Installation and Getting Started

• Environment Setup
  • Prerequisites
  • Installation
  • Connecting to a Rigetti QVM
• Basic Usage
  • Some Program Construction Features
  • Fixing a Mistaken Instruction
  • The Standard Gate Set
  • Defining New Gates
• Advanced Usage
  • Quantum Fourier Transform (QFT)
  • Classical Control Flow
  • Parametric Depolarizing Noise
  • Pauli Operator Algebra
• Exercises
  • Exercise 1 - Quantum Dice
  • Exercise 2 - Controlled Gates
  • Exercise 3 - Grover's Algorithm

This toolkit provides some simple libraries for writing quantum programs using the quantum instruction language Quil. pyQuil is part of the Forest suite of tools for quantum programming and is currently in private beta.

import pyquil.quil as pq
import pyquil.forest as forest
from pyquil.gates import *
qvm = forest.Connection()
p = pq.Program()
p.inst(H(0), CNOT(0, 1))
    <pyquil.pyquil.Program object at 0x101ebfb50>
qvm.wavefunction(p)
    [(0.7071067811865475+0j), 0j, 0j, (0.7071067811865475+0j)]

It comes with a few parts:

1. 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.
2. QVM: A Quantum Virtual Machine, which is an implementation of the quantum abstract machine on classical hardware. The QVM lets you use a regular computer to simulate a small quantum computer. You can access the Rigetti QVM running in the cloud with your API key. Sign up here to get your key.
3. pyQuil: A Python library to help write and run Quil code and quantum programs.

Environment Setup

Prerequisites

Before starting, ensure that you have an installation of Python 2.7 (version 2.7.10 or greater) and the Python package manager pip. We recommend installing Anaconda for an all-in-one installation of Python 2.7. If pip is not installed, it can be installed with easy_install pip.

Installation

After obtaining pyQuil from GitHub or from a source distribution, navigate into its directory in a terminal and run

pip install pyquil

The library will now be available globally.

Connecting to a Rigetti QVM

In order to connect to a Rigetti QVM you need to configure your pyQuil installation with your QVM API key. For permanent one-time setup, you can do this by creating a file in your home directory with the following lines:

[Rigetti Forest]
url: <FOREST_URL>
key: <YOUR_FOREST_API_KEY>

Look here to see more information about setting up your connection to Forest.

If this configuration is not set, pyQuil will default to looking for a local QVM at https://api.rigetti.com/qvm.

Basic Usage

To ensure that your installation is working correctly, try running the following Python commands interactively. First, import the quil module (which constructs quantum programs) and the forest module (which allows connections to the Rigetti QVM). We'll also import some basic gates for pyQuil.

Next, we want to open a connection to the QVM.

Now we can make a program by adding some Quil instruction using the inst method on a Program object.

This program simply applies the $X$-gate to the zeroth qubit, measures that qubit, and stores the measurement result in the zeroth classical register. We can look at the Quil code that makes up this program simply by printing it.

Most importantly, of course, we can see what happens if we run this program on the QVM:

We see that the result of this program is that the classical register [0] now stores the state of qubit 0, which should be $\left\vert 1\right\rangle$ after an $X$-gate. We can of course ask for more classical registers:

The classical registers are initialized to zero, so registers [1] and [2] come out as zero. If we stored the measurement in a different classical register we would obtain:

We can also run programs multiple times and accumulate all the results in a single list:

Try running the above code several times. You will, with very high probability, get different results each time.

As the QVM is a virtual machine, we can also inspect the wavefunction of a program directly, even without measurements:

It is important to remember that this wavefunction method is just a useful debugging tool for small quantum systems, and it cannot be feasibly obtained on a quantum processor.

Some Program Construction Features

Multiple instructions can be applied at once or chained together. The following are all valid programs:

Fixing a Mistaken Instruction

If an instruction was appended to a program incorrectly, one can pop it off.

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: 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.

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.

Quil in general supports defining parametric gates, though right now only static gates are supported by pyQuil. Below we show how we can define $X_1\otimes \sqrt{X_0} $ as a single gate.

Advanced Usage

Quantum Fourier Transform (QFT)

Let's do an example that includes multi-qubit parameterized gates.

Here we wish to compute the discrete Fourier transform of [0, 1, 0, 0, 0, 0, 0, 0]. We do this in three steps:

1. Write a function called qft3 to make a 3-qubit QFT quantum program.
2. Write a state preparation quantum program.
3. Execute state preparation followed by the QFT on the QVM.

First we define a function to make a 3-qubit QFT quantum program. This is a mix of Hadamard and CPHASE gates, with a final bit reversal correction at the end consisting of a single SWAP gate.

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 by doing the following:

Next, we want to prepare a state that corresponds to the sequence we want to compute the discrete Fourier transform of. Fortunately, this is easy, we just apply an $X$-gate to the zeroth qubit.

We can verify that this works by computing its wavefunction. However, we need to add some "dummy" qubits, because otherwise wavefunction would return a two-element vector.

If we have two quantum programs a and b, we can concatenate them by doing a + b. Using this, all we need to do is compute the QFT after state preparation to get our final result.

We can verify this works by computing the (inverse) FFT from NumPy.

Classical Control Flow

Here are a couple quick examples that show how much richer the classical control of a Quil program can be. In this first example, we have a register called classical_flag_register which we use for looping. Then we construct the loop in the following steps:

1. We first initialize this register to 1 with the init_register program so our while loop will execute. This is often called the loop preamble or loop initialization.

2. Next, we write body of the loop in a program itself. This will be a program that computes an $X$ followed by an $H$ on our qubit.

3. Lastly, we put it all together using the while_do method.

Notice that the init_register program applied a Quil instruction directly to a classical register. There are several classical commands that can be used in this fashion:

• TRUE which sets a single classical bit to be 1
• FALSE which sets a single classical bit to be 0
• NOT which flips a classical bit
• AND which operates on two classical bits
• OR which operates on two classical bits
• MOVE which moves the value of a classical bit at one classical address into another
• EXCHANGE 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.

We can run this program a few times to see what we get in the answer_register.

Parametric Depolarizing Noise

The Rigetti QVM has support for emulating certain types of noise models. One such model is parametric depolarizing 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.

We can test this by applying an $X$ gate and measuring. Nominally, we should always measure 1.

Parametric Programs

A big advantage of working in pyQuil is that you are able to leverage all the functionality of Python to generate Quil programs. In quantum/classical hybrid algorithms this often leads to situations where complex classical functions are used to generate Quil programs. pyQuil provides a convenient construction to allow you to use Python functions to generate templates of Quil programs, called ParametricPrograms:

The parametric program par_p now takes the same arguments as rotator:

We can think of ParametricPrograms as a sort of template for Quil programs. They cache computations that happen in Python functions so that templates in Quil can be efficiently substituted.

Pauli Operator Algebra

Many algorithms require manipulating sums of Pauli combinations, such as $$\sigma = \tfrac{1}{2}I - \tfrac{3}{4}X_0Y_1Z_3 + (5-2i)Z_1X_2,$$ 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:

There are two primary things one can do with Pauli terms and sums:

1. A Pauli sum's fully "tensored up" form can be computed with the tensor_up function.

2. Quil code can be generated to compute the exponentiation of a Pauli term, i.e., $\exp[-i\sigma]$.

When arithmetic is done with Pauli sums, simplification is automatically done.

The following shows an instructive example of all three.

Exercises

Exercise 1 - Quantum Dice

Write a quantum program to simulate throwing an 8-sided 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 and SciPy to construct new gate matrices.

1. 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.

2. 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 single-shot 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: