Installation and Getting Started

This toolkit provides some simple libraries for writing quantum programs.

import pyquil.quil as pq
import pyquil.api as api
from pyquil.gates import *
qvm = api.SyncConnection()
p = pq.Program()
p.inst(H(0), CNOT(0, 1))
    <pyquil.pyquil.Program object at 0x101ebfb50>
wvf, _ = qvm.wavefunction(p)
print(wvf)
    (0.7071067812+0j)|00> + (0.7071067812+0j)|11>

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.
  4. QPUConnection: pyQuil also includes some a special connection which lets you run experiments on Rigetti’s prototype superconducting quantum processors over the cloud. These experiments are described in more detail here.

Environment Setup

Prerequisites

Before you can start writing quantum programs, you will need Python 2.7 (version 2.7.10 or greater) or Python 3.6 and the Python package manager pip. We recommend installing Anaconda for an all-in-one installation of Python (2.7). If you don’t have pip, it can be installed with easy_install pip.

Installation

You can install pyQuil directly from the Python package manager pip using:

pip install pyquil

To instead install the bleeding-edge version from source, clone the pyquil GitHub repository, navigate into its directory in a terminal, and run:

pip install -e .

On Mac/Linux, if this command does not succeed because of permissions errors, then instead run:

sudo pip install -e .

This will also install pyQuil’s dependencies (requests >= 2.4.2 and NumPy >= 1.10) if you do not already have them.

The library will now be available globally.

Connecting to the Rigetti Forest

pyQuil can be used to build and manipulate Quil programs without restriction. However, to run programs (e.g., to get wavefunctions, get multishot experiment data), you will need an API key for Rigetti Forest. This will allow you to run your programs on the Rigetti QVM or QPU.

Once you have your key, you need to set up configuration in the file .pyquil_config which pyQuil will attempt to find in your home directory by default. You can change this location by setting the environment variable PYQUIL_CONFIG to the path of the file.

Note

When setting the environment variable make sure to use the full path or include the HOME variable. E.g. export PYQUIL_CONFIG=$HOME/<CONFIG_NAME>

Loading the pyquil.forest module will print a warning if this is not found. The configuration file is in INI format and should contain all the information required to connect to Forest:

[Rigetti Forest]
url: <URL to Rigetti Forest or QVM endpoint>
key: <Rigetti Forest API key>

Look here to learn more about the Forest toolkit.

If url is not set, pyQuil will default to looking for a local endpoint at 127.0.0.1:5000.

You may also create the .pyquil_config automatically by running the following command, which will prompt you for the required information (URL, key, and user id). The script will then create a file in the proper location (the user’s root directory):

pyquil-config-setup

Alternatively, connection information can be provided in environment variables.

export QVM_URL=<URL to Rigetti Forest or QVM endpoint>
export QVM_API_KEY=<Rigetti Forest API key>
export QVM_USER_ID=<Rigetti User ID>

Endpoints

There are two important endpoints to keep in mind. You will use different ones for different types of jobs.

https://api.rigetti.com/qvm is used for making synchronous calls to the QVM. You should use this for most of the getting started materials unless otherwise instructed.

https://job.rigetti.com/beta is used for large async QVM jobs or for running jobs on a QPU.

Running your first quantum program

pyQuil is a Python library that helps you write programs in the Quantum Instruction Language (Quil). It also ships with a simple script examples/run_quil.py that runs Quil code directly. You can test your connection to Forest using this script by executing the following on your command line

cd examples/
python run_quil.py hello_world.quil

You should see the following output array [[1, 0, 0, 0, 0, 0, 0, 0]]. This indicates that you have a good connection to our API.

You can continue to write more Quil code in files and run them using the run_quil.py script. The following sections describe how to use the pyQuil library directly to build quantum programs in Python.

Basic pyQuil 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 will also import some basic gates for pyQuil as well as numpy.

import pyquil.quil as pq
import pyquil.api as api
from pyquil.gates import *
import numpy as np

Next, we want to open a connection to the QVM. Forest supports two types of connections through pyQuil. The first is a synchronous connection that immediately runs requested jobs against the QVM. This will time out on longer jobs that run for more than 30 seconds. Synchronous connections are good for experimenting interactively as they give quick feedback.

