¶
The simplest case that is usually taught in introductory quantum mechanics and quantum information courses are Born’s rule and the projection postulate which state that there exist a complete set of orthogonal projection operators
i.e., one for each measurement outcome. Any projection operator must satisfy \(\Pi_x^\dagger = \Pi_x = \Pi_x^2\) and for an orthogonal set of projectors any two members satisfy
and for a complete set we additionally demand that \(\sum_{x\in\mathcal{O}} \Pi_x = 1\). Following our introduction to gate noise, we write quantum states as density matrices as this is more general and in closer correspondence with classical probability theory.
With these the probability of outcome \(x\) is given by \(p(x) = \tr{\Pi_x \rho \Pi_x} = \tr{\Pi_x^2 \rho} = \tr{\Pi_x \rho}\) and the post measurement state is
which is the projection postulate applied to mixed states.
If we were a sloppy quantum programmer and accidentally erased the measurement outcome then our best guess for the post measurement state would be given by something that looks an awful lot like a Kraus map:
The completeness of the projector set ensures that the trace of the post measurement is still 1 and the Kraus map form of this expression ensures that \(\rho_{\text{post measurement}}\) is a positive (semi)definite operator.
Classical Readout BitFlip Error¶
Consider now the ideal measurement as above, but where the outcome \(x\) is transmitted across a noisy classical channel that produces a final outcome \(x'\in \mathcal{O}' = \{0', 1'\}\) according to some conditional probabilities \(p(x'x)\) that can be recorded in the assignment probability matrix
Note that this matrix has only two independent parameters as each column must be a valid probability distribution, i.e. all elements are nonnegative and each column sums to 1.
This matrix allows us to obtain the probabilities \(\mathbf{p}' := (p(x'=0), p(x'=1))^T\) from the original outcome probabilities \(\mathbf{p} := (p(x=0), p(x=1))^T\) via \(\mathbf{p}' = P_{x'x}\mathbf{p}\). The difference relative to the ideal case above is that now an outcome \(x' = 0\) does not necessarily imply that the post measurement state is truly \(\Pi_{0} \rho \Pi_{0} / p(x=0)\). Instead, the post measurement state given a noisy outcome \(x'\) must be
where
where we have exploited the cyclical property of the trace \(\tr{ABC}=\tr{BCA}\) and the projection property \(\Pi_x^2 = \Pi_x\). This has allowed us to derive the noisy outcome probabilities from a set of positive operators
that must sum to 1:
The above result is a type of generalized Bayes’ theorem that is extremely useful for this type of (slightly) generalized measurement and the family of operators \(\{E_{x'} x' \in \mathcal{O}'\}\) whose expectations give the probabilities is called a positive operator valued measure (POVM). These operators are not generally orthogonal nor valid projection operators but they naturally arise in this scenario. This is not yet the most general type of measurement, but it will get us pretty far.
How to Model \(T_1\) Error¶
T1 type errors fall outside our framework so far as they involve a scenario in which the quantum state itself is corrupted during the measurement process in a way that potentially erases the premeasurement information as opposed to a loss of purely classical information. The most appropriate framework for describing this is given by that of measurement instruments, but for the practical purpose of arriving at a relatively simple description, we propose describing this by a T1 damping Kraus map followed by the noisy readout process as described above.
Further Reading¶
Chapter 3 of John Preskill’s lecture notes http://www.theory.caltech.edu/people/preskill/ph229/notes/chap3.pdf
Working with Readout Noise¶
Come up with a good guess for your readout noise parameters \(p(00)\) and \(p(11)\), the offdiagonals then follow from the normalization of \(P_{x'x}\). If your assignment fidelity \(F\) is given, and you assume that the classical bit flip noise is roughly symmetric, then a good approximation is to set \(p(00)=p(11)=F\).
For your QUIL program
p
, and a qubit indexq
call:p.define_noisy_readout(q, p00, p11)
where you should replace
p00
andp11
with the assumed probabilities.
Scroll down for some examples!
from __future__ import print_function, division
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from pyquil.quil import Program, MEASURE, Pragma
from pyquil.api.qvm import QVMConnection
from pyquil.gates import I, X, RX, H, CNOT
from pyquil.noise import (estimate_bitstring_probs, correct_bitstring_probs,
bitstring_probs_to_z_moments, estimate_assignment_probs)
DARK_TEAL = '#48737F'
FUSCHIA = '#D6619E'
BEIGE = '#EAE8C6'
cxn = QVMConnection()
Example 1: Rabi Sequence with Noisy Readout¶
%%time
# number of angles
num_theta = 101
# number of program executions
trials = 200
thetas = np.linspace(0, 2*np.pi, num_theta)
p00s = [1., 0.95, 0.9, 0.8]
results_rabi = np.zeros((num_theta, len(p00s)))
for jj, theta in enumerate(thetas):
for kk, p00 in enumerate(p00s):
cxn.random_seed = hash((jj, kk))
p = Program(RX(theta, 0))
# assume symmetric noise p11 = p00
p.define_noisy_readout(0, p00=p00, p11=p00)
p.measure(0, 0)
res = cxn.run(p, [0], trials=trials)
results_rabi[jj, kk] = np.sum(res)
CPU times: user 1.2 s, sys: 73.6 ms, total: 1.27 s
Wall time: 3.97 s
plt.figure(figsize=(14, 6))
for jj, (p00, c) in enumerate(zip(p00s, [DARK_TEAL, FUSCHIA, "k", "gray"])):
plt.plot(thetas, results_rabi[:, jj]/trials, c=c, label=r"$p(00)=p(11)={:g}$".format(p00))
plt.legend(loc="best")
plt.xlim(*thetas[[0,1]])
plt.ylim(.1, 1.1)
plt.grid(alpha=.5)
plt.xlabel(r"RX angle $\theta$ [radian]", size=16)
plt.ylabel(r"Excited state fraction $n_1/n_{\rm trials}$", size=16)
plt.title("Effect of classical readout noise on Rabi contrast.", size=18)
<matplotlib.text.Text at 0x104314250>
Example 2: Estimate the Assignment Probabilities¶
Here we will estimate \(P_{x'x}\) ourselves! You can run some simple experiments to estimate the assignment probability matrix directly from a QPU.
On a perfect quantum computer
estimate_assignment_probs(0, 1000, cxn, Program())
array([[ 1., 0.],
[ 0., 1.]])
On an imperfect quantum computer
cxn.seed = None
header0 = Program().define_noisy_readout(0, .85, .95)
header1 = Program().define_noisy_readout(1, .8, .9)
header2 = Program().define_noisy_readout(2, .9, .85)
ap0 = estimate_assignment_probs(0, 100000, cxn, header0)
ap1 = estimate_assignment_probs(1, 100000, cxn, header1)
ap2 = estimate_assignment_probs(2, 100000, cxn, header2)
print(ap0, ap1, ap2, sep="\n")
[[ 0.84967 0.04941]
[ 0.15033 0.95059]]
[[ 0.80058 0.09993]
[ 0.19942 0.90007]]
[[ 0.90048 0.14988]
[ 0.09952 0.85012]]
Example 3: Correct for Noisy Readout¶
3a) Correcting the Rabi Signal from Above¶
ap_last = np.array([[p00s[1], 1  p00s[1]],
[1  p00s[1], p00s[1]]])
corrected_last_result = [correct_bitstring_probs([1p, p], [ap_last])[1] for p in results_rabi[:, 1] / trials]
plt.figure(figsize=(14, 6))
for jj, (p00, c) in enumerate(zip(p00s, [DARK_TEAL, FUSCHIA, "k", "gray"])):
if jj not in [0, 3]:
continue
plt.plot(thetas, results_rabi[:, jj]/trials, c=c, label=r"$p(00)=p(11)={:g}$".format(p00), alpha=.3)
plt.plot(thetas, corrected_last_result, c="red", label=r"Corrected $p(00)=p(11)={:g}$".format(p00s[1]))
plt.legend(loc="best")
plt.xlim(*thetas[[0,1]])
plt.ylim(.1, 1.1)
plt.grid(alpha=.5)
plt.xlabel(r"RX angle $\theta$ [radian]", size=16)
plt.ylabel(r"Excited state fraction $n_1/n_{\rm trials}$", size=16)
plt.title("Corrected contrast", size=18)
<matplotlib.text.Text at 0x1055e7310>
We find that the corrected signal is fairly noisy (and sometimes exceeds the allowed interval \([0,1]\)) due to the overall very small number of samples \(n=200\).
3b) Corrupting and Correcting GHZ State Correlations¶
In this example we will create a GHZ state \(\frac{1}{\sqrt{2}}\left[\left000\right\rangle + \left111\right\rangle \right]\) and measure its outcome probabilities with and without the above noise model. We will then see how the PauliZ moments that indicate the qubit correlations are corrupted (and corrected) using our API.
ghz_prog = Program(H(0), CNOT(0, 1), CNOT(1, 2),
MEASURE(0, 0), MEASURE(1, 1), MEASURE(2, 2))
print(ghz_prog)
results = cxn.run(ghz_prog, [0, 1, 2], trials=10000)
H 0
CNOT 0 1
CNOT 1 2
MEASURE 0 [0]
MEASURE 1 [1]
MEASURE 2 [2]
header = header0 + header1 + header2
noisy_ghz = header + ghz_prog
print(noisy_ghz)
noisy_results = cxn.run(noisy_ghz, [0, 1, 2], trials=10000)
PRAGMA READOUTPOVM 0 "(0.85 0.050000000000000044 0.15000000000000002 0.95)"
PRAGMA READOUTPOVM 1 "(0.8 0.09999999999999998 0.19999999999999996 0.9)"
PRAGMA READOUTPOVM 2 "(0.9 0.15000000000000002 0.09999999999999998 0.85)"
H 0
CNOT 0 1
CNOT 1 2
MEASURE 0 [0]
MEASURE 1 [1]
MEASURE 2 [2]
probs = estimate_bitstring_probs(results)
probs[0, 0, 0], probs[1, 1, 1]
(0.50419999999999998, 0.49580000000000002)
As expected the outcomes 000
and 111
each have roughly
probability \(1/2\).
noisy_probs = estimate_bitstring_probs(noisy_results)
noisy_probs[0, 0, 0], noisy_probs[1, 1, 1]
(0.30869999999999997, 0.3644)
The noisecorrupted outcome probabilities deviate significantly from their ideal values!
corrected_probs = correct_bitstring_probs(noisy_probs, [ap0, ap1, ap2])
corrected_probs[0, 0, 0], corrected_probs[1, 1, 1]
(0.50397601453064977, 0.49866843912900716)
The corrected outcome probabilities are much closer to the ideal value.
We expect these to all be very small
zmoments = bitstring_probs_to_z_moments(probs)
zmoments[1, 0, 0], zmoments[0, 1, 0], zmoments[0, 0, 1]
(0.0083999999999999631, 0.0083999999999999631, 0.0083999999999999631)
We expect these to all be close to 1.
zmoments[1, 1, 0], zmoments[0, 1, 1], zmoments[1, 0, 1]
(1.0, 1.0, 1.0)
zmoments_corr = bitstring_probs_to_z_moments(corrected_probs)
zmoments_corr[1, 0, 0], zmoments_corr[0, 1, 0], zmoments_corr[0, 0, 1]
(0.0071476770049732075, 0.0078641261685578612, 0.0088462563282706852)
zmoments_corr[1, 1, 0], zmoments_corr[0, 1, 1], zmoments_corr[1, 0, 1]
(0.99477496902638118, 1.0008376440216553, 1.0149652015905912)
Overall the correction can restore the contrast in our multiqubit observables, though we also see that the correction can lead to slightly nonphysical expectations. This effect is reduced the more samples we take.
Alternative: A global Pauli error model¶
The Rigetti QVM has support for emulating certain types of noise models. One such model is parametric Pauli noise, which is defined by a set of 6 probabilities:
 The probabilities \(P_X\), \(P_Y\), and \(P_Z\) which define respectively the probability of a Pauli \(X\), \(Y\), or \(Z\) gate getting applied to each qubit after every gate application. These probabilities are called the gate noise probabilities.
 The probabilities \(P_X'\), \(P_Y'\), and \(P_Z'\) which define respectively the probability of a Pauli \(X\), \(Y\), or \(Z\) gate getting applied to the qubit being measured before it is measured. These probabilities are called the measurement noise probabilities.
We can instantiate a noisy QVM by creating a new connection with these probabilities specified.
# 20% chance of a X gate being applied after gate applications and before measurements.
gate_noise_probs = [0.2, 0.0, 0.0]
meas_noise_probs = [0.2, 0.0, 0.0]
noisy_qvm = qvm(gate_noise=gate_noise_probs, measurement_noise=meas_noise_probs)
We can test this by applying an \(X\)gate and measuring. Nominally,
we should always measure 1
.
p = Program().inst(X(0)).measure(0, 0)
print("Without Noise: {}".format(qvm.run(p, [0], 10)))
print("With Noise : {}".format(noisy_qvm.run(p, [0], 10)))
Without Noise: [[1], [1], [1], [1], [1], [1], [1], [1], [1], [1]]
With Noise : [[0], [0], [0], [0], [0], [1], [1], [1], [1], [0]]
Advanced Usage¶
Note
If you’re running locally, remember set up the QVM and quilc in server mode before trying to use them: Setting Up Server Mode for PyQuil.
PyQuil Configuration Files¶
Network endpoints for the Rigetti Forest infrastructure and information pertaining to QPU access are
stored in a pair of configuration files. These files are located by default at ~/.qcs_config
and ~/.forest_config
.
The location can be changed by setting the environment variables QCS_CONFIG
or FOREST_CONFIG
to point to the new
location.
When running on a QMI, the values in these configuration files are automatically managed so as to point to the correct endpoints. When running locally, configuration files are not necessary. Thus, the average user will not have to do any work to get their configuration files set up.
If for some reason you want to use an atypical configuration, you may need to modify these files.
The default QCS config file on any QMI looks similar to the following:
# .qcs_config
[Rigetti Forest]
url = https://forestserver.qcs.rigetti.com
key = 4fd1239111eb52ec35c2262765ae4c4f
user_id = 4fd1239111eb52ec35c2262765ae4c4f
[QPU]
exec_on_engage = bash exec_on_engage.sh
where
url
is the endpoint that pyQuil hits for device information and for the 2.0 endpoints,key
stores the Forest 1.X API key,user_id
stores a Forest 2.0 user ID, andexec_on_engage
specifies the shell command that the QMI will launch when the QMI becomes QPUengaged. It would have no effect if you are running locally, but is important if you are running on the QMI. By default, it runs theexec_on_engage.sh
shell script. It’s best to leave the configuration as is, and edit that script. More documentation aboutexec_on_engage.sh
can be found in the QCS docs here.
The Forest config file on any QMI has these contents, with specific IP addresses filled in:
# .forest_config
[Rigetti Forest]
qpu_endpoint_address = None
qvm_address = http://10.1.165.XX:5000
compiler_server_address = tcp://10.1.165.XX:5555
where
qpu_endpoint_address
is the endpoint where pyQuil will try to communicate with the QPU orchestrating service during QPUengagement. It may not appear until your QMI engages, and furthermore will have no effect if you are running locally. It’s best to leave this alone. If you obtain access to one of our QPUs, we will fill it in for you.qvm_address
is the endpoint where pyQuil will try to communicate with the Rigetti Quantum Virtual Machine. On a QMI, this points to the provided QVM instance. On a local installation, this should be set to the server endpoint for a locally running QVM instance. However, pyQuil will use the default valuehttp://localhost:5000
if this file isn’t found, which is the correct endpoint when you run the QVM locally withqvm S
.compiler_server_address
: This is the endpoint where pyQuil will try to communicate with the compiler server. On a QMI, this points to a provided compiler server instance. On a local installation, this should be set to the server endpoint for a locally runningquilc
instance. However, pyQuil will use the default valuetcp://localhost:5555
if this isn’t set, which is the correct endpoint when you runquilc
locally withquilc S
.
Note
PyQuil itself reads these values out using the helper class pyquil._config.PyquilConfig
. PyQuil users should not
ever need to touch this class directly.
Using Qubit Placeholders¶
Note
The functionality provided inline by QubitPlaceholders
is similar to writing a function which returns a
Program
, with qubit indices taken as arguments to the function.
In pyQuil, we typically use integers to identify qubits
from pyquil import Program
from pyquil.gates import CNOT, H
print(Program(H(0), CNOT(0, 1)))
H 0
CNOT 0 1
However, when running on real, nearterm QPUs we care about what
particular physical qubits our program will run on. In fact, we may want
to run the same program on an assortment of different qubits. This is
where using QubitPlaceholder
s comes in.
from pyquil.quilatom import QubitPlaceholder
q0 = QubitPlaceholder()
q1 = QubitPlaceholder()
p = Program(H(q0), CNOT(q0, q1))
print(p)
H {q4402789176}
CNOT {q4402789176} {q4402789120}
If you try to use this program directly, it will not work
print(p.out())
RuntimeError: Qubit q4402789176 has not been assigned an index
Instead, you must explicitly map the placeholders to physical qubits. By
default, the function address_qubits
will address qubits from 0 to
N.
from pyquil.quil import address_qubits
print(address_qubits(p))
H 0
CNOT 0 1
The real power comes into play when you provide an explicit mapping:
print(address_qubits(prog, qubit_mapping={
q0: 14,
q1: 19,
}))
H 14
CNOT 14 19
Register¶
Usually, your algorithm will use an assortment of qubits. You can use
the convenience function QubitPlaceholder.register()
to request a
list of qubits to build your program.
qbyte = QubitPlaceholder.register(8)
p_evens = Program(H(q) for q in qbyte)
print(address_qubits(p_evens, {q: i*2 for i, q in enumerate(qbyte)}))
H 0
H 2
H 4
H 6
H 8
H 10
H 12
H 14
Classical Control Flow¶
Note
Classical control flow is not yet supported on the QPU.
Here are a couple quick examples that show how much richer a Quil program can be with classical control flow. In this first example, we create a while loop by following these steps:
 Declare a register called
flag_register
to use as a boolean test for looping.  Initialize this register to
1
program so our while loop will execute. This is often called the loop preamble or loop initialization.  Write the body of the loop in its own
Program
. This will be a program that applies an \(X\) gate followed by a \(H\) gate on our qubit.  Using the
while_do()
method to add control flow.
from pyquil import Program
from pyquil.gates import *
# Initialize the Program and declare a 1 bit memory space for our boolean flag
outer_loop = Program()
flag_register = outer_loop.declare('flag_register', 'BIT')
# Set the initial flag value to 1
outer_loop += MOVE(flag_register, 1)
# Define the body of the loop with a new Program
inner_loop = Program()
inner_loop += Program(X(0), H(0))
inner_loop += MEASURE(0, flag_register)
# Run inner_loop in a loop until flag_register is 0
outer_loop.while_do(flag_register, inner_loop)
print(outer_loop)
DECLARE flag_register BIT[1]
MOVE flag_register 1
LABEL @START1
JUMPUNLESS @END2 flag_register
X 0
H 0
MEASURE 0 flag_register
JUMP @START1
LABEL @END2
Notice that the outer_loop
program applied a Quil instruction directly to a
classical register. There are several classical commands that can be used in this fashion:
NOT
which flips a classical bitAND
which operates on two classical bitsIOR
which operates on two classical bitsMOVE
which moves the value of a classical bit at one classical address into anotherEXCHANGE
which swaps the value of two classical bits
In this next example, we show how to do conditional branching in the
form of the traditional if
construct as in many programming
languages. Much like the last example, we construct programs for each
branch of the if
, and put it all together by using the if_then()
method.
# Declare our memory spaces
branching_prog = Program()
test_register = branching_prog.declare('test_register', 'BIT')
ro = branching_prog.declare('ro', 'BIT')
# Construct each branch of our ifstatement. We can have empty branches
# simply by having empty programs.
then_branch = Program(X(0))
else_branch = Program()
# Construct our program so that the result in test_register is equally likely to be a 0 or 1
branching_prog += H(1)
branching_prog += MEASURE(1, test_register)
# Add the conditional branching
branching_prog.if_then(test_register, then_branch, else_branch)
# Measure qubit 0 into our readout register
branching_prog += MEASURE(0, ro)
print(branching_prog)
DECLARE test_register BIT[1]
DECLARE ro BIT[1]
H 1
MEASURE 1 test_register
JUMPWHEN @THEN1 test_register
JUMP @END2
LABEL @THEN1
X 0
LABEL @END2
MEASURE 0 ro
We can run this program a few times to see what we get in the readout register ro
.
from pyquil import get_qc
qc = get_qc("2qqvm")
branching_prog.wrap_in_numshots_loop(10)
qc.run(branching_prog)
[[1], [1], [1], [0], [1], [0], [0], [1], [1], [0]]
Pauli Operator Algebra¶
Many algorithms require manipulating sums of Pauli combinations, such as
\(\sigma = \frac{1}{2}I  \frac{3}{4}X_0Y_1Z_3 + (52i)Z_1X_2,\) where
\(G_n\) indicates the gate \(G\) acting on qubit \(n\). We
can represent such sums by constructing PauliTerm
and PauliSum
.
The above sum can be constructed as follows:
from pyquil.paulis import ID, sX, sY, sZ
# Pauli term takes an operator "X", "Y", "Z", or "I"; a qubit to act on, and
# an optional coefficient.
a = 0.5 * ID()
b = 0.75 * sX(0) * sY(1) * sZ(3)
c = (52j) * sZ(1) * sX(2)
# Construct a sum of Pauli terms.
sigma = a + b + c
print(f"sigma = {sigma}")
sigma = (0.5+0j)*I + (0.75+0j)*X0*Y1*Z3 + (52j)*Z1*X2
Right now, the primary thing one can do with Pauli terms and sums is to construct the exponential of the Pauli term, i.e., \(\exp[i\beta\sigma]\). This is accomplished by constructing a parameterized Quil program that is evaluated when passed values for the coefficients of the angle \(\beta\).
Related to exponentiating Pauli sums we provide utility functions for finding the commuting subgroups of a Pauli sum and approximating the exponential with the SuzukiTrotter approximation through fourth order.
When arithmetic is done with Pauli sums, simplification is automatically done.
The following shows an instructive example of all three.
from pyquil.paulis import exponential_map
sigma_cubed = sigma * sigma * sigma
print(f"Simplified: {sigma_cubed}\n")
# Produce Quil code to compute exp[iX]
H = 1.0 * sX(0)
print(f"Quil to compute exp[iX] on qubit 0:\n"
f"{exponential_map(H)(1.0)}")
Simplified: (32.4687530j)*I + (16.734375+15j)*X0*Y1*Z3 + (71.5625144.625j)*Z1*X2
Quil to compute exp[iX] on qubit 0:
H 0
RZ(2.0) 0
H 0
exponential_map
returns a function allowing you to fill in a multiplicative
constant later. This commonly occurs in variational algorithms. The function
exponential_map
is used to compute \(\exp[i \alpha H]\) without explicitly filling in a
value for \(\alpha\).
expH = exponential_map(H)
print(f"0:\n{expH(0.0)}\n")
print(f"1:\n{expH(1.0)}\n")
print(f"2:\n{expH(2.0)}")
0:
H 0
RZ(0) 0
H 0
1:
H 0
RZ(2.0) 0
H 0
2:
H 0
RZ(4.0) 0
H 0
To take it one step further, you can use Parametric Compilation with exponential_map
. For instance:
ham = sZ(0) * sZ(1)
prog = Program()
theta = prog.declare('theta', 'REAL')
prog += exponential_map(ham)(theta)
Exercises¶
Exercise 1: Quantum Dice¶
Write a quantum program to simulate throwing an 8sided die. The Python function you should produce is:
def throw_octahedral_die():
# return the result of throwing an 8 sided die, an int between 1 and 8, by running a quantum program
Next, extend the program to work for any kind of fair die:
def throw_polyhedral_die(num_sides):
# return the result of throwing a num_sides sided die by running a quantum program
Exercise 2: Controlled Gates¶
We can use the full generality of NumPy to construct new gate matrices.
 Write a function
controlled
which takes a \(2\times 2\) matrix \(U\) representing a single qubit operator, and makes a \(4\times 4\) matrix which is a controlled variant of \(U\), with the first argument being the control qubit.  Write a Quil program to define a controlled\(Y\) gate in this manner. Find the wavefunction when applying this gate to qubit 1 controlled by qubit 0.
Exercise 3: Grover’s Algorithm¶
Write a quantum program for the singleshot Grover’s algorithm. The Python function you should produce is:
# data is an array of 0's and 1's such that there are exactly three times as many
# 0's as 1's
def single_shot_grovers(data):
# return an index that contains the value 1
As an example: single_shot_grovers([0,0,1,0])
should return 2.
HINT  Remember that the Grover’s diffusion operator is:
Exercise 4: Prisoner’s Dilemma¶
A classic strategy game is the prisoner’s dilemma where two prisoners get the minimal penalty if they collaborate and stay silent, get zero penalty if one of them defects and the other collaborates (incurring maximum penalty) and get intermediate penalty if they both defect. This game has an equilibrium where both defect and incur intermediate penalty.
However, things change dramatically when we allow for quantum strategies leading to the Quantum Prisoner’s Dilemma.
Can you design a program that simulates this game?
Exercise 5: Quantum Fourier Transform¶
The quantum Fourier transform (QFT) is a quantum implementation of the discrete Fourier transform. The Fourier transform can be used to transform a function from the time domain into the frequency domain.
 Compute the discrete Fourier transform of
[0, 1, 0, 0, 0, 0, 0, 0]
, using pyQuil:  Write a state preparation quantum program.
 Write a function to make a 3qubit QFT program, taking qubit indices as arguments.
 Combine your solutions to part a and b into one program and use the
WavefunctionSimulator
to get the solution.
Note
For a more challenging initial state, try 01100100
.
Solution¶
Part a: Prepare the initial state¶
We are going to apply the QFT on the amplitudes of the states.
We want to prepare a state that corresponds to the sequence for which we want to compute the discrete Fourier transform. As the exercise hinted in part b, we need 3 qubits to transform an 8 bit sequence. It is simplest to understand if we think of the qubits as three digits in a binary string (aka bitstring). There are 8 possible values the bitstring can have, and in our quantum state, each of these possibilities has an amplitude. Our 8 indices in the QFT sequence label each of these states. For clarity:
\(000\rangle\) => 10000000
\(001\rangle\) => 01000000
…
\(111\rangle\) > 00000001
The sequence we want to compute is 01000000
, so our initial state is simply \(001\rangle\). For a bitstring with more
than one 1
, we would want an equal superposition over all the selected states. (E.g. 01100000
would be an
equal superposition of \(001\rangle\) and \(010\rangle\)).
To set up the \(001\rangle\) state, we only have to apply one \(X\)gate to the zeroth qubit.
from pyquil import Program
from pyquil.gates import *
state_prep = Program(X(0))
We can verify that this works by computing its wavefunction with the
Wavefunction Simulator. However, we need to add some “dummy” qubits,
because otherwise wavefunction
would return a twoelement vector for only qubit 0.
from pyquil.api import WavefunctionSimulator
add_dummy_qubits = Program(I(1), I(2)) # The identity gate I has no affect
wf_sim = WavefunctionSimulator()
wavefunction = wf_sim.wavefunction(state_prep + add_dummy_qubits)
print(wavefunction)
(1+0j)001>
We’ll need wf_sim
for part c, too.
Part b: Three qubit QFT program¶
In this part, we define a function, qft3
, to make a 3qubit QFT quantum program. The algorithm
is nicely described on this page.
It is a mix of Hadamard and CPHASE gates, with a SWAP gate for bit reversal correction.
from math import pi
def qft3(q0, q1, q2):
p = Program()
p += [SWAP(q0, q2),
H(q0),
CPHASE(pi / 2.0, q0, q1),
H(q1),
CPHASE(pi / 4.0, q0, q2),
CPHASE(pi / 2.0, q1, q2),
H(q2)]
return p
There is a very important detail to recognize here: The function
qft3
doesn’t compute the QFT, but rather it makes a quantum
program to compute the QFT on qubits q0
, q1
, and q2
.
We can see what this program looks like in Quil notation with print(qft(0, 1, 2))
.
SWAP 0 2
H 0
CPHASE(pi/2) 0 1
H 1
CPHASE(pi/4) 0 2
CPHASE(pi/2) 1 2
H 2
Part c: Execute the QFT¶
Combining parts a and b:
compute_qft_prog = state_prep + qft3(0, 1, 2)
wavefunction = wf_sim.wavefunction(compute_qft_prog)
print(wavefunction.amplitudes)
array([ 3.53553391e01+0.j , 2.50000000e010.25j ,
2.16489014e170.35355339j, 2.50000000e010.25j ,
3.53553391e01+0.j , 2.50000000e01+0.25j ,
2.16489014e17+0.35355339j, 2.50000000e01+0.25j ])
We can verify this works by computing the inverse FFT on the output with NumPy and seeing that we get back our input (with some floating point error).
from numpy.fft import ifft
ifft(wavefunction.amplitudes, norm="ortho")
array([0.+0.00000000e+00j, 1.+9.38127079e17j, 0.+0.00000000e+00j,
0.1.53080850e17j, 0.+0.00000000e+00j, 0.6.31965379e17j,
0.+0.00000000e+00j, 0.1.53080850e17j])
After ignoring the terms that are on the order of 1e17
, we get [0, 1, 0, 0, 0, 0, 0, 0]
, which was our input!
Example: The MeyerPenny Game¶
To create intuition for quantum algorithms, it is useful (and fun) to play with the abstraction that the software provides.
The MeyerPenny Game [1] is a simple example we’ll use from quantum game theory. The interested reader may want to read more about quantum game theory in the article Toward a general theory of quantum games [2]. The MeyerPenny Game goes as follows:
The Starship Enterprise, during one of its deepspace missions, is facing an immediate calamity at the edge of a wormhole, when a powerful alien suddenly appears. The alien, named Q, offers to help Picard, the captain of the Enterprise, under the condition that Picard beats Q in a simple game of heads or tails.
The rules¶
Picard is to place a penny heads up into an opaque box. Then Picard and Q take turns to flip or not flip the penny without being able to see it; first Q then P then Q again. After this the penny is revealed; Q wins if it shows heads (H), while tails (T) makes Picard the winner.
Picard vs. Q¶
Picard quickly estimates that his chance of winning is 50% and agrees to play the game. He loses the first round and insists on playing again. To his surprise Q agrees, and they continue playing several rounds more, each of which Picard loses. How is that possible?
What Picard did not anticipate is that Q has access to quantum tools. Instead of flipping the penny, Q puts the penny into a superposition of heads and tails proportional to the quantum state \(H\rangle+T\rangle\). Then no matter whether Picard flips the penny or not, it will stay in a superposition (though the relative sign might change). In the third step Q undoes the superposition and always finds the penny to show heads.
Let’s see how this works!
To simulate the game we first construct the corresponding quantum circuit, which takes two qubits: one to simulate Picard’s choice whether or not to flip the penny, and the other to represent the penny. The initial state for all qubits is \(0\rangle\) (which is mapped to \(T\rangle\), tails). To simulate Picard’s decision, we assume that he chooses randomly whether or not to flip the coin, in agreement with the optimal strategy for the classic pennyflip game. This random choice can be created by putting one qubit into an equal superposition, e.g. with the Hadamard gate \(H\), and then measure its state. The measurement will show heads or tails with equal probability \(p_h=p_t=0.5\).
To simulate the penny flip game we take the second qubit and put it into its excited state \(1\rangle\) (which is mapped to \(H\rangle\), heads) by applying the X (or NOT) gate. Q’s first move is to apply the Hadamard gate H. Picard’s decision about the flip is simulated as a CNOT operation where the control bit is the outcome of the random number generator described above. Finally Q applies a Hadamard gate again, before we measure the outcome. The full circuit is shown in the figure below.
In pyQuil¶
We first import and initialize the necessary tools [3]
from pyquil import Program
from pyquil.api import WavefunctionSimulator
from pyquil.gates import *
wf_sim = WavefunctionSimulator()
p = Program()
and then wire it all up into the overall measurement circuit; remember that qubit 0 is the penny, and qubit 1 represents Picard’s choice.
p += X(0)
p += H(0)
p += H(1)
p += CNOT(1, 0)
p += H(0)
We use the quantum mechanics principle of deferred measurement to keep all the measurement logic separate from the gates.
Our method call to the WavefunctionSimulator
will handle measuring for us [4].
Finally, we play the game several times. (Remember to run your qvm server.)
wf_sim.run_and_measure(p, trials=10)
array([[1, 1],
[1, 1],
[1, 1],
[1, 1],
[1, 1],
[1, 0],
[1, 1],
[1, 1],
[1, 1],
[1, 0]])
In each trial, the first number is the outcome of the game, whereas the second number represents Picard’s choice to flip or not flip the penny.
Inspecting the results, we see that no matter what Picard does, Q will always win!
[1]  https://link.aps.org/doi/10.1103/PhysRevLett.82.1052 
[2]  https://arxiv.org/abs/quantph/0611234 
[3]  See more: Programs and Gates 
[4]  More about measurements and run_and_measure : Measurement 
Changelog¶
v2.8 (May 20, 2019)¶
Improvements and Changes:
 PyQuil now verifies that you are using the correct version of the QVM and quilc (gh913).
 Added support for defining permutation gates for use with the latest version of quilc (gh891).
Bugfixes:
 Preserve modifiers during
address_qubits
(gh907).
v2.7.2 (May 3, 2019)¶
Bugfixes:
 An additional backwardsincompatible change from gh870 snuck through 2.7.1, and is addressed in this patch release.
v2.7.1 (April 30, 2019)¶
Bugfixes:
 The changes to operator estimation (gh870, gh896) were not made in a backwardscompatible fashion, and therefore this patch release aims to remedy that. Going forward, there will be much more stringent requirements around backwards compatibility and deprecation.
v2.7 (April 29, 2019)¶
Improvements and Changes:
 Standard deviation > standard error in operator estimation (gh870).
 Update what pyQuil expects from quilc in terms of rewiring pragmas – they are now comments rather than distinct instructions (gh878).
 Allow users to deprioritize QPU jobs – mostly a Rigettiinternal feature (gh877).
 Remove the
qubits
field from theTomographyExperiment
dataclass (gh896).
Bugfixes:
 Ensure that shots aren’t lost when passing a
Program
throughaddress_qubits
(gh895).
v2.6 (March 29, 2019)¶
Improvements and Changes:
 Added a CODEOWNERS file for default reviewers (gh855).
 Bifurcated the
QPUCompiler
endpoint parameter into two –quilc_endpoint
andqpu_compiler_endpoint
– to reflect changes in Quantum Cloud Services (gh856).  Clarified documentation around the DELAY pragma (gh862).
 Added information about the
local_qvm
context manager to the getting started documentation (gh851).
Bugfixes:
 Added a nonNone default timeout to the
QVMCompiler
object (gh850) and theget_benchmarker
function (gh854).  Fixed the docstring for the
apply_clifford_to_pauli
function (gh836).  Allowed the
apply_clifford_to_pauli
function to now work with the Identity as input (gh849).  Updated a stale link to the Rigetti Forest Slack workspace (gh860).
 Fixed a notation typo in the documentation for noise (gh861).
Special thanks to willzeng for all the contributions this release!
v2.5 (March 6, 2019)¶
Improvements and Changes:
 PyQuil’s Gate objects now expose
.controlled(q)
and.dagger()
modifiers, which turn a gate respectively into its controlled variant, conditional on the qubitq
, or into its inverse.  The operator estimation suite’s
measure_observables
method now exposes areadout_symmetrize
argument, which helps mitigate a machine’s fidelity asymmetry between recognizing a qubit in the ground state versus the excited state.  The
MEASURE
instruction in pyQuil now has a mandatory second argument. Previously, the second argument could be omitted to induce “measurement for effect”, without storing the readout result to a classical register, but users found this to be a common source of accidental error and a generally rude surprise. To ensure the user really intends to measure only for effect, we now require that they supply an explicitNone
as the second argument.
Bugfixes:
 Some stale tests have been brought into the modern era.
v2.4 (February 14, 2019)¶
Announcements:
 The Quil Compiler (quilc) and
the Quantum Virtual Machine (QVM),
which are part of the Forest SDK, have been open sourced! In addition to
downloading the binaries, you can now build these applications locally from source,
or run them via the Docker images rigetti/quilc
and rigetti/qvm. These Docker images are now
used as the
services
in the GitLab CI build plan YAML (gh792, gh794, gh795).
Improvements and Changes:
 The
WavefunctionSimulator
now supports the use of parametric Quil programs, via thememory_map
parameter for its various methods (gh787).  Operator estimation data structures introduced in v2.2 have changed. Previously,
ExperimentSettings
had two members:in_operator
andout_operator
. Theout_operator
is unchanged, butin_operator
has been renamed toin_state
and its data type is nowTensorProductState
instead ofPauliTerm
. It was always an abuse of notation to interpret pauli operators as defining initial states. Analogous to the Pauli helper functions sI, sX, sY, and sZ,TensorProductState
objects are constructed by multiplying together terms generated by the helper functions plusX, minusX, plusY, minusY, plusZ, and minusZ. This functionality enables process tomography and process DFE (gh770).  Operator estimation now offers a “greedy” method for grouping tomographylike experiments that share a natural tensor product basis (ntpb), as an alternative to the clique cover version (gh754).
 The
quilc
endpoint for rewriting Quil parameter arithmetic has been changed fromresolve_gate_parameter_arithmetic
torewrite_arithmetic
(gh802).  The difference between ProtoQuil and QPUsupported Quil is now better defined (gh798).
Bugfixes:
 Resolved an issue with postgate noise in the pyQVM (gh801).
 A
TypeError
with a useful error message is now raised when aProgram
object is run on a QPUbackedQuantumComputer
, rather than a confusingAttributeError
(gh799).
v2.3 (January 28, 2019)¶
PyQuil 2.3 is the latest release of pyQuil, Rigetti’s toolkit for constructing and running quantum programs. A major new feature is the release of a new suite of simulators:
 We’re proud to introduce the first iteration of a Pythonbased quantum virtual machine (QVM)
called PyQVM. This QVM is completely contained within pyQuil and does not need any external
dependencies. Try using it with
get_qc("9qsquarepyqvm")
or explore thepyquil.pyqvm.PyQVM
object directly. Underthehood, there are three quantum simulator backends:ReferenceWavefunctionSimulator
uses standard matrixvector multiplication to evolve a statevector. This includes a suite of tools inpyquil.unitary_tools
for dealing with unitary matrices.NumpyWavefunctionSimulator
uses numpy’s tensordot functionality to efficiently evolve a statevector. For most simulations, performance is quite good.ReferenceDensitySimulator
uses matrixmatrix multiplication to evolve a density matrix.
 Matrix representations of Quil standard gates are included in
pyquil.gate_matrices
(gh552).  The density simulator has extremely limited support for Krausoperator based noise models. Let us know if you’re interested in contributing more robust noisemodel support.
 This functionality should be considered experimental and may undergo minor API changes.
Important changes to note:
 Quil math functions (like COS, SIN, …) used to be ambiguous with respect to case sensitivity. They are now casesensitive and should be uppercase (gh774).
 In the next release of pyQuil, communication with quilc will happen exclusively via the rpcq
protocol.
LocalQVMCompiler
andLocalBenchmarkConnection
will be removed in favor of a unifiedQVMCompiler
andBenchmarkConnection
. This change should be transparent if you useget_qc
andget_benchmarker
, respectively. In anticipation of this change we recommend that you upgrade your version of quilc to 1.3, released Jan 30, 2019 (gh730).  When using a paramaterized gate, the QPU control electronics only allowed multiplying
parameters by powers of two. If you only ever multiply a parameter by the same constant,
this isn’t too much of a problem because you can fold the multiplicative constant
into the definition of the parameter. However, if you are multiplying the same variable
(e.g.
gamma
in QAOA) by different constants (e.g. weighted maxcut edge weights) it doesn’t work. PyQuil will now transparently handle the latter case by expanding to a vector of parameters with the constants folded in, allowing you to multiply variables by whatever you want (gh707).
As always, this release contains bug fixes and improvements:
 The CZ gate fidelity metric available in the Specs object now has its associated standard
error, which is accessible from the method
Specs.fCZ_std_errs
(gh751).  Operator estimation code now correctly handles identity terms with coefficients. Previously, it would always estimate these terms as 1.0 (gh758).
 Operator estimation results include the total number of counts (shots) taken.
 Operator estimation JSON serialization uses utf8. Please let us know if this causes problems (gh769).
 The example quantum die program now can roll dice that are not powers of two (gh749).
 The teleportation and Meyer penny game examples had a syntax error (gh778, gh772).
 When running on the QPU, you could get into trouble if the QPU name passed to
get_qc
did not match the lattice you booked. This is now validated (gh771).
We extend thanks to community member estamm12 for their contribution to this release.
v2.2 (January 4, 2019)¶
PyQuil 2.2 is the latest release of pyQuil, Rigetti’s toolkit for constructing and running quantum programs. Bug fixes and improvements include:
pauli.is_zero
andpaulis.is_identity
would sometimes return erroneous answers (gh710). Parameter expressions involving addition and subtraction are now converted to Quil with spaces
around the operators, e.g.
theta + 2
instead oftheta+2
. This disambiguates subtracting two parameters, e.g.alpha  beta
is not one variable namedalphabeta
(gh743).  T1 is accounted for in T2 noise models (gh745).
 Documentation improvements (gh723, gh719, gh720, gh728, gh732, gh742).
 Support for PNG generation of circuit diagrams via LaTeX (gh745).
 We’ve started transitioning to using Gitlab as our continuous integration provider for pyQuil (gh741, gh752).
This release includes a new module for facilitating the estimation of quantum observables/operators (gh682). Firstclass support for estimating observables should make it easier to express nearterm algorithms. This release includes:
 data structures for expressing tomographylike experiments and their results
 grouping of experiment settings that can be simultaneously estimated
 functionality to executing a tomographylike experiment on a quantum computer
Please look forward to more features and polish in future releases. Don’t hesitate to submit feedback or suggestions as GitHub issues.
We extend thanks to community member petterwittek for their contribution to this release.
Bugfix release 2.2.1 was released January 11 to maintain compatibility with the latest version of the quilc compiler (gh759).
v2.1 (November 30, 2018)¶
PyQuil 2.1 is an incremental release of pyQuil, Rigetti’s toolkit for constructing and running quantum programs. Changes include:
 Major documentation improvements.
QuantumComputer.run()
accepts an optionalmemory_map
parameter to facilitate running parametric executables (gh657).QuantumComputer.reset()
will reset the state of a QAM to recover from an error condition (gh703). Bug fixes (gh674, gh696).
 Quil parser improvements (gh689, gh685).
 Optional interleaver argument when generating RB sequences (gh673).
 Our GitHub organization name has changed from
rigetticomputing
torigetti
(gh713).
v2.0 (November 1, 2018)¶
PyQuil 2.0 is a major release of pyQuil, Rigetti’s toolkit for constructing and running quantum programs. This release contains many major changes including:
 The introduction of Quantum Cloud Services. Access Rigetti’s QPUs from colocated classical compute resources for minimal latency. The web API for running QVM and QPU jobs has been deprecated and cannot be accessed with pyQuil 2.0
 Advances in classical control systems and compilation allowing the precompilation of parametric binary executables for rapid hybrid algorithm iteration.
 Changes to Quil—our quantum instruction language—to provide easier ways of interacting with classical memory.
The new QCS access model and features will allow you to execute hybrid quantum algorithms several orders of magnitude (!) faster than the previous web endpoint. However, to fully exploit these speed increases you must update your programs to use the latest pyQuil features and APIs. Please read New in Forest 2  Other for a comprehensive migration guide.
An incomplete list of significant changes:
 Python 2 is no longer supported. Please use Python 3.6+
 Parametric gates are now normal functions. You can no longer write
RX(pi/2)(0)
to get a QuilRX(pi/2) 0
instruction. Just useRX(pi/2, 0)
.  Gates support keyword arguments, so you can write
RX(angle=pi/2, qubit=0)
.  All
async
methods have been removed fromQVMConnection
andQVMConnection
is deprecated.QPUConnection
has been removed in accordance with the QCS access model. Usepyquil.get_qc()
as the primary means of interacting with the QVM or QPU. WavefunctionSimulator
allows unfettered access to wavefunction properties and routines. These methods and properties previously lived onQVMConnection
and have been deprecated there. Classical memory in Quil must be declared with a name and type. Please read New in Forest 2  Other for more.
 Compilation has changed. There are now different
Compiler
objects that target either the QPU or QVM. You must explicitly compile your programs to run on a QPU or a realistic QVM.
Version 2.0.1 was released on November 9, 2018 and includes documentation changes only. This release is only available as a git tag. We have not pushed a new package to PyPI.
v1.9 (June 6, 2018)¶
We’re happy to announce the release of pyQuil 1.9. PyQuil is Rigetti’s toolkit for constructing and running quantum programs. This release is the latest in our series of regular releases, and it’s filled with convenience features, enhancements, bug fixes, and documentation improvements.
Special thanks to community members sethuiyer, vtomole, rht, akarazeev, ejdanderson, markf94, playadust, and kadora626 for contributing to this release!
Qubit placeholders¶
One of the focuses of this release is a reworked concept of “Qubit Placeholders”. These are
logical qubits that can be used to construct programs. Now, a program containing qubit placeholders
must be “addressed” prior to running on a QPU or QVM. The addressing stage involves mapping
each qubit placeholder to a physical qubit (represented as an integer). For example, if you have
a 3 qubit circuit that you want to run on different sections of the Agave chip, you now can
prepare one Program and address it to many different subgraphs of the chip topology.
Check out the QubitPlaceholder
example notebook for more.
To support this idea, we’ve refactored parts of Pyquil to remove the assumption that qubits
can be “sorted”. While true for integer qubit labels, this probably isn’t true in general.
A notable change can be found in the construction of a PauliSum
: now terms will stay in the
order they were constructed.
PauliTerm
now remembers the order of its operations.sX(1)*sZ(2)
will compile to different Quil code thansZ(2)*sX(1)
, although the terms will still be equal according to the__eq__
method. DuringPauliSum
combination of like terms, a warning will be emitted if two terms are combined that have different orders of operation.PauliTerm.id()
takes an optional argumentsort_ops
which defaults to True for backwards compatibility. However, this function should not be used for comparing termtype like it has been used previously. UsePauliTerm.operations_as_set()
instead. In the future,sort_ops
will default to False and will eventually be removed.Program.alloc()
has been deprecated. Please instantiateQubitPlaceholder()
directly or request a “register” (list) ofn
placeholders by using the class constructorQubitPlaceholder.register(n)()
. Programs must contain either (1) all instantiated qubits with integer indexes or (2) all
placeholder qubits of type
QubitPlaceholder
. We have found that most users use (1) but (2) will become useful with larger and more diverse devices.  Programs that contain qubit placeholders must be explicitly addressed prior to execution.
Previously, qubits would be assigned “under the hood” to integers 0…N. Now, you must use
address_qubits()
which returns a new program with all qubits indexed depending on thequbit_mapping
argument. The original program is unaffected and can be “readdressed” multiple times. PauliTerm
can now acceptQubitPlaceholder
in addition to integers.QubitPlaceholder
is no longer a subclass ofQubit
.LabelPlaceholder
is no longer a subclass ofLabel
.QuilAtom
subclasses’ hash functions have changed.
Randomized benchmarking sequence generation¶
Pyquil now includes support for performing a simple benchmarking routine  randomized
benchmarking. There is a new method in the CompilerConnection
that will return
sequences of pyquil programs, corresponding to elements of the Clifford group. These programs
are uniformly randomly sampled, and have the property that they compose to the identity. When
concatenated and run as one program, these programs can be used in a procedure called randomized
benchmarking to gain insight about the fidelity of operations on a QPU.
In addition, the CompilerConnection
has another new method,
apply_clifford_to_pauli()
which conjugates PauliTerms
by
Program
that are composed of Clifford gates. That is to say, given a circuit C,
that contains only gates corresponding to elements of the Clifford group, and a tensor product of
elements P, from the Pauli group, this method will compute $PCP^{dagger}$. Such a procedure can
be used in various ways. An example is predicting the effect a Clifford circuit will have on an
input state modeled as a density matrix, which can be written as a sum of Pauli matrices.
Ease of Use¶
This release includes some qualityoflife improvements such as the ability to initialize
programs with generator expressions, sensible defaults for Program.measure_all()
,
and sensible defaults for classical_addresses
in run()
methods.
Program
can be initiated with a generator expression.Program.measure_all()
(with no arguments) will measure all qubits in a program.classical_addresses
is now optional in QVM and QPUrun()
methods. By default, any classical addresses targeted byMEASURE
will be returned.QVMConnection.pauli_expectation()
acceptsPauliSum
as arguments. This offers a more sensible API compared toQVMConnection.expectation()
. pyQuil will now retry jobs every 10 seconds if the QPU is retuning.
CompilerConnection.compile()
now takes an optional argumentisa
that allows percompilation specification of the target ISA. An empty program will trigger an exception if you try to run it.
Supported versions of Python¶
We strongly support using Python 3 with Pyquil. Although this release works with Python 2, we are dropping official support for this legacy language and moving to community support for Python 2. The next major release of Pyquil will introduce Python 3.5+ only features and will no longer work without modification for Python 2.
Bug fixes¶
shift_quantum_gates
has been removed. Users who relied on this functionality should useQubitPlaceholder
andaddress_qubits()
to achieve the same result. Users should also doublecheck data resulting from use of this function as there were several edge cases which would cause the shift to be applied incorrectly resulting in badlyaddressed qubits. Slightly perturbed angles when performing RX gates under a Kraus noise model could result in incorrect behavior.
 The quantum die example returned incorrect values when
n = 2^m
.
Introduction to Quantum Computing¶
With every breakthrough in science there is the potential for new technology. For over twenty years, researchers have done inspiring work in quantum mechanics, transforming it from a theory for understanding nature into a fundamentally new way to engineer computing technology. This field, quantum computing, is beautifully interdisciplinary, and impactful in two major ways:
 It reorients the relationship between physics and computer science. Physics does not just place restrictions on what computers we can design, it also grants new power and inspiration.
 It can simulate nature at its most fundamental level, allowing us to solve deep problems in quantum chemistry, materials discovery, and more.
Quantum computing has come a long way, and in the next few years there will be significant breakthroughs in the field. To get here, however, we have needed to change our intuition for computation in many ways. As with other paradigms — such as objectoriented programming, functional programming, distributed programming, or any of the other marvelous ways of thinking that have been expressed in code over the years — even the basic tenants of quantum computing opens up vast new potential for computation.
However, unlike other paradigms, quantum computing goes further. It requires an extension of classical probability theory. This extension, and the core of quantum computing, can be formulated in terms of linear algebra. Therefore, we begin our investigation into quantum computing with linear algebra and probability.
From Bit to Qubit¶
Probabilistic Bits as Vector Spaces¶
From an operational perspective, a bit is described by the results of measurements performed on it. Let the possible results of measuring a bit (0 or 1) be represented by orthonormal basis vectors \(\vec{0}\) and \(\vec{1}\). We will call these vectors outcomes. These outcomes span a twodimensional vector space that represents a probabilistic bit. A probabilistic bit can be represented as a vector
where \(a\) represents the probability of the bit being 0 and \(b\) represents the probability of the bit being 1. This clearly also requires that \(a+b=1\). In this picture the system (the probabilistic bit) is a twodimensional real vector space and a state of a system is a particular vector in that vector space.
import numpy as np
import matplotlib.pyplot as plt
outcome_0 = np.array([1.0, 0.0])
outcome_1 = np.array([0.0, 1.0])
a = 0.75
b = 0.25
prob_bit = a * outcome_0 + b * outcome_1
X, Y = prob_bit
plt.figure()
ax = plt.gca()
ax.quiver(X, Y, angles='xy', scale_units='xy', scale=1)
ax.set_xlim([0, 1])
ax.set_ylim([0, 1])
plt.draw()
plt.show()
Given some state vector, like the one plotted above, we can find the probabilities associated with each outcome by projecting the vector onto the basis outcomes. This gives us the following rule:
where Pr(0) and Pr(1) are the probabilities of the 0 and 1 outcomes respectively.
Dirac Notation¶
Physicists have introduced a convenient notation for the vector transposes and dot products we used in the previous example. This notation, called Dirac notation in honor of the great theoretical physicist Paul Dirac, allows us to define
Thus, we can rewrite our “measurement rule” in this notation as
We will use this notation throughout the rest of this introduction.
Multiple Probabilistic Bits¶
This vector space interpretation of a single probabilistic bit can be straightforwardly extended to multiple bits. Let us take two coins as an example (labelled 0 and 1 instead of H and T since we are programmers). Their states can be represented as
where \(1_u\) represents the outcome 1 on coin \(u\). The combined system of the two coins has four possible outcomes \(\{ 0_u0_v,\;0_u1_v,\;1_u0_v,\;1_u1_v \}\) that are the basis states of a larger fourdimensional vector space. The rule for constructing a combined state is to take the tensor product of individual states, e.g.
Then, the combined space is simply the space spanned by the tensor products of all pairs of basis vectors of the two smaller spaces.
Similarly, the combined state for \(n\) such probabilistic bits is a vector of size \(2^n\) and is given by \(\bigotimes_{i=0}^{n1}\,v_i\rangle\). We will talk more about these larger spaces in the quantum case, but it is important to note that not all composite states can be written as tensor products of substates (e.g. consider the state \(\frac{1}{2}\,0_u0_v\rangle + \frac{1}{2}\,1_u1_v\rangle\)). The most general composite state of \(n\) probabilistic bits can be written as \(\sum_{j=0}^{2^n  1} a_{j} (\bigotimes_{i=0}^{n1}\,b_{ij}\rangle\)) where each \(b_{ij} \in \{0, 1\}\) and \(a_j \in \mathbb{R}\), i.e. as a linear combination (with real coefficients) of tensor products of basis states. Note that this still gives us \(2^n\) possible states.
Qubits¶
Quantum mechanics rewrites these rules to some extent. A quantum bit, called a qubit, is the quantum analog of a bit in that it has two outcomes when it is measured. Similar to the previous section, a qubit can also be represented in a vector space, but with complex coefficients instead of real ones. A qubit system is a twodimensional complex vector space, and the state of a qubit is a complex vector in that space. Again we will define a basis of outcomes \(\{\,0\rangle, \,1\rangle\}\) and let a generic qubit state be written as
Since these coefficients can be imaginary, they cannot be simply interpreted as probabilities of their associated outcomes. Instead we rewrite the rule for outcomes in the following manner:
and as long as \(\alpha^2 + \beta^2 = 1\) we are able to recover acceptable probabilities for outcomes based on our new complex vector.
This switch to complex vectors means that rather than representing a state vector in a plane, we instead represent the vector on a sphere (called the Bloch sphere in quantum mechanics literature). From this perspective the quantum state corresponding to an outcome of 0 is represented by:
Notice that the two axes in the horizontal plane have been labeled \(x\) and \(y\), implying that \(z\) is the vertical axis (not labeled). Physicists use the convention that a qubit’s \(\{\,0\rangle, \,1\rangle\}\) states are the positive and negative unit vectors along the z axis, respectively. These axes will be useful later in this document.
Multiple qubits are represented in precisely the same way, by taking linear combinations (with complex coefficients, now) of tensor products of basis states. Thus \(n\) qubits have \(2^n\) possible states.
An Important Distinction¶
The probabilistic states described above represent ignorance of an underlying state, like 0 or 1 for probabilistic bits. This is not true for quantum states. The nature of quantum states is a deep topic with no full scientific consensus. However, nogo theorems like Bell’s Theorem have ruled out the option of local hidden variable theories for quantum mechanics. Effectively, these say that quantum states can’t be interpreted as purely representing ignorance of an underlying local objective state. In practice this means that a pure quantum state simply is the complex vector described in the last section, and we consider it just as “real” as a headsup coin. This distinction between quantum and classical states is foundational for understanding quantum computing.
Some Code¶
Let us take a look at some code in pyQuil to see how these quantum states play out. We will dive deeper into quantum operations and pyQuil in the following sections. Note that in order to run these examples you will need to install pyQuil and download the QVM and Compiler. Each of the code snippets below will be immediately followed by its output.
# Imports for pyQuil (ignore for now)
import numpy as np
from pyquil.quil import Program
from pyquil.api import WavefunctionSimulator
# create a WavefunctionSimulator object
wavefunction_simulator = WavefunctionSimulator()
# pyQuil is based around operations (or gates) so we will start with the most
# basic one: the identity operation, called I. I takes one argument, the index
# of the qubit that it should be applied to.
from pyquil.gates import I
# Make a quantum program that allocates one qubit (qubit #0) and does nothing to it
p = Program(I(0))
# Quantum states are called wavefunctions for historical reasons.
# We can run this basic program on our connection to the simulator.
# This call will return the state of our qubits after we run program p.
# This api call returns a tuple, but we'll ignore the second value for now.
wavefunction = wavefunction_simulator.wavefunction(p)
# wavefunction is a Wavefunction object that stores a quantum state as a list of amplitudes
alpha, beta = wavefunction
print("Our qubit is in the state alpha={} and beta={}".format(alpha, beta))
print("The probability of measuring the qubit in outcome 0 is {}".format(abs(alpha)**2))
print("The probability of measuring the qubit in outcome 1 is {}".format(abs(beta)**2))
Our qubit is in the state alpha=(1+0j) and beta=0j
The probability of measuring the qubit in outcome 0 is 1.0
The probability of measuring the qubit in outcome 1 is 0.0
Applying an operation to our qubit affects the probability of each outcome.
# We can import the qubit "flip" operation, called X, and see what it does.
# We will learn more about this operation in the next section.
from pyquil.gates import X
p = Program(X(0))
wavefunc = wavefunction_simulator.wavefunction(p)
alpha, beta = wavefunc
print("Our qubit is in the state alpha={} and beta={}".format(alpha, beta))
print("The probability of measuring the qubit in outcome 0 is {}".format(abs(alpha)**2))
print("The probability of measuring the qubit in outcome 1 is {}".format(abs(beta)**2))
Our qubit is in the state alpha=0j and beta=(1+0j)
The probability of measuring the qubit in outcome 0 is 0.0
The probability of measuring the qubit in outcome 1 is 1.0
In this case we have flipped the probability of outcome 0 into the probability of outcome 1 for our qubit. We can also investigate what happens to the state of multiple qubits. We’d expect the state of multiple qubits to grow exponentially in size, as their vectors are tensored together.
# Multiple qubits also produce the expected scaling of the state.
p = Program(I(0), I(1))
wavefunction = wavefunction_simulator.wavefunction(p)
print("The quantum state is of dimension:", len(wavefunction.amplitudes))
p = Program(I(0), I(1), I(2), I(3))
wavefunction = wavefunction_simulator.wavefunction(p)
print("The quantum state is of dimension:", len(wavefunction.amplitudes))
p = Program()
for x in range(10):
p += I(x)
wavefunction = wavefunction_simulator.wavefunction(p)
print("The quantum state is of dimension:", len(wavefunction.amplitudes))
The quantum state is of dimension: 4
The quantum state is of dimension: 16
The quantum state is of dimension: 1024
Let’s look at the actual value for the state of two qubits combined. The resulting dictionary of this method contains outcomes as keys and the probabilities of those outcomes as values.
# wavefunction(Program) returns a coefficient array that corresponds to outcomes in the following order
wavefunction = wavefunction_simulator.wavefunction(Program(I(0), I(1)))
print(wavefunction.get_outcome_probs())
{'00': 1.0, '01': 0.0, '10': 0.0, '11': 0.0}
Qubit Operations¶
In the previous section we introduced our first two operations: the I
(or Identity) operation and the X
(or NOT) operation. In this section we will get into some
more details on what these operations are.
Quantum states are complex vectors on the Bloch sphere, and quantum operations are matrices with two properties:
 They are reversible.
 When applied to a state vector on the Bloch sphere, the resulting vector is also on the Bloch sphere.
Matrices that satisfy these two properties are called unitary matrices. Such matrices have the characteristic property that their complex conjugate transpose is equal to their inverse, a property directly linked to the requirement that the probabilities of measuring qubits in any of the allowed states must sum to 1. Applying an operation to a quantum state is the same as multiplying a vector by one of these matrices. Such an operation is called a gate.
Since individual qubits are twodimensional vectors, operations on individual qubits are 2x2 matrices. The identity matrix leaves the state vector unchanged:
so the program that applies this operation to the zero state is just
p = Program(I(0))
print(wavefunction_simulator.wavefunction(p))
(1+0j)0>
Pauli Operators¶
Let’s revisit the X
gate introduced above. It is one of three important singlequbit gates,
called the Pauli operators:
from pyquil.gates import X, Y, Z
p = Program(X(0))
wavefunction = wavefunction_simulator.wavefunction(p)
print("X0> = ", wavefunction)
print("The outcome probabilities are", wavefunction.get_outcome_probs())
print("This looks like a bit flip.\n")
p = Program(Y(0))
wavefunction = wavefunction_simulator.wavefunction(p)
print("Y0> = ", wavefunction)
print("The outcome probabilities are", wavefunction.get_outcome_probs())
print("This also looks like a bit flip.\n")
p = Program(Z(0))
wavefunction = wavefunction_simulator.wavefunction(p)
print("Z0> = ", wavefunction)
print("The outcome probabilities are", wavefunction.get_outcome_probs())
print("This state looks unchanged.")
X0> = (1+0j)1>
The outcome probabilities are {'0': 0.0, '1': 1.0}
This looks like a bit flip.
Y0> = 1j1>
The outcome probabilities are {'0': 0.0, '1': 1.0}
This also looks like a bit flip.
Z0> = (1+0j)0>
The outcome probabilities are {'0': 1.0, '1': 0.0}
This state looks unchanged.
The Pauli matrices have a visual interpretation: they perform 180degree rotations of
qubit state vectors on the Bloch sphere. They operate about their respective axes
as shown in the Bloch sphere depicted above. For example, the X
gate performs a 180degree
rotation about the \(x\) axis. This explains the results of our code above: for a state vector
initially in the +\(z\) direction, both X
and Y
gates will rotate it to \(z\),
and the Z
gate will leave it unchanged.
However, notice that while the X
and Y
gates produce the same outcome probabilities, they
actually produce different states. These states are not distinguished if they are measured
immediately, but they produce different results in larger programs.
Quantum programs are built by applying successive gate operations:
# Composing qubit operations is the same as multiplying matrices sequentially
p = Program(X(0), Y(0), Z(0))
wavefunction = wavefunction_simulator.wavefunction(p)
print("ZYX0> = ", wavefunction)
print("With outcome probabilities\n", wavefunction.get_outcome_probs())
ZYX0> = [ 0.1.j 0.+0.j]
With outcome probabilities
{'0': 1.0, '1': 0.0}
MultiQubit Operations¶
Operations can also be applied to composite states of multiple qubits.
One common example is the controlledNOT or CNOT
gate that works on two
qubits. Its matrix form is:
Let’s take a look at how we could use a CNOT
gate in pyQuil.
from pyquil.gates import CNOT
p = Program(CNOT(0, 1))
wavefunction = wavefunction_simulator.wavefunction(p)
print("CNOT00> = ", wavefunction)
print("With outcome probabilities\n", wavefunction.get_outcome_probs(), "\n")
p = Program(X(0), CNOT(0, 1))
wavefunction = wavefunction_simulator.wavefunction(p)
print("CNOT01> = ", wavefunction)
print("With outcome probabilities\n", wavefunction.get_outcome_probs(), "\n")
p = Program(X(1), CNOT(0, 1))
wavefunction = wavefunction_simulator.wavefunction(p)
print("CNOT10> = ", wavefunction)
print("With outcome probabilities\n", wavefunction.get_outcome_probs(), "\n")
p = Program(X(0), X(1), CNOT(0, 1))
wavefunction = wavefunction_simulator.wavefunction(p)
print("CNOT11> = ", wavefunction)
print("With outcome probabilities\n", wavefunction.get_outcome_probs(), "\n")
CNOT00> = (1+0j)00>
With outcome probabilities
{'00': 1.0, '01': 0.0, '10': 0.0, '11': 0.0}
CNOT01> = (1+0j)11>
With outcome probabilities
{'00': 0.0, '01': 0.0, '10': 0.0, '11': 1.0}
CNOT10> = (1+0j)10>
With outcome probabilities
{'00': 0.0, '01': 0.0, '10': 1.0, '11': 0.0}
CNOT11> = (1+0j)01>
With outcome probabilities
{'00': 0.0, '01': 1.0, '10': 0.0, '11': 0.0}
The CNOT
gate does what its name implies: the state of the second qubit is flipped
(negated) if and only if the state of the first qubit is 1 (true).
Another twoqubit gate example is the SWAP
gate, which swaps the \( 01\rangle \)
and \(10\rangle \) states:
from pyquil.gates import SWAP
p = Program(X(0), SWAP(0,1))
wavefunction = wavefunction_simulator.wavefunction(p)
print("SWAP01> = ", wavefunction)
print("With outcome probabilities\n", wavefunction.get_outcome_probs())
SWAP01> = (1+0j)10>
With outcome probabilities
{'00': 0.0, '01': 0.0, '10': 1.0, '11': 0.0}
In summary, quantum computing operations are composed of a series of complex matrices applied to complex vectors. These matrices must be unitary (meaning that their complex conjugate transpose is equal to their inverse) because the overall probability of all outcomes must always sum to one.
The Quantum Abstract Machine¶
We now have enough background to introduce the programming model that underlies Quil. This is a hybrid quantumclassical model in which \(N\) qubits interact with \(M\) classical bits:
These qubits and classical bits come with a defined gate set, e.g. which gate operations can be applied to which qubits. Different kinds of quantum computing hardware place different limitations on what gates can be applied, and the fixed gate set represents these limitations.
Full details on the Quantum Abstract Machine and Quil can be found in the Quil whitepaper.
The next section on measurements will describe the interaction between the classical and quantum parts of a Quantum Abstract Machine (QAM).
Qubit Measurements¶
Measurements have two effects:
 They project the state vector onto one of the basic outcomes
 (optional) They store the outcome of the measurement in a classical bit.
Here’s a simple example:
# Create a program that stores the outcome of measuring qubit #0 into classical register [0]
p = Program()
classical_register = p.declare('ro', 'BIT', 1)
p += Program(I(0)).measure(0, classical_register[0])
Up until this point we have used the quantum simulator to cheat a little bit — we have
actually looked at the wavefunction that comes back. However, on real
quantum hardware, we are unable to directly look at the wavefunction.
Instead we only have access to the classical bits that are affected by
measurements. This functionality is emulated by QuantumComputer.run()
. Note that the run
command is to be applied on the compiled version of the program.
from pyquil import get_qc
qc = get_qc('9qsquareqvm')
print (qc.run(qc.compile(p)))
[[0]]
We see that the classical register reports a value of zero. However, if we had flipped the qubit before measurement then we obtain:
p = Program()
classical_register = p.declare('ro', 'BIT', 1)
p += Program(X(0)) # Flip the qubit
p.measure(0, classical_register[0]) # Measure the qubit
print (qc.run(qc.compile(p)))
[[1]]
These measurements are deterministic, e.g. if we make them multiple times then we always get the same outcome:
p = Program()
classical_register = p.declare('ro', 'BIT', 1)
p += Program(X(0)) # Flip the qubit
p.measure(0, classical_register[0]) # Measure the qubit
trials = 10
p.wrap_in_numshots_loop(shots=trials)
print (qc.run(qc.compile(p)))
[[1], [1], [1], [1], [1], [1], [1], [1], [1], [1]]
Classical/Quantum Interaction¶
However this is not the case in general — measurements can affect the quantum state as well. In fact, measurements act like projections onto the outcome basis states. To show how this works, we first introduce a new singlequbit gate, the Hadamard gate. The matrix form of the Hadamard gate is:
The following pyQuil code shows how we can use the Hadamard gate:
from pyquil.gates import H
# The Hadamard produces what is called a superposition state
coin_program = Program(H(0))
wavefunction = wavefunction_simulator.wavefunction(coin_program)
print("H0> = ", wavefunction)
print("With outcome probabilities\n", wavefunction.get_outcome_probs())
H0> = (0.7071067812+0j)0> + (0.7071067812+0j)1>
With outcome probabilities
{'0': 0.49999999999999989, '1': 0.49999999999999989}
A qubit in this state will be measured half of the time in the \( 0\rangle \) state, and half of the time in the \( 1\rangle \) state. In a sense, this qubit truly is a random variable representing a coin. In fact, there are many wavefunctions that will give this same operational outcome. There is a continuous family of states of the form
that represent the outcomes of an unbiased coin. Being able to work with all of these different new states is part of what gives quantum computing extra power over regular bits.
p = Program()
ro = p.declare('ro', 'BIT', 1)
p += Program(H(0)).measure(0, ro[0])
# Measure qubit #0 a number of times
p.wrap_in_numshots_loop(shots=10)
# We see probabilistic results of about half 1's and half 0's
print (qc.run(qc.compile(p)))
[[0], [1], [1], [0], [1], [0], [0], [1], [0], [0]]
pyQuil allows us to look at the wavefunction after a measurement as well:
coin_program = Program(H(0))
print ("Before measurement: H0> = ", wavefunction_simulator.wavefunction(coin_program), "\n")
ro = coin_program.declare('ro', 'BIT', 1)
coin_program.measure(0, ro[0])
for _ in range(5):
print ("After measurement: ", wavefunction_simulator.wavefunction(coin_program))
Before measurement: H0> = (0.7071067812+0j)0> + (0.7071067812+0j)1>
After measurement: (1+0j)1>
After measurement: (1+0j)1>
After measurement: (1+0j)1>
After measurement: (1+0j)1>
After measurement: (1+0j)1>
We can clearly see that measurement has an effect on the quantum state independent of what is stored classically. We begin in a state that has a 5050 probability of being \( 0\rangle \) or \( 1\rangle \). After measurement, the state changes into being entirely in \( 0\rangle \) or entirely in \( 1\rangle \) according to which outcome was obtained. This is the phenomenon referred to as the collapse of the wavefunction. Mathematically, the wavefunction is being projected onto the vector of the obtained outcome and subsequently rescaled to unit norm.
# This happens with bigger systems too, as can be seen with this program,
# which prepares something called a Bell state (a special kind of "entangled state")
bell_program = Program(H(0), CNOT(0, 1))
wavefunction = wavefunction_simulator.wavefunction(bell_program)
print("Before measurement: Bell state = ", wavefunction, "\n")
classical_regs = bell_program.declare('ro', 'BIT', 2)
bell_program.measure(0, classical_regs[0]).measure(1, classical_regs[1])
for _ in range(5):
wavefunction = wavefunction_simulator.wavefunction(bell_program)
print("After measurement: ", wavefunction.get_outcome_probs())
Before measurement: Bell state = (0.7071067812+0j)00> + (0.7071067812+0j)11>
After measurement: {'00': 0.0, '01': 0.0, '10': 0.0, '11': 1.0}
After measurement: {'00': 0.0, '01': 0.0, '10': 0.0, '11': 1.0}
After measurement: {'00': 0.0, '01': 0.0, '10': 0.0, '11': 1.0}
After measurement: {'00': 0.0, '01': 0.0, '10': 0.0, '11': 1.0}
After measurement: {'00': 0.0, '01': 0.0, '10': 0.0, '11': 1.0}
The above program prepares entanglement because, even though there are random outcomes, after every measurement both qubits are in the same state. They are either both \( 0\rangle \) or both \( 1\rangle \). This special kind of correlation is part of what makes quantum mechanics so unique and powerful.
Classical Control¶
There are also ways of introducing classical control of quantum programs. For example, we can use the state of classical bits to determine what quantum operations to run.
true_branch = Program(X(7)) # if branch
false_branch = Program(I(7)) # else branch
# Branch on ro[1]
p = Program()
ro = p.declare('ro', 'BIT', 8)
p += Program(X(0)).measure(0, ro[1]).if_then(ro[1], true_branch, false_branch)
# Measure qubit #7 into ro[7]
p.measure(7, ro[7])
# Run and check register [7]
print (qc.run(qc.compile(p)))
[[1 1]]
The second [1] here means that qubit 7 was indeed flipped.
Example: The Probabilistic Halting Problem¶
A fun example is to create a program that has an exponentially increasing chance of halting, but that may run forever!
p = Program()
ro = p.declare('ro', 'BIT', 1)
inside_loop = Program(H(0)).measure(0, ro[0])
p.inst(X(0)).while_do(ro[0], inside_loop)
qc = get_qc('9qsquareqvm')
print (qc.run(qc.compile(p)))
[[0]]
Next Steps¶
We hope that you have enjoyed your whirlwind tour of quantum computing. You are now ready to check out the Installation and Getting Started guide!
If you would like to learn more, Nielsen and Chuang’s Quantum Computation and Quantum Information is a particularly excellent resource for newcomers to the field.
If you’re interested in learning about the software behind quantum computing, take a look at our blog posts on The Quantum Software Challenge.
Program¶

