Estimating pi ($\pi$) using Quantum Phase Estimation Algorithm
Quick overview of the quantum phase estimation algorithm
Quantum Phase Estimation (QPE) is a quantum algorithm that forms the building block of many more complex quantum algorithms. At its core, QPE solves a fairly straightforward problem: given an operator $U$ and a quantum state $\vert\psi\rangle$ that is an eigenvalue of $U$ with $U\vert\psi\rangle = \exp\left(2 \pi i \theta\right)\vert\psi\rangle$, can we obtain an estimate of $\theta$?
The answer is yes. The QPE algorithm gives us $2^n\theta$, where $n$ is the number of qubits we use to estimate the phase $\theta$.
Estimating $\pi$
In this demo, we choose $$U = u_1(\theta), \vert\psi\rangle = \vert1\rangle$$ where $$ u_1(\theta) = \begin{bmatrix} 1 & 0\\ 0 & \exp(i\theta) \end{bmatrix} $$ is one of the quantum gates available in Qiskit, and $$u_1(\theta)\vert1\rangle = \exp(i\theta)\vert1\rangle.$$
By choosing the phase for our gate to be $\theta = 1$, we can solve for $\pi$ using the following two relations:
- From the output of the QPE algorithm, we measure an estimate for $2^n\theta$. Then, $\theta = \text{measured} / 2^n$
- From the definition of the $u_1(\theta)$ gate above, we know that $2\pi\theta = 1 \Rightarrow \pi = 1 / 2\theta$
Combining these two relations, $\pi = 1 / \left(2 \times (\text{(measured)}/2^n)\right)$.
For detailed understanding of the QPE algorithm, please refer to the chapter dedicated to it in the Qiskit Textbook located at qiskit.org/textbook.
We begin by importing the necessary libraries.
## import the necessary tools for our work
%matplotlib inline
from IPython.display import clear_output
from qiskit import *
from qiskit.visualization import plot_histogram
import numpy as np
import matplotlib.pyplot as plotter
from qiskit.tools.monitor import job_monitor
import seaborn as sns, operator
sns.set_style("dark")
pi = np.pi
The function qft_dagger computes the inverse Quantum Fourier Transform. For a detailed understanding of this algorithm, see the dedicated chapter for it in the Qiskit Textbook.
## Code for inverse Quantum Fourier Transform
## adapted from Qiskit Textbook at
## qiskit.org/textbook
def qft_dagger(circ_, n_qubits):
"""n-qubit QFTdagger the first n qubits in circ"""
for qubit in range(int(n_qubits/2)):
circ_.swap(qubit, n_qubits-qubit-1)
for j in range(0,n_qubits):
for m in range(j):
circ_.cu1(-np.pi/float(2**(j-m)), m, j)
circ_.h(j)
The next function, qpe_pre, prepares the initial state for the estimation. Note that the starting state is created by applying a Hadamard gate on the all but the last qubit, and setting the last qubit to $\vert1\rangle$.
## Code for initial state of Quantum Phase Estimation
## adapted from Qiskit Textbook at qiskit.org/textbook
## Note that the starting state is created by applying
## H on the first n_qubits, and setting the last qubit to |psi> = |1>
def qpe_pre(circ_, n_qubits):
circ_.h(range(n_qubits))
circ_.x(n_qubits)
for x in reversed(range(n_qubits)):
for _ in range(2**(n_qubits-1-x)):
circ_.cu1(1, n_qubits-1-x, n_qubits)
Next, we write a quick function, run_job, to run a quantum circuit and return the results.
## Run a Qiskit job on either hardware or simulators
def run_job(circ_, backend_, shots_=1000, optimization_level_=0):
job = execute(circ_, backend=backend_, shots=shots_, optimization_level=optimization_level_)
job_monitor(job)
return job.result().get_counts(circ_)
Then, load your account to use the cloud simulator or real devices.
## Load your IBMQ account if
## you'd like to use the cloud simulator or real quantum devices
IBMQ.load_account()
simulator_cloud = IBMQ.get_provider(hub='ibm-q',group='open',project='main').get_backend('ibmq_qasm_simulator')
simulator = Aer.get_backend('qasm_simulator')
device = IBMQ.get_provider(hub='ibm-q',group='open',project='main').get_backend('ibmq_16_melbourne')
Finally, we bring everything together in a function called get_pi_estimate that uses n_qubits to get an estimate for $\pi$.
## Function to estimate pi
## Summary: using the notation in the Qiskit textbook (qiskit.org/textbook),
## do quantum phase estimation with the operator U = u1(theta) and |psi> = |1>
## such that u1(theta)|1> = exp(2 x pi x i x theta)|1>
## By setting theta = 1 radian, we can solve for pi
## using 2^n x 1 radian = most frequently measured count = 2 x pi
def get_pi_estimate(n_qubits):
# create the circuit
circ = QuantumCircuit(n_qubits + 1, n_qubits)
# create the input state
qpe_pre(circ, n_qubits)
# apply a barrier
circ.barrier()
# apply the inverse fourier transform
qft_dagger(circ, n_qubits)
# apply a barrier
circ.barrier()
# measure all but the last qubits
circ.measure(range(n_qubits), range(n_qubits))
if n_qubits < 10:
circ.draw(output='mpl').savefig(str(n_qubits)+'_qubit_circuit.png')
# run the job and get the results
counts = run_job(circ, backend_=simulator, shots_=10000, optimization_level_=0)
# print(counts)
# get the count that occurred most frequently
max_counts_result = max(counts, key=counts.get)
max_counts_result = int(max_counts_result, 2)
# solve for pi from the measured counts
theta = max_counts_result/2**n_qubits
return (1./(2*theta))
Now, run the get_pi_estimate function with different numbers of qubits and print the estimates.
# estimate pi using different numbers of qubits
nqs = list(range(2,12+1))
pi_estimates = []
for nq in nqs:
thisnq_pi_estimate = get_pi_estimate(nq)
pi_estimates.append(thisnq_pi_estimate)
print(f"{nq} qubits, pi ≈ {thisnq_pi_estimate}")
And plot all the results.
plotter.plot(nqs, [pi]*len(nqs), '--r')
plotter.plot(nqs, pi_estimates, '.-', markersize=12)
plotter.xlim([1.5, 12.5])
plotter.ylim([1.5, 4.5])
plotter.legend(['$\pi$', 'estimate of $\pi$'])
plotter.xlabel('Number of qubits', fontdict={'size':20})
plotter.ylabel('$\pi$ and estimate of $\pi$', fontdict={'size':20})
plotter.tick_params(axis='x', labelsize=12)
plotter.tick_params(axis='y', labelsize=12)
plotter.show()