# open a synchronous connection
qvm = api.SyncConnection()

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

p = pq.Program()
p.inst(X(0)).measure(0, 0)
<pyquil.quil.Program at 0x101d45a90>

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.

print(p)
X 0
MEASURE 0 [0]

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

classical_regs = [0] # A list of which classical registers to return the values of.

qvm.run(p, classical_regs)
[[1]]

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:

qvm.run(p, [0, 1, 2])
[[1, 0, 0]]

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:

p = pq.Program()   # clear the old program
p.inst(X(0)).measure(0, 1)
qvm.run(p, [0, 1, 2])
[[0, 1, 0]]

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

coin_flip = pq.Program().inst(H(0)).measure(0, 0)
num_flips = 5
qvm.run(coin_flip, [0], num_flips)
[[0], [1], [0], [1], [0]]

Try running the above code several times. You will see that 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:

coin_flip = pq.Program().inst(H(0))
qvm.wavefunction(coin_flip)
(<pyquil.wavefunction.Wavefunction at 0x1088a2c10>, [])

The first element in the returned tuple is a Wavefunction object that stores the amplitudes of the quantum state at the conclusion of the program. We can print this object

coin_flip = pq.Program().inst(H(0))
wvf, _ = qvm.wavefunction(coin_flip)
print(wvf)
(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 wvf[0] == 1 / np.sqrt(2)
# The amplitudes are stored as a numpy array on the Wavefunction object
print(wvf.amplitudes)
prob_dict = wvf.get_outcome_probs() # extracts the probabilities of outcomes as a dict
print(prob_dict)
prob_dict.keys() # these stores the bitstring outcomes
assert len(wvf) == 1 # gives the number of qubits
[ 0.70710678+0.j  0.70710678+0.j]
{'1': 0.49999999999999989, '0': 0.49999999999999989}

The second element returned from a wavefunction call is an optional amount of classical memory to check:

coin_flip = pq.Program().inst(H(0)).measure(0,0)
wavf, classical_mem = qvm.wavefunction(coin_flip, classical_addresses=range(9))

Additionally, we can pass a random seed to the Connection object. This allows us to reliably reproduce measurement results for the purpose of testing:

seeded_cxn = api.SyncConnection(random_seed=17)
print(seeded_cxn.run(pq.Program(H(0)).measure(0, 0), [0], 20))

seeded_cxn = api.SyncConnection(random_seed=17)
# This will give identical output to the above
print(seeded_cxn.run(pq.Program(H(0)).measure(0, 0), [0], 20))

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:

print("Multiple inst arguments with final measurement:")
print(pq.Program().inst(X(0), Y(1), Z(0)).measure(0, 1))

print("Chained inst with explicit MEASURE instruction:")
print(pq.Program().inst(X(0)).inst(Y(1)).measure(0, 1).inst(MEASURE(1, 2)))

print("A mix of chained inst and measures:")
print(pq.Program().inst(X(0)).measure(0, 1).inst(Y(1), X(0)).measure(0, 0))

print("A composition of two programs:")
print(pq.Program(X(0)) + pq.Program(Y(0)))
Multiple inst arguments with final measurement:
X 0
Y 1
Z 0
MEASURE 0 [1]

Chained inst with explicit MEASURE instruction:
X 0
Y 1
MEASURE 0 [1]
MEASURE 1 [2]

A mix of chained inst and measures:
X 0
MEASURE 0 [1]
Y 1
X 0
MEASURE 0 [0]

A composition of two programs:
X 0
Y 0

Fixing a Mistaken Instruction

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

p = pq.Program().inst(X(0))
p.inst(Y(1))
print("Oops! We have added Y 1 by accident:")
print(p)

print("We can fix by popping:")
p.pop()
print(p)

print("And then add it back:")
p += pq.Program(Y(1))
print(p)
Oops! We have added Y 1 by accident:
X 0
Y 1