class
pyquil.quil.
Program
(*instructions)[source]¶ A list of pyQuil instructions that comprise a quantum program.
>>> from pyquil import Program >>> from pyquil.gates import * >>> p = Program() >>> p += H(0) >>> p += CNOT(0, 1)
Attributes
instructions
Fill in any placeholders and return a list of quil AbstractInstructions. defined_gates
A list of defined gates on the program. out
()Serializes the Quil program to a string suitable for submitting to the QVM or QPU. get_qubits
([indices])Returns all of the qubit indices used in this program, including gate applications and allocated qubits. is_protoquil
()Protoquil programs may only contain gates, Pragmas, and RESET. Program Construction
__iadd__
(other)Concatenate two programs together using +=, returning a new one. __add__
(other)Concatenate two programs together, returning a new one. inst
(*instructions)Mutates the Program object by appending new instructions. gate
(name, params, qubits)Add a gate to the program. defgate
(name, matrix[, parameters])Define a new static gate. define_noisy_gate
(name, qubit_indices, kraus_ops)Overload a static ideal gate with a noisy one defined in terms of a Kraus map. define_noisy_readout
(qubit, p00, p11)For this program define a classical bit flip readout error channel parametrized by p00
andp11
.no_noise
()Prevent a noisy gate definition from being applied to the immediately following Gate instruction. measure
(qubit_index, classical_reg)Measures a qubit at qubit_index and puts the result in classical_reg reset
([qubit_index])Reset all qubits or just a specific qubit at qubit_index. measure_all
(*qubit_reg_pairs)Measures many qubits into their specified classical bits, in the order they were entered. alloc
()Get a new qubit. declare
(name[, memory_type, memory_size, …])DECLARE a quil variable wrap_in_numshots_loop
(shots)Wraps a Quil program in a loop that reruns the same program many times. Control Flow
while_do
(classical_reg, q_program)While a classical register at index classical_reg is 1, loop q_program if_then
(classical_reg, if_program[, …])If the classical register at index classical reg is 1, run if_program, else run else_program. Utility Methods
copy
()Perform a shallow copy of this program. pop
()Pops off the last instruction. dagger
([inv_dict, suffix])Creates the conjugate transpose of the Quil program. __getitem__
(index)Allows indexing into the program to get an action.
Utility Functions¶

