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arXiv:2111.10430 · Journal of Physics A Mathematical and Theoretical

Some error analysis for the quantum phase estimation algorithms

Xiantao Li

Published Jul 7, 2022• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ac7f6c

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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.
Phase estimation, due to Kitaev \cite{Kit95}, is one of the most fundamental subroutines in quantum computing. In the basic scenario, one is given black-box access to a unitary U , and an eigenstate | ψ ⟩ of U with unknown eigenvalue e i θ , and the task is to estimate the eigenphase θ within ± δ , with high probability. The cost of an algorithm for us is the number of applications of U and U − 1 .We tightly characterize the cost of several variants of phase estimation where we are no longer given an eigenstate, but are required to estimate the maximum eigenphase of U , aided by advice in the form of states (or a unitary preparing those states) which are promised to have at least a certain overlap γ with the top eigenspace.We give algorithms and nearly matching lower bounds for all ranges of parameters.We show that a small number of copies of the advice state (or of an advice-preparing unitary) are not significantly better than having no advice at all. We also show that having lots of advice (applications of the advice-preparing unitary) does not significantly reduce cost, and neither does knowledge of the eigenbasis of U . We immediately obtain a lower bound on the complexity of the Unitary recurrence time problem, resolving an open question of She and Yuen \cite{SY23}.Lastly, we study how efficiently one can reduce the error probability in the basic phase-estimation scenario. We show that a phase-estimation algorithm with precision δ and error probability ϵ has cost Ω ( 1 δ log ⁡ 1 ϵ ) , matching an easy upper bound. This contrasts with some other scenarios in quantum computing (e.g., search) where error-probability reduction costs only a factor O ( log ⁡ ( 1 / ϵ ) ) . Our lower bound uses a variant of the polynomial method with trigonometric polynomials.
Quantum phase estimation is one of the most fundamental and useful quantum computing algorithms available to researchers, but traditional methods for estimating the phase of a unitary operator acting on its eigenvalue leave information about the phase contained in all of the output from a quantum computer unused. Mean squared error (MSE) for such a scheme is 0.03 for simulated data with and without noise. Machine learning with classical methods, like linear regression (MSE=0.011 for noisy data and 0.0043 for noise-free data) and random forests (MSE=0.01 for noisy data and 0.006 for noise-free data) increase the accuracy of the estimation drastically without requiring additional qubits like the traditional approach. However, neural networks reduce this number further (MSE=0.0077) in the case of noisy data and with noise-free data (MSE=0.003). Examining the performance of neural networks with different modifications also yields insight into their performance on quantum computing output that could be generalized to output from other algorithms and on quantum computers with different numbers of qubits.
A quantum algorithm for estimating the phase, which determines the eigenvalue of a unitary operator, is discussed. It is assumed that the eigenvector of this operator and the corresponding quantum circuit are given. The memory register where the approximate phase value is stored consists of n qubits, which makes it possible to determine the phase accurate to 2 −n with the probability greater than 8/π 2 . By way of example, computations for the case of the quantum phase shift operator are performed. The simulation of the quantum algorithm and the computation of the eigenvalue are performed using the QuantumCircuit package written in the Wolfram Mathematica computer algebra system. This system is also used to perform all the computations and visualize the results.
The quantum phase estimation (QPE) is one of the fundamental algorithms based on the quantum Fourier transform. It has applications in order-finding, factoring, and finding the eigenvalues of unitary operators. The major challenge in running QPE and other quantum algorithms is the noise in quantum computers. In the present work, we study the impact of incoherent noise on QPE, modeled as trace-preserving and completely positive quantum channels. Different noise models such as depolarizing, phase flip, bit flip, and bit-phase flip are taken to understand the performance of the QPE in the presence of noise. The simulation results indicate that the standard deviation of the eigenvalue of the unitary operator has strong exponential dependence upon the error probability of individual qubits. However, the standard deviation increases only linearly with the number of qubits for fixed error probability when that error probability is small.
We numerically implement full quantum circuits of optimal tapered Quantum Phase Estimation (in the form of an implementable approximate bandwidth-limited DPSS taper) for the purpose of minimum eigenvalue computation, where the Hamiltonian time-evolution unitary is implemented using standard Trotterization, and the taper state is constructed using an optimized bandwidth-limited taper. We illustrate, on representative quantum Hamiltonians, with tractable exact classical numerical quantum circuit simulations, how tapered QPE performs with respect to optimal phase sampling rate, absolute error, Shannon entropy of the optimal phase distribution, when the tQPE algorithm parameters are changed. Those algorithm parameters are Trotter error, finite sampling, imperfect initial states, total evolution time, and digitization error (number of phase qubits). The reported numerical experiments include up to ℓ=10\ell=10ℓ=10 phase qubits of precision with m=3m=3m=3 additional phase-register error suppression qubits, on a 444-qubit Heisenberg quantum magnet model Hamiltonian, using up to 12th-order Trotterization. We numerically show i) the steady-state optimal phase sampling rate is determined by the initial state overlap with the ground-state, ii) with respect to phase sampling probability, or overall eigenvalue sampling error rate, there is no strong evolution time dependence for tQPE unless there is high Trotter error, iii) the approximated time-evolution unitaries in tQPE, like in standard QPE, can result in substantial ringing, which leads to non-physical eigenvalue estimates.
We provide a modification to the quantum phase estimation algorithm (QPEA) [Abrams and Lloyd, Phys. Rev. Lett. 83, 5162 (1999); Cleve et al., Proc. R. Soc. A 454, 339 (1998); Nielsen and Chuang, Quantum computation and quantum information, 2002.] inspired by classical windowing methods for spectral density estimation. From this modification we obtain an upper bound in the cost that implies a cubic improvement with respect to the algorithm's error rate. Numerical evaluation of the costs also demonstrates an improvement. Moreover, with similar techniques, we detail an iterative projective measurement method for ground state preparation that gives an exponential improvement over previous bounds using QPEA. Numerical tests that confirm the expected scaling behavior are also obtained. For these numerical tests we have used a lattice Thirring model as testing ground. Using well-known perturbation theory results, we also show how to more appropriately estimate the cost scaling with respect to state error instead of evolution operator error.