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arXiv:quant-ph/0305038 · Journal of Physics A Mathematical and General

Quantum phase estimation algorithms with delays: effects of dynamical phases

L. F. Wei, Franco Nori

Published Apr 5, 2004• 11 citations
arXivHTMLPDF

Abstract

The unavoidable finite time intervals between the sequential operations needed for performing practical quantum computing can degrade the performance of quantum computers. During these delays, unwanted relative dynamical phases are produced due to the free evolution of the superposition wave-function of the qubits. In general, these coherent "errors" modify the desired quantum interferences and thus spoil the correct results, compared to the ideal standard quantum computing that does not consider the effects of delays between successive unitary operations. Here, we show that, in the framework of the quantum phase estimation algorithm, these coherent phase "errors", produced by the time delays between sequential operations, can be avoided by setting up the delay times to satisfy certain matching conditions.

DOI
10.1088/0305-4470/37/16/010

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The unavoidable finite time intervals between the sequential operations needed for performing practical quantum computing can degrade the performance of quantum computers. During these delays, unwanted relative dynamical phases are produced due to the free evolution of the superposition wave-function of the qubits. In general, these coherent "errors" modify the desired quantum interferences and thus spoil the correct results, compared to the ideal standard quantum computing that does not consider the effects of delays between successive unitary operations. Here, we show that, in the framework of the quantum phase estimation algorithm, these coherent phase "errors", produced by the time delays between sequential operations, can be avoided by setting up the delay times to satisfy certain matching conditions.
Ideal quantum algorithms usually assume that quantum computing is performed continuously by a sequence of unitary transformations. However, there always exist idle finite time intervals between consecutive operations in a realistic quantum computing process. During these delays, coherent errors will accumulate from the dynamical phases of the superposed wave functions. Here we explore the sensitivity of Shor's quantum factoring algorithm to such errors. Our results clearly show a severe sensitivity of Shor's factorization algorithm to the presence of delay times between successive unitary transformations. Specifically, in the presence of these coherent errors, the probability of obtaining the correct answer decreases exponentially with the number of qubits of the work register. A particularly simple phase-matching approach is proposed in this paper to avoid or suppress these coherent errors when using Shor's algorithm to factorize integers. The robustness of this phase-matching condition is evaluated analytically and numerically for the factorization of several integers: 4, 15, 21, and 33.
Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary N tot times, the variance of the estimates can scale as O ( 1 / N tot 2 ) , compared to the best “classical” strategy with O ( 1 / N tot ) . The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without interprobe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O ( 1 / N tot 2 ) . Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analizes the results of previous steps. Published by the American Physical Society 2024
Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary NtotN_{\mathrm{tot}}Ntot​ times, the variance of the estimates can scale as O(1/Ntot2)O(1/{N^2_{\mathrm{tot}}})O(1/Ntot2​), compared to the best ``classical'' strategy with O(1/Ntot)O(1/{N_{\mathrm{tot}}})O(1/Ntot​). The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without inter-probe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O(1/Ntot2)O(1/{N^2_{\mathrm{tot}}})O(1/Ntot2​). Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analyses the results of previous steps.
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.
Abstract This paper is concerned with the phase estimation algorithm in quantum computing, especially the scenarios where (1) the input vector is not an eigenvector; (2) the unitary operator is approximated by Trotter or Taylor expansion methods; (3) random approximations are used for the unitary operator. We characterize the probability of computing the phase values in terms of the consistency error, including the residual error, Trotter splitting error, or statistical mean-square error. In the first two cases, we show that in order to obtain the phase value with error less or equal to 2 − n and probability at least 1 − ϵ , the required number of qubits is t ⩾ n + log 2 + δ 2 2 ϵ Δ E 2 . The parameter δ quantifies the error associated with the inexact eigenvector and/or the unitary operator, and Δ E characterizes the spectral gap, i.e., the separation from the rest of the phase values. This analysis generalizes the standard result (Cleve et al 1998 Phys. Rev X 11 011020; Nielsen and Chuang 2002 Quantum Computation and Quantum Information ) by including these effects. More importantly, it shows that when δ < Δ E , the complexity remains the same. For the third case, we found a similar estimate, but the number of random steps has to be sufficiently large.
Quantum phase estimation is one of the most important tools in quantum algorithms. It can be made non-adaptive (meaning all applications of the unitary UφU_φUφ​ happen simultaneously) without using more applications of UφU_φUφ​, albeit at the expense of using many more qubits. It is also known that there is no advantage for adaptive algorithms in the case where the phase that needs to be estimated is arbitrary or is uniformly random. Here we give examples of a special case of phase estimation, with a promise on the values that the unknown phase can take, where adaptive methods are provably better than non-adaptive methods by a factor of nearly 2 in the number of uses of UφU_φUφ​. We also prove some upper bounds on the maximum advantage that adaptive algorithms for phase estimation can achieve over non-adaptive ones.