pyquil.quil.
get_default_qubit_mapping
(program)[source]¶ Takes a program which contains qubit placeholders and provides a mapping to the integers 0 through N1.
The output of this function is suitable for input to
address_qubits()
.Parameters: program – A program containing qubit placeholders Returns: A dictionary mapping qubit placeholder to an addressed qubit from 0 through N1.

pyquil.quil.
address_qubits
(program, qubit_mapping=None)[source]¶ Takes a program which contains placeholders and assigns them all defined values.
Either all qubits must be defined or all undefined. If qubits are undefined, you may provide a qubit mapping to specify how placeholders get mapped to actual qubits. If a mapping is not provided, integers 0 through N are used.
This function will also instantiate any label placeholders.
Parameters:  program – The program.
 qubit_mapping – A dictionarylike object that maps from
QubitPlaceholder
toQubit
orint
(but not both).
Returns: A new Program with all qubit and label placeholders assigned to real qubits and labels.

pyquil.quil.
instantiate_labels
(instructions)[source]¶ Takes an iterable of instructions which may contain label placeholders and assigns them all defined values.
Returns: list of instructions with all label placeholders assigned to real labels.

pyquil.quil.
implicitly_declare_ro
(instructions)[source]¶ Implicitly declare a register named
ro
for backwards compatibility with Quil 1.There used to be one unnamed hunk of classical memory. Now there are variables with declarations. Instead of:
MEASURE 0 [0]
You must now measure into a named register, idiomatically:
MEASURE 0 ro[0]
The
MEASURE
instruction will emit this (with a deprecation warning) if you’re still using bare integers for classical addresses. However, you must also declare memory in the new scheme:DECLARE ro BIT[8] MEASURE 0 ro[0]
This method will determine if you are in “backwards compatibility mode” and will declare a readout
ro
register for you. If you program contains any DECLARE commands or if it does not have any MEASURE x ro[x], this will not do anything.This behavior is included for backwards compatibility and will be removed in future releases of PyQuil. Please DECLARE all memory including
ro
.

