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Quantum Electronics

Iterative quantum phase estimation on an IBM quantum processor

N.A. Zhuravlev, Ilya I. Beterov

Published May 19, 2021• 1 citation
Publisher

Abstract

Abstract An elementary algorithm of quantum phase estimation based on the modified Kitaev algorithm is implemented on two qubits of an IBM quantum processor. This algorithm includes adiabatic preparation of the initial state, controlled phase shift with allowance for the results of previous measurements of qubit states, and single measurement of the qubit quantum state in order to obtain each significant bit of the measured phase. Classical error correction is applied to determine the correct sequence of bits, which makes it possible to eliminate the influence of limited accuracy of two-qubit gates.

DOI
10.1070/qel17586

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Abstract An elementary algorithm of quantum phase estimation based on the modified Kitaev algorithm is implemented on two qubits of an IBM quantum processor. This algorithm includes adiabatic preparation of the initial state, controlled phase shift with allowance for the results of previous measurements of qubit states, and single measurement of the qubit quantum state in order to obtain each significant bit of the measured phase. Classical error correction is applied to determine the correct sequence of bits, which makes it possible to eliminate the influence of limited accuracy of two-qubit gates.
Quantum phase estimation (QPE) is one of the core algorithms for quantum computing. It has been extensively studied and applied in a variety of quantum applications such as the Shor's factoring algorithm, quantum sampling algorithms and the calculation of the eigenvalues of unitary matrices. The QPE algorithm has been combined with Kitaev's algorithm and the inverse quantum Fourier transform (IQFT) which are utilized as a fundamental component of such quantum algorithms. In this paper, we explore the computational challenges of implementing QPE algorithms on noisy intermediate-scale quantum (NISQ) machines using the IBM Q Experience (e.g., the IBMQX4, 5-qubit quantum computing hardware platform). Our experimental results indicate that the accuracy of finding the phase using these QPE algorithms is severely constrained by the NISQ computer's physical characteristics such as coherence time and error rates. To mitigate these physical limitations, we propose implementing a modified solution by reducing the number of controlled rotation gates and phase shift operations, thereby increasing the accuracy of the finding phase in near-term quantum computers.
Many researchers have been heavily investigated on quantum phase estimation (QPE) algorithms to find the unknown phase, since QPE is the core building block of the most quantum algorithms such as the Shor's factoring algorithm, quantum sampling algorithms, and finding the eigenvalues of unitary matrices. Kitaev's algorithm and QPE algorithms using inverse Quantum Fourier transform were proposed and widely used by researchers as a key component for their quantum algorithms. In this paper, we explore the experimental challenges of QPE algorithms on Noisy Intermediate-Scale Quantum (NISQ) computers by implementing various QPE algorithms on the state-of-the-art IBM quantum computer. Our experimental results demonstrate that the accuracy of finding the phase using these algorithms are severely constrained by NISQ's physical characteristics such as coherence time and error rates. To mitigate such physical limitations, we propose modified solutions of these algorithms by reducing the number of control gates and phase shift operations. Our experimental results showed that our solutions can significantly increase the accuracy of the finding phase in near-term quantum computers.
Quantum computing is the field that studies computation using quantum mechanical systems, exploiting properties such as superposition or entanglement. The focus of this project will be on quantum phase estimation algorithms. The formalism commonly used to represent quantum algorithms as quantum circuits composed by qubits and quantum gates will be presented. Then, we will be able to explain the quantum Fourier transform (that is exponentially faster than its classical analogue) and three quantum phase estimation algorithms: Kitaev’s phase estimation (KPEA), the phase estimation algorithm based on QFT (PEA) and the iterative quantum phase estimation algorithm based on a semiclassical Fourier transform (IPEA). We will use the Qiskit framework in order to run simulations of such algorithms, and even run them in real IBM Q Experience quantum computers (specifically, we will use the 5 qubit computer ibmq_essex). Statistical post-processing techniques will be essential in order to extract a phase estimation from the results obtained. We will use two post-processing techniques: the majority and the average rule. The performance of these methods applied to the results of several simulations and executions in ibmq_essex will be studied. At last, we will talk about the possible applications of phase estimation algorithms. Phase estimation is used as a subroutine in the field of quantum simulation, allowing us to obtain several magnitudes of a quantum system such as the energy levels. We will present an efficient quantum algorithm for factoring called Shor’s algorithm. The cost of factorization with a classical computer increases exponentially with the size of the input, so this is a example of a task that cannot be solved efficiently on a classical computer, but can be solved efficiently by a quantum computer. Phase estimation is also used in the field of quantum metrology. Specifically, an N-V center can be used to measure magnetic fields with high sensitivity and spatial resolution operating at room temperature thanks to a measuring protocol based on phase estimation.
In this work we consider Kitaev's algorithm for quantum phase estimation. We analyze the use of phase shifts that simplify the estimation of successive bits in the estimation of unknown phase φ, By using increasingly accurate shifts we reduce the number of measurements to the point where only a single measurement is needed for each additional bit. This results in an algorithm that can estimate φ to m + 2 bits of accuracy with probability at least 1- ϵ using Nϵ+ m measurements, where Nϵis a quantity that depends only on ϵ and the particular sampling algorithm. We present different sampling algorithms and study the exact number of measurements needed through careful numerical evaluation, and provide theoretical bounds and numerical values for Nϵ.
In the process of implementing a quantum algorithm as a quantum circuit, the error rate of an actual quantum computer should be considered. In this paper, we confirm through simulation that the one control qubit quantum phase estimation algorithm performs better than the many control qubit quantum phase estimation algorithm on the noisy quantum processor. We use QISKit to simulate a quantum phase estimation algorithm. We performed the simulation considering the error rate of the IBM Q quantum processor. The error rate of the one control qubit quantum phase estimation algorithm is compared with the many control qubit quantum phase estimation algorithm.