Quantum Phase Estimation with Qiskit v2.x

In the previous articles, we covered the following topics:

In this article, we will implement Quantum Phase Estimation with Qiskit v2.x.

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Quantum Phase Estimation with Qiskit 2.0

In this example, we’ll use the $S$ gate as the target unitary matrix and $\ket{1}$ as its eigenstate. In other words, we aim to estimate the eigenvalue of the $S$ gate with respect to $\ket{1}$ using Quantum Phase Estimation. While it might be possible to compute this eigenvalue mentally, let’s pretend we don’t know it and proceed with the implementation. First, we’ll import the necessary functions.

from qiskit import QuantumCircuit
from qiskit.primitives import StatevectorSampler
from qiskit.circuit.library import QFT
from qiskit.circuit.library import SGate

Let’s define the sampler.

sampler = StatevectorSampler()

Let’s define the quantum circuit. In this example, we set the number of qubits in the readout register to 3.

n_qubits = 4
n_readout_qubits = n_qubits - 1
qc = QuantumCircuit(n_qubits, n_readout_qubits)

As explained in previous article, we can apply Hadamard gates to the qubits in the readout register based on the theory.

for qubit in list(range(n_readout_qubits)):
    qc.h(qubit)

Additionally, the fourth qubit, which is the last qubit, is prepared in the eigenstate $\ket{1}$.

qc.x(n_readout_qubits)

Next, we apply the controlled-$S$ gate. As shown in previous article, note that the controlled-$S$ gate must be applied $2^{n-1}$ times depending on the qubit.

ctrl_s = SGate().control(1)
for ctl_bit in list(range(n_readout_qubits)):
    for i in list(range(2**(ctl_bit))):
        qc.append(ctrl_s, [ctl_bit, n_readout_qubits])

Next, we will perform the inverse Quantum Fourier Transform, but before that, let’s take a moment to review. When executing QFT(num_qubits=3, inverse=True, do_swaps=True).decompose().draw('mpl'), the inverse Quantum Fourier Transform for 3 qubits results in the following circuit:

Since the state of the readout register needs to be arranged according to the order of this transformation, we don’t apply SWAP gate this time.

iqft = QFT(num_qubits=n_readout_qubits, inverse=True, do_swaps=True)
qc.append(iqft, [0, 1, 2])
qc.measure([0, 1, 2], [0, 1, 2])

When implementing the Quantum Fourier Transform and its inverse, it is important to pay close attention to the order in which operations are applied. In this case, by constructing the inverse Quantum Fourier Transform circuit while taking into account the pattern of controlled-$S$ gates applied to each qubit, we can obtain the expected result.

Printing the circuit by qc.draw('mpl') yields the following:

Now, let’s execute the quantum circuit.

pub = (qc, None)
job = sampler.run([pub], shots=1024)
result=job.result()[0]
print(result.data.c.get_counts())
# Output
# Counter({'010': 1024})

By measuring only the readout register, we obtain 010. Since the number of qubits in the readout register was $3$, this means that $\phi = 2/2^3 = 1/4$ was estimated by this circuit.

In other words, the eigenvalue of the $S$ gate with respect to $\ket{1}$ is estimated as $e^{2\pi i \phi} = e^{\pi i /2}$. This is indeed consistent with the theoretical expression below.
Quantum phase estimation has been successfully executed!

$$ S |1\rangle = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/2} \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = e^{i\pi/2} |1\rangle $$

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IBM Certified Quantum Computation Qiskit v2.x Practice Exams

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Quantum Computing on AWS for Beginners: Amazon Braket