pyquil.quil.
merge_with_pauli_noise
(prog_list, probabilities, qubits)[source]¶ Insert pauli noise channels between each item in the list of programs. This noise channel is implemented as a single noisy identity gate acting on the provided qubits. This method does not rely on merge_programs and so avoids the inclusion of redundant Kraus Pragmas that would occur if merge_programs was called directly on programs with distinct noisy gate definitions.
Parameters:  prog_list (
Iterable
[+T_co]) – an iterable such as a program or a list of programs. If a program is provided, a single noise gate will be applied after each gate in the program. If a list of programs is provided, the noise gate will be applied after each program.  probabilities (
List
[~T]) – The 4^num_qubits list of probabilities specifying the desired pauli channel. There should be either 4 or 16 probabilities specified in the order I, X, Y, Z or II, IX, IY, IZ, XI, XX, XY, etc respectively.  qubits (
List
[~T]) – a list of the qubits that the noisy gate should act on.
Returns: A single program with noisy gates inserted between each element of the program list.
Return type:  prog_list (

pyquil.quil.
merge_programs
(prog_list)[source]¶ Merges a list of pyQuil programs into a single one by appending them in sequence. If multiple programs in the list contain the same gate and/or noisy gate definition with identical name, this definition will only be applied once. If different definitions with the same name appear multiple times in the program list, each will be applied once in the order of last occurrence.
Parameters: prog_list (list) – A list of pyquil programs Returns: a single pyQuil program Return type: Program

pyquil.quil.
get_classical_addresses_from_program
(program)[source]¶ Returns a sorted list of classical addresses found in the MEASURE instructions in the program.
Parameters: program (Program) – The program from which to get the classical addresses. Return type: Dict
[str
,List
[int
]]Returns: A mapping from memory region names to lists of offsets appearing in the program.

pyquil.quil.
percolate_declares
(program)[source]¶ Move all the DECLARE statements to the top of the program. Return a fresh obejct.
Parameters: program ( Program
) – Perhaps jumbled program.Return type: Program
Returns: Program with DECLAREs all at the top and otherwise the same sorted contents.

pyquil.quil.
validate_protoquil
(program)[source]¶ Ensure that a program is valid ProtoQuil, otherwise raise a ValueError. Protoquil is a subset of Quil which excludes control flow and classical instructions.
Parameters: program ( Program
) – The Quil program to validate.Return type: None
Gates and Instructions¶
A Program
is effectively a list of gates and instructions which
can be created using the function documented in this section:
>>> p = Program()
>>> p += H(qubit=0)
>>> p += RY(angle=pi/3, qubit=1)
>>> p += CNOT(0, 1)
...
Native gates for Rigetti QPUs¶
Physical quantum processors can enact a subset of all named gates. Luckily,
a small set of gates is universal for quantum computation, so all named gates
can be enacted by suitable combinations of physically realizable gates. Rigetti’s
superconducting quantum processors can perform RX()
with angle=+pi/2
or
angle=+pi
, RZ()
with an arbitrary angle, and CZ()
interactions
between neighboring qubits. Rigetti QPUs can natively measure in the computational (Z) basis.
RX (angle, qubit) 
Produces the RX gate. 
RZ (angle, qubit) 
Produces the RZ gate. 
CZ (control, target) 
Produces a controlledZ gate. 
MEASURE (qubit, classical_reg) 
Produce a MEASURE instruction. 
All gates and instructions¶
In general, you will write a quantum program using the full suite of Quil gates and instructions and use the Quil compiler to compile programs into the target instruction set (ISA). The full list of quantum gates and classical Quil instructions is enumerated here.
Singlequbit gates
I (qubit) 
Produces the I identity gate. 
X (qubit) 
Produces the X (“NOT”) gate. 
Y (qubit) 
Produces the Y gate. 
Z (qubit) 
Produces the Z gate. 
H (qubit) 
Produces the Hadamard gate. 
S (qubit) 
Produces the S gate. 
T (qubit) 
Produces the T gate. 
RX (angle, qubit) 
Produces the RX gate. 
RY (angle, qubit) 
Produces the RY gate. 
RZ (angle, qubit) 
Produces the RZ gate. 
PHASE (angle, qubit) 
Produces the PHASE gate. 
Multiqubit gates
CZ (control, target) 
Produces a controlledZ gate. 
CNOT (control, target) 
Produces a controlledNOT (controlledX) gate. 
CCNOT (control1, control2, target) 
Produces a doublycontrolled NOT gate. 
CPHASE00 (angle, control, target) 
Produces a controlledphase gate that phases the 00> state. 
CPHASE01 (angle, control, target) 
Produces a controlledphase gate that phases the 01> state. 
CPHASE10 (angle, control, target) 
Produces a controlledphase gate that phases the 10> state. 
CPHASE (angle, control, target) 
Produces a controlledphase instruction. 
SWAP (q1, q2) 
Produces a SWAP gate which swaps the state of two qubits. 
CSWAP (control, target_1, target_2) 
Produces a controlledSWAP gate. 
ISWAP (q1, q2) 
Produces an ISWAP gate. 
PSWAP (angle, q1, q2) 
Produces a parameterized SWAP gate. 
Classical instructions
WAIT 
This instruction tells the quantum computation to halt. 
RESET ([qubit_index]) 
Reset all qubits or just a specific qubit at qubit_index. 
NOP 
This instruction applies no operation at that timestep. 
HALT 
This instruction ends the program. 
MEASURE (qubit, classical_reg) 
Produce a MEASURE instruction. 
NEG (classical_reg) 
Produce a NEG instruction. 
NOT (classical_reg) 
Produce a NOT instruction. 
AND (classical_reg1, classical_reg2) 
Produce an AND instruction. 
OR (classical_reg1, classical_reg2) 
Produce an OR instruction. 
IOR (classical_reg1, classical_reg2) 
Produce an inclusive OR instruction. 
XOR (classical_reg1, classical_reg2) 
Produce an exclusive OR instruction. 
MOVE (classical_reg1, classical_reg2) 
Produce a MOVE instruction. 
EXCHANGE (classical_reg1, classical_reg2) 
Produce an EXCHANGE instruction. 
LOAD (target_reg, region_name, offset_reg) 
Produce a LOAD instruction. 
STORE (region_name, offset_reg, source) 
Produce a STORE instruction. 
CONVERT (classical_reg1, classical_reg2) 
Produce a CONVERT instruction. 
ADD (classical_reg, right) 
Produce an ADD instruction. 
SUB (classical_reg, right) 
Produce a SUB instruction. 
MUL (classical_reg, right) 
Produce a MUL instruction. 
DIV (classical_reg, right) 
Produce an DIV instruction. 
EQ (classical_reg1, classical_reg2, …) 
Produce an EQ instruction. 
LT (classical_reg1, classical_reg2, …) 
Produce an LT instruction. 
LE (classical_reg1, classical_reg2, …) 
Produce an LE instruction. 
GT (classical_reg1, classical_reg2, …) 
Produce an GT instruction. 
GE (classical_reg1, classical_reg2, …) 
Produce an GE instruction. 
Collections
QUANTUM_GATES 
Dictionary of quantum gate functions keyed by gate names. 
STANDARD_INSTRUCTIONS 
Dictionary of standard instruction functions keyed by instruction names. 
Pauli Operators¶
Quantum operators can be expressed as combinations of Pauli operators I, X, Y, Z:
>>> operator = sZ(0)*sZ(1) + sX(2)*sY(3)
>>> print(operator)
(1+0j)*Z0*Z1 + (1+0j)*X2*Y3
Construction functions¶
sX (q) 
A function that returns the sigma_X operator on a particular qubit. 
sY (q) 
A function that returns the sigma_Y operator on a particular qubit. 
sZ (q) 
A function that returns the sigma_Z operator on a particular qubit. 
sI ([q]) 
A function that returns the identity operator, optionally on a particular qubit. 
ID () 
The identity operator. 
ZERO () 
The zero operator. 
Working with operators¶
simplify_pauli_sum (pauli_sum) 
Simplify the sum of Pauli operators according to Pauli algebra rules. 
check_commutation (pauli_list, pauli_two) 
Check if commuting a PauliTerm commutes with a list of other terms by natural calculation. 
commuting_sets (pauli_terms) 
Gather the Pauli terms of pauli_terms variable into commuting sets 
is_identity (term) 
Tests to see if a PauliTerm or PauliSum is a scalar multiple of identity 
is_zero (pauli_object) 
Tests to see if a PauliTerm or PauliSum is zero. 
exponentiate (term) 
Creates a pyQuil program that simulates the unitary evolution exp(1j * term) 
exponential_map (term) 
Returns a function f(alpha) that constructs the Program corresponding to exp(1j*alpha*term). 
exponentiate_commuting_pauli_sum (pauli_sum) 
Returns a function that maps all substituent PauliTerms and sums them into a program. 
suzuki_trotter (trotter_order, trotter_steps) 
Generate trotterization coefficients for a given number of Trotter steps. 
trotterize (first_pauli_term, second_pauli_term) 
Create a Quil program that approximates exp( (A + B)t) where A and B are PauliTerm operators. 
Classes¶

class
pyquil.paulis.
PauliSum
(terms)[source]¶ A sum of one or more PauliTerms.
Parameters: terms (Sequence) – A Sequence of PauliTerms. Methods
get_qubits
()The support of all the operators in the PauliSum object. simplify
()Simplifies the sum of Pauli operators according to Pauli algebra rules. get_programs
()Get a Pyquil Program corresponding to each term in the PauliSum and a coefficient for each program

class
pyquil.paulis.
PauliTerm
(op, index, coefficient=1.0)[source]¶ A term is a product of Pauli operators operating on different qubits.
Create a new Pauli Term with a Pauli operator at a particular index and a leading coefficient.
Parameters: Methods
id
([sort_ops])Returns an identifier string for the PauliTerm (ignoring the coefficient). operations_as_set
()Return a frozenset of operations in this term. copy
()Properly creates a new PauliTerm, with a completely new dictionary of operators program
from_list
(terms_list[, coefficient])Allocates a Pauli Term from a list of operators and indices. pauli_string
([qubits])Return a string representation of this PauliTerm without its coefficient and with implicit qubit indices.
Quantum Computer¶

