Quantum Fourier Transoform in Qiskit

Theory

Note: Quantum Computing (3)

As we mentioned before on that page, QFT circuit is represented as,

\left( \ket{0} + e^{i2\pi 0.j_n} \ket{1} \right) \otimes \left( \ket{0} + e^{i2\pi 0.j_{n-1}j_n} \ket{1} \right) \otimes \cdots \otimes \left( \ket{0} + e^{i2\pi 0.j_1j_2\cdots j_n} \ket{1} \right)

So, hadamal gate and general rotation gates are frequently and repeatedly used.

R_l = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\frac{2\pi}{2^l}} \end{pmatrix}

Now, we are going to see how to implement that circuit using qiskit

Code

import argparse
import numpy as np
import qiskit
from qiskit import visualization


def qft_simple(qc, num_qubits):
    for i in range(num_qubits):
        for j in range(i):
            qc.cp(
                2 * np.pi / (2**(i-j)),
                j, i
            )
        qc.h(i)    
    return qc

def qft_general(qc, num_qubits):
    for j in range(num_qubits):
        qc.h(j)
        for k in range(j+1, num_qubits):
            qc.cp(2 * np.pi / 2**(k-j+1), k, j)
    for j in range(num_qubits//2):
        qc.swap(j, num_qubits-j-1)
    return qc


def main(args):

    # initialize circuit
    name='ft'
    num_qubits = args.qubits_size
    qc = qiskit.QuantumCircuit(num_qubits, name=name)
    signal = np.ones(2**num_qubits).tolist()
    signal = np.array(signal) / np.linalg.norm(signal)
    qc.initialize(signal, range(num_qubits))

    # define circuit
    ft_circuit = qft_general(qc, num_qubits)
    print(ft_circuit, len(ft_circuit))
    fig = ft_circuit.draw('mpl')
    fig.savefig('./circuit-test1.png')

    # simulate circuit
    simulator = qiskit.Aer.get_backend('statevector_simulator')
    compiled_circuit = qiskit.transpile(
        ft_circuit,
        simulator,
        output_name='ft_circuit'
    )
    job = qiskit.execute(compiled_circuit, simulator)
    result = job.result()

    statevector = result.get_statevector()
    vis1 = visualization.plot_bloch_multivector(statevector)
    vis1.savefig('./bloch_multivector-1.png')

    # measure state vector
    ft_circuit.measure_all()
    compiled_meas_circuit = qiskit.transpile(
        ft_circuit,
        simulator,
        output_name='ft_circuit_m'
    )
    job_meas = qiskit.execute(compiled_meas_circuit, simulator, shots=1024)
    result_meas = job_meas.result()
    counts = result_meas.get_counts()
    vis2 = visualization.plot_histogram(counts)
    vis2.savefig('./histogram-1.png')


if __name__ == '__main__':

    parser = argparse.ArgumentParser(
        description='What this program is going to do.'
    )
    parser.add_argument(
        '--qubits_size', '-QS', type=int, default=4, help=''
    )
    args = parser.parse_args()

    main(args)

The Structure of Quantum Circuits

The following illustrates the quantum Fourier transform circuit generated by the code mentioned above. The initial state is Fourier transformed as a vector with constant values.

The state vector come out from the circuit

The following represents the state vector after quantum Fourier transformation, depicted on the Bloch sphere. Since a constant value is Fourier transformed, all elements result in the |0⟩ state.

QFT using Qiskit Circuit Library

But actually, we have another simpler implementation using Qiskit Circuit Library, means that Qiskit provides QFT function.

The structure of Quantum Circuit is written as below.

import argparse
import numpy as np
import qiskit
from qiskit import visualization
from qiskit.circuit.library import QFT


def main(args):
    num_qubits = args.qubits_size
    name='ft'

    initial_state = [1/np.sqrt(2**num_qubits)] * 2**num_qubits
    qc = qiskit.QuantumCircuit(num_qubits, name=name)
    qc.initialize(initial_state, range(num_qubits))

    qc.append(QFT(num_qubits), range(num_qubits))
    qc.measure_all()

    fig = qc.draw('mpl')
    fig.savefig('./circuit-test2.png')

    simulator = qiskit.Aer.get_backend('statevector_simulator')
    compiled_circuit = qiskit.transpile(
        qc,
        simulator,
        output_name='ft_circuit'
    )
    job = qiskit.execute(compiled_circuit, simulator)
    result = job.result()
    statevector = result.get_statevector()

    vis1 = visualization.plot_bloch_multivector(statevector)
    vis1.savefig('./bloch_multivector-2.png')


    compiled_meas_circuit = qiskit.transpile(
        qc,
        simulator,
        output_name='ft_circuit2'
    )
    job_meas = qiskit.execute(compiled_meas_circuit, simulator, shots=1024)
    result_meas = job_meas.result()
    counts = result_meas.get_counts()

    vis2 = visualization.plot_histogram(counts)
    vis2.savefig('./histogram-2.png')


if __name__ == '__main__':

    parser = argparse.ArgumentParser(
        description='What this program is going to do.'
    )
    parser.add_argument(
        '--qubits_size', '-QS', type=int, default=4, help=''
    )
    args = parser.parse_args()

    main(args)