We can fix by popping:
X 0

And then add it back:
X 0
Y 1

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.

import numpy as np

# First we define the new gate from a matrix
x_gate_matrix = np.array(([0.0, 1.0], [1.0, 0.0]))
sqrt_x = np.array([[ 0.5+0.5j,  0.5-0.5j],
                   [ 0.5-0.5j,  0.5+0.5j]])
p = pq.Program().defgate("SQRT-X", sqrt_x)

# Then we can use the new gate,
p.inst(("SQRT-X", 0))
print(p)
DEFGATE SQRT-X:
    0.5+0.5i, 0.5-0.5i
    0.5-0.5i, 0.5+0.5i

SQRT-X 0
print(qvm.wavefunction(p)[0])
(0.5+0.5j)|0> + (0.5-0.5j)|1>

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

# A multi-qubit defgate example
x_gate_matrix = np.array(([0.0, 1.0], [1.0, 0.0]))
sqrt_x = np.array([[ 0.5+0.5j,  0.5-0.5j],
                [ 0.5-0.5j,  0.5+0.5j]])
x_sqrt_x = np.kron(x_gate_matrix, sqrt_x)
p = pq.Program().defgate("X-SQRT-X", x_sqrt_x)

# Then we can use the new gate
p.inst(("X-SQRT-X", 0, 1))
wavf, _ = qvm.wavefunction(p)
print(wavf)
(0.5+0.5j)|01> + (0.5-0.5j)|11>

Advanced Usage

Quantum Fourier Transform (QFT)

Let us 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.

from math import pi

def qft3(q0, q1, q2):
    p = pq.Program()
    p.inst( H(q2),
            CPHASE(pi/2.0, q1, q2),
            H(q1),
            CPHASE(pi/4.0, q0, q2),
            CPHASE(pi/2.0, q0, q1),
            H(q0),
            SWAP(q0, 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 by doing the following:

print(qft3(0, 1, 2))
H 2
CPHASE(1.5707963267948966) 1 2
H 1
CPHASE(0.7853981633974483) 0 2
CPHASE(1.5707963267948966) 0 1
H 0
SWAP 0 2

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.

state_prep = pq.Program().inst(X(0))

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.

add_dummy_qubits = pq.Program().inst(I(1), I(2))
wavf, _ = qvm.wavefunction(state_prep + add_dummy_qubits)
print(wavf)
(1+0j)|001>

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.

wavf, _ = qvm.wavefunction(state_prep + qft3(0, 1, 2))
print(wavf.amplitudes)
array([  3.53553391e-01+0.j        ,   2.50000000e-01+0.25j      ,
         2.16489014e-17+0.35355339j,  -2.50000000e-01+0.25j      ,
        -3.53553391e-01+0.j        ,  -2.50000000e-01-0.25j      ,
        -2.16489014e-17-0.35355339j,   2.50000000e-01-0.25j      ])

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

from numpy.fft import ifft
ifft([0,1,0,0,0,0,0,0], norm="ortho")
array([ 0.35355339+0.j        ,  0.25000000+0.25j      ,
        0.00000000+0.35355339j, -0.25000000+0.25j      ,
       -0.35355339+0.j        , -0.25000000-0.25j      ,
        0.00000000-0.35355339j,  0.25000000-0.25j      ])

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.
# Name our classical registers:
classical_flag_register = 2

# Write out the loop initialization and body programs:
init_register = pq.Program(TRUE([classical_flag_register]))
loop_body = pq.Program(X(0), H(0)).measure(0, classical_flag_register)

# Put it all together in a loop program:
loop_prog = init_register.while_do(classical_flag_register, loop_body)

print(loop_prog)
TRUE [2]
LABEL @START1
JUMP-UNLESS @END2 [2]
X 0
H 0
MEASURE 0 [2]
JUMP @START1
LABEL @END2

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.

# Name our classical registers:
test_register = 1
answer_register = 0

# Construct each branch of our if-statement. We can have empty branches
# simply by having empty programs.
then_branch = pq.Program(X(0))
else_branch = pq.Program()

# Make a program that will put a 0 or 1 in test_register with 50% probability:
branching_prog = pq.Program(H(1)).measure(1, test_register)

# Add the conditional branching:
branching_prog.if_then(test_register, then_branch, else_branch)

# Measure qubit 0 into our answer register:
branching_prog.measure(0, answer_register)

print(branching_prog)
H 1
MEASURE 1 [1]
JUMP-WHEN @THEN3 [1]
JUMP @END4
LABEL @THEN3
X 0
LABEL @END4
MEASURE 0 [0]

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

qvm.run(branching_prog, [answer_register], 10)
[[1], [1], [1], [0], [1], [0], [0], [1], [1], [0]]

Parametric Depolarizing Noise

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 = api.SyncConnection(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 = pq.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]]

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:

# This function returns a quantum circuit with different rotation angles on a gate on qubit 0
def rotator(angle):
    return pq.Program(RX(angle, 0))

from pyquil.parametric import ParametricProgram
par_p = ParametricProgram(rotator) # This produces a new type of parameterized program object

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

print(par_p(0.5))
RX(0.5) 0

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 \(\sigma = \frac{1}{2}I - \frac{3}{4}X_0Y_1Z_3 + (5-2i)Z_1X_2,\) 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:

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 = (5-2j) * sZ(1) * sX(2)

# Construct a sum of Pauli terms.
sigma = a + b + c
print("sigma = {}".format(sigma))
sigma = 0.5*I + -0.75*X0*Y1*Z3 + (5-2j)*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 Suzuki-Trotter approximation through fourth order.

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

The following shows an instructive example of all three.

import pyquil.paulis as pl

# Simplification
sigma_cubed = sigma * sigma * sigma
print("Simplified  : {}".format(sigma_cubed))
print()

#Produce Quil code to compute exp[iX]
H = -1.0 * sX(0)
print("Quil to compute exp[iX] on qubit 0:")
print(pl.exponential_map(H)(1.0))
Simplified  : (32.46875-30j)*I + (-16.734375+15j)*X0*Y1*Z3 + (71.5625-144.625j)*Z1*X2

Quil to compute exp[iX] on qubit 0:
H 0
RZ(-2.0) 0
H 0

A more sophisticated feature of pyQuil is that it can create templates of Quil programs in ParametricProgram objects. An example use of these templates is in exponentiating a Hamiltonian that is parametrized by a constant. 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.

parametric_prog = pl.exponential_map(H)
print parametric_prog(0.0)
print parametric_prog(1.0)
print parametric_prog(2.0)

This ParametricProgram now acts as a template, caching the result of the exponential_map calculation so that it can be used later with new values.

JobConnections

Larger pyQuil programs can take longer than 30 seconds to run. These jobs can be posted into the cloud job queue using a different connection object. The mode of interaction with the API is asynchronous. This means that there is a seperate query to post a job and to get the result.

from pyquil.quil import Program
from pyquil.gates import X, H, I
from pyquil.api import JobConnection

job_qvm = JobConnection(endpoint="https://job.rigetti.com/beta")
res = job_qvm.run(Program(X(0)).measure(0, 0), [0])

The res is an instance of a JobResult object. It has an id and allows you to make queries to see if the job result is finished.

zz = res.get()
print type(zz), zz
<class 'pyquil.job_results.JobResult'> {u'status': u'Submitted', u'jobId': u'BLSLJCBGNP'}

is_done updates the JobResult object once, and returns True if the job has completed. Once the job is finished, then the results can be retrieved from the JobResult object:

import time

while not res.is_done():
    time.sleep(1)
print res
answer = res.decode()
print answer
{u'result': u'[[1]]', u'jobId': u'BLSLJCBGNP'}

<type 'list'> [[1]]

This same pattern applies to the wavefunction, expectation, and run_and_measure calls on the JobConnection object.

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 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:

\[\begin{split}\begin{pmatrix} 2/N - 1 & 2/N & \cdots & 2/N \\ 2/N & & &\\ \vdots & & \ddots & \\ 2/N & & & 2/N-1 \end{pmatrix}\end{split}\]