pyquil.
get_qc
(name, *, as_qvm=None, noisy=None, connection=None)[source]¶ Get a quantum computer.
A quantum computer is an object of type
QuantumComputer
and can be backed either by a QVM simulator (“Quantum/Quil Virtual Machine”) or a physical Rigetti QPU (“Quantum Processing Unit”) made of superconducting qubits.You can choose the quantum computer to target through a combination of its name and optional flags. There are multiple ways to get the same quantum computer. The following are equivalent:
>>> qc = get_qc("Aspen116QAnoisyqvm") >>> qc = get_qc("Aspen116QA", as_qvm=True, noisy=True)
and will construct a simulator of an Aspen1 lattice with a noise model based on device characteristics. We also provide a means for constructing generic quantum simulators that are not related to a given piece of Rigetti hardware:
>>> qc = get_qc("9qsquareqvm") >>> qc = get_qc("9qsquare", as_qvm=True)
Finally, you can get request a QVM with “no” topology of a given number of qubits (technically, it’s a fully connected graph among the given number of qubits) with:
>>> qc = get_qc("5qqvm") # or "6qqvm", or "34qqvm", ...
These lessrealistic, fullyconnected QVMs will also be more lenient on what types of programs they will
run
. Specifically, you do not need to do any compilation. For the other, realistic QVMs you must useqc.compile()
orqc.compiler.native_quil_to_executable()
prior toqc.run()
.The Rigetti QVM must be downloaded from https://www.rigetti.com/forest and run as a server alongside your python program. To use pyQuil’s builtin QVM, replace all
"qvm"
suffixes with"pyqvm"
:>>> qc = get_qc("5qpyqvm")
Redundant flags are acceptable, but conflicting flags will raise an exception:
>>> qc = get_qc("9qsquareqvm") # qc is fully specified by its name >>> qc = get_qc("9qsquareqvm", as_qvm=True) # redundant, but ok >>> qc = get_qc("9qsquareqvm", as_qvm=False) # Error!
Use
list_quantum_computers()
to retrieve a list of known qc names.This method is provided as a convenience to quickly construct and use QVM’s and QPU’s. Power users may wish to have more control over the specification of a quantum computer (e.g. custom noise models, bespoke topologies, etc.). This is possible by constructing a
QuantumComputer
object by hand. Please refer to the documentation onQuantumComputer
for more information.Parameters:  name (
str
) – The name of the desired quantum computer. This should correspond to a name returned bylist_quantum_computers()
. Names ending in “qvm” will return a QVM. Names ending in “pyqvm” will return aPyQVM
. Names ending in “noisyqvm” will return a QVM with a noise model. Otherwise, we will return a QPU with the given name.  as_qvm (
Optional
[bool
]) – An optional flag to force construction of a QVM (instead of a QPU). If specified and set toTrue
, a QVMbacked quantum computer will be returned regardless of the name’s suffix  noisy (
Optional
[bool
]) – An optional flag to force inclusion of a noise model. If specified and set toTrue
, a quantum computer with a noise model will be returned regardless of the name’s suffix. The noise model for QVMs based on a real QPU is an empirically parameterized model based on real device noise characteristics. The generic QVM noise model is simple T1 and T2 noise plus readout error. Seedecoherence_noise_with_asymmetric_ro()
.  connection (
Optional
[ForestConnection
]) – An optionalForestConnection
object. If not specified, the default values for URL endpoints will be used. If you deign to change any of these parameters, pass your ownForestConnection
object.
Return type: QuantumComputer
Returns: A preconfigured QuantumComputer
 name (

pyquil.
list_quantum_computers
(connection=None, qpus=True, qvms=True)[source]¶ List the names of available quantum computers
Parameters:  connection (
Optional
[ForestConnection
]) – An optional :py:class:ForestConnection` object. If not specified, the default values for URL endpoints will be used, and your API key will be read from ~/.pyquil_config. If you deign to change any of these parameters, pass your ownForestConnection
object.  qpus (
bool
) – Whether to include QPU’s in the list.  qvms (
bool
) – Whether to include QVM’s in the list.
Return type:  connection (

class
pyquil.api.
QuantumComputer
(*, name, qam, device, compiler, symmetrize_readout=False)[source]¶ A quantum computer for running quantum programs.
A quantum computer has various characteristics like supported gates, qubits, qubit topologies, gate fidelities, and more. A quantum computer also has the ability to run quantum programs.
A quantum computer can be a real Rigetti QPU that uses superconducting transmon qubits to run quantum programs, or it can be an emulator like the Rigetti QVM with noise models and mimicked topologies.
Parameters:  name (
str
) – A string identifying this particular quantum computer.  qam (
QAM
) – A quantum abstract machine which handles executing quantum programs. This dispatches to a QVM or QPU.  device (
AbstractDevice
) – A collection of connected qubits and associated specs and topology.  symmetrize_readout (
bool
) – Whether to apply readout error symmetrization. Seerun_symmetrized_readout()
for a complete description.
Methods
run
(executable[, memory_map])Run a quil executable. run_and_measure
(program, trials)Run the provided state preparation program and measure all qubits. run_symmetrized_readout
(program, trials)Run a quil program in such a way that the readout error is made collectively symmetric qubits
()Return a sorted list of this QuantumComputer’s device’s qubits qubit_topology
()Return a NetworkX graph representation of this QuantumComputer’s device’s qubit connectivity. get_isa
([oneq_type, twoq_type])Return a target ISA for this QuantumComputer’s device. compile
(program[, to_native_gates, optimize])A highlevel interface to program compilation.  name (
Compilers¶
An appropriate compiler is automatically created when using get_qc()
and it is
stored on the QuantumComputer
object as the compiler
attribute.
The exact process for compilation depends on whether you’re targeting a QPU or a QVM, and
you can conceive of other compilation strategies than those included with pyQuil by default.
Therefore, we define an abstract interface that all compilers must follow. See
AbstractCompiler
for more, or use one of the listed compilers below.
_qac.AbstractCompiler 
The abstract interface for a compiler. 
QVMCompiler (endpoint, device[, timeout]) 
Client to communicate with the Compiler Server. 
QPUCompiler (quilc_endpoint, …[, timeout, name]) 
Client to communicate with the Compiler Server. 
QAMs¶
An appropriate QAM is automatically created when using get_qc()
and it is
stored on the QuantumComputer
object as the qam
attribute.
The Quantum Abstract Machine (QAM) provides an abstract interface for running hybrid quantum/classical quil programs on either a Quantum Virtual Machine (QVM, a classical simulator) or a Quantum Processor Unit (QPU, a real quantum device).
QAM () 
The platonic ideal of this class is as a generic interface describing how a classical computer interacts with a live quantum computer. 
QPU (endpoint[, user, priority]) 
A connection to the QPU. 
QVM (connection[, noise_model, gate_noise, …]) 
A virtual machine that classically emulates the execution of Quil programs. 
PyQVM (n_qubits[, quantum_simulator_type, …]) 
PyQuil’s builtin Quil virtual machine. 
Devices¶
An appropriate Device is automatically created when using get_qc()
and it is
stored on the QuantumComputer
object as the device
attribute.
There are properties of real quantum computers that go beyond the quantum abstract machine
(QAM) abstraction. Real devices have performance specs, limited ISAs, and restricted topologies.
AbstractDevice
provides an abstract interface for accessing
properties of a real quantum device or for mocking out relevant properties for a more realistic
QVM.
AbstractDevice 

Device (name, raw) 
A device (quantum chip) that can accept programs. 
NxDevice (topology) 
A shim over the AbstractDevice API backed by a NetworkX graph. 
The data structures used are documented here
ISA 
Basic Instruction Set Architecture specification. 
Specs 
Basic specifications for the device, such as gate fidelities and coherence times. 
Utility functions¶

pyquil.device.
isa_from_graph
(graph, oneq_type='Xhalves', twoq_type='CZ')[source]¶ Generate an ISA object from a NetworkX graph.
Parameters:  graph (
Graph
) – The graph  oneq_type – The type of 1qubit gate. Currently ‘Xhalves’
 twoq_type – The type of 2qubit gate. One of ‘CZ’ or ‘CPHASE’.
Return type:  graph (

pyquil.device.
specs_from_graph
(graph)[source]¶ Generate a Specs object from a NetworkX graph with placeholder values for the actual specs.
Parameters: graph ( Graph
) – The graph
Simulators¶
QVMs promise to behave like a real QPU. However, underthehood there is usually a simulation that has fewer constraints than a physical device. For example, in a wavefunction (or statevector) simulation, you can directly inspect amplitudes and probabilities.
WavefunctionSimulator ([connection, random_seed]) 
A simulator that propagates a wavefunction representation of a quantum state. 
ReferenceWavefunctionSimulator (n_qubits[, rs]) 
A wavefunction simulator that prioritizes readability over performance. 
ReferenceDensitySimulator (n_qubits[, rs]) 
A density matrix simulator that prioritizes readability over performance. 
NumpyWavefunctionSimulator (n_qubits[, rs]) 
A wavefunction simulator that uses numpy’s tensordot or einsum to update a state vector 
Reference Utilities¶
lifted_pauli (pauli_sum, qubits) 
Takes a PauliSum object along with a list of qubits and returns a matrix corresponding the tensor representation of the object. 
lifted_gate (gate, n_qubits) 
Lift a pyquil Gate in a full n_qubits qubit Hilbert space. 
program_unitary (program, n_qubits) 
Return the unitary of a pyQuil program. 
all_bitstrings (n_bits) 
All bitstrings in lexicographical order as a 2d np.ndarray. 
Numpy Utilities¶
targeted_einsum (gate, wf, wf_target_inds) 
Leftmultiplies the given axes of the wf tensor by the given gate matrix. 
targeted_tensordot (gate, wf, wf_target_inds) 
Leftmultiplies the given axes of the wf tensor by the given gate matrix. 
Noise Models¶
Functions¶
append_kraus_to_gate (kraus_ops, gate_matrix) 
Follow a gate gate_matrix by a Kraus map described by kraus_ops . 
pauli_kraus_map (probabilities) 
Generate the Kraus operators corresponding to a pauli channel. 
damping_kraus_map ([p]) 
Generate the Kraus operators corresponding to an amplitude damping noise channel. 
dephasing_kraus_map ([p]) 
Generate the Kraus operators corresponding to a dephasing channel. 
tensor_kraus_maps (k1, k2) 
Generate the Kraus map corresponding to the composition of two maps on different qubits. 
combine_kraus_maps (k1, k2) 
Generate the Kraus map corresponding to the composition of two maps on the same qubits with k1 being applied to the state after k2. 
damping_after_dephasing (T1, T2, gate_time) 
Generate the Kraus map corresponding to the composition of a dephasing channel followed by an amplitude damping channel. 
get_noisy_gate (gate_name, params) 
Look up the numerical gate representation and a proposed ‘noisy’ name. 
_decoherence_noise_model (gates[, T1, T2, …]) 
The default noise parameters 
decoherence_noise_with_asymmetric_ro (gates) 
Similar to _decoherence_noise_model() , but with asymmetric readout. 
apply_noise_model (prog, noise_model) 
Apply a noise model to a program and generated a ‘noisyfied’ version of the program. 
add_decoherence_noise (prog[, T1, T2, …]) 
Add generic damping and dephasing noise to a program. 
estimate_bitstring_probs (results) 
Given an array of single shot results estimate the probability distribution over all bitstrings. 
corrupt_bitstring_probs (p, …) 
Given a 2d array of true bitstring probabilities (outer axis iterates over shots, inner axis over bits) and a list of assignment probability matrices (one for each bit in the readout, ordered like the inner axis of results) compute the corrupted probabilities. 
bitstring_probs_to_z_moments (p) 
Convert between bitstring probabilities and joint Z moment expectations. 
estimate_assignment_probs (q, trials, cxn[, p0]) 
Estimate the readout assignment probabilities for a given qubit q . 
Classes¶

class
pyquil.noise.
NoiseModel
[source]¶ Encapsulate the QPU noise model containing information about the noisy gates.
Variables:  gates (Sequence[KrausModel]) – The tomographic estimates of all gates.
 assignment_probs (Dict[int,np.array]) – The single qubit readout assignment probability matrices keyed by qubit id.
Create new instance of _NoiseModel(gates, assignment_probs)
Methods
gates_by_name
(name)Return all defined noisy gates of a particular gate name. to_dict
()Create a JSON serializable representation of the noise model. from_dict
(d)Recreate the noise model from a dictionary representation.

class
pyquil.noise.
KrausModel
[source]¶ Encapsulate a single gate’s noise model.
Variables:  gate (str) – The name of the gate.
 params (Sequence[float]) – Optional parameters for the gate.
 targets (Sequence[int]) – The target qubit ids.
 kraus_ops (Sequence[np.array]) – The Kraus operators (must be square complex numpy arrays).
 fidelity (float) – The average gate fidelity associated with the Kraus map relative to the ideal operation.
Create new instance of _KrausModel(gate, params, targets, kraus_ops, fidelity)
Methods
unpack_kraus_matrix
(m)Helper to optionally unpack a JSON compatible representation of a complex Kraus matrix. unpack_kraus_matrix
(m)Helper to optionally unpack a JSON compatible representation of a complex Kraus matrix. to_dict
()Create a dictionary representation of a KrausModel. from_dict
(d)Recreate a KrausModel from the dictionary representation.
Operator Estimation¶
Data structures¶
ExperimentSetting (in_state, out_operator) 
Input and output settings for a tomographylike experiment. 
TomographyExperiment (settings, program[, qubits]) 
A tomographylike experiment. 
ExperimentResult (setting, expectation, …) 
An expectation and standard deviation for the measurement of one experiment setting in a tomographic experiment. 
Methods¶
group_experiments (experiments[, method]) 
Group experiments that are diagonal in a shared tensor product basis (TPB) to minimize number of QPU runs. 
measure_observables (qc, tomo_experiment[, …]) 
Measure all the observables in a TomographyExperiment. 