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arXiv:2002.11649 · Physical Review A

Efficient phase-factor evaluation in quantum signal processing

Yulong Dong, Xiang Meng, K. Birgitta Whaley, Lin Lin

Published Apr 22, 2021• 140 citations
arXivHTMLPDF

Abstract

Quantum signal processing (QSP) is a powerful quantum algorithm to exactly implement matrix polynomials on quantum computers. Asymptotic analysis of quantum algorithms based on QSP has shown that asymptotically optimal results can in principle be obtained for a range of tasks, such as Hamiltonian simulation and the quantum linear system problem. A further benefit of QSP is that it uses a minimal number of ancilla qubits, which facilitates its implementation on near-to-intermediate term quantum architectures. However, there is so far no classically stable algorithm allowing computation of the phase factors that are needed to build QSP circuits. Existing methods require the use of variable precision arithmetic and can only be applied to polynomials of a relatively low degree. We present here an optimization-based method that can accurately compute the phase factors using standard double precision arithmetic operations. We demonstrate the performance of this approach with applications to Hamiltonian simulation, eigenvalue filtering, and quantum linear system problems. Our numerical results show that the optimization algorithm can find phase factors to accurately approximate polynomials of a degree larger than $10\phantom{\rule{0.16em}{0ex}}000$ with errors below ${10}^{\ensuremath{-}12}$.

DOI
10.1103/physreva.103.042419

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Quantum signal processing (QSP) is a powerful quantum algorithm to exactly implement matrix polynomials on quantum computers. Asymptotic analysis of quantum algorithms based on QSP has shown that asymptotically optimal results can in principle be obtained for a range of tasks, such as Hamiltonian simulation and the quantum linear system problem. A further benefit of QSP is that it uses a minimal number of ancilla qubits, which facilitates its implementation on near-to-intermediate term quantum architectures. However, there is so far no classically stable algorithm allowing computation of the phase factors that are needed to build QSP circuits. Existing methods require the use of variable precision arithmetic and can only be applied to polynomials of a relatively low degree. We present here an optimization-based method that can accurately compute the phase factors using standard double precision arithmetic operations. We demonstrate the performance of this approach with applications to Hamiltonian simulation, eigenvalue filtering, and quantum linear system problems. Our numerical results show that the optimization algorithm can find phase factors to accurately approximate polynomials of a degree larger than 1000010\phantom{\rule{0.16em}{0ex}}00010000 with errors below 10\ensuremath−12{10}^{\ensuremath{-}12}10\ensuremath−12.
Quantum signal processing (QSP) provides a representation of scalar polynomials of degree ddd as products of matrices in SU(2)\mathrm{SU}(2)SU(2), parameterized by (d+1)(d+1)(d+1) real numbers known as phase factors. QSP is the mathematical foundation of quantum singular value transformation (QSVT), which is often regarded as one of the most important quantum algorithms of the past decade, with a wide range of applications in scientific computing, from Hamiltonian simulation to solving linear systems of equations and eigenvalue problems. In this article we survey recent advances in the mathematical and numerical analysis of QSP. In particular, we focus on its generalization beyond polynomials, the computational complexity of algorithms for phase factor evaluation, and the numerical stability of such algorithms. The resolution to some of these problems relies on an unexpected interplay between QSP, nonlinear Fourier analysis on SU(2)\mathrm{SU}(2)SU(2), fast polynomial multiplications, and Gaussian elimination for matrices with displacement structure.
Quantum signal processing (QSP) requires the efficient computation of phase factors associated with a polynomial transformation. One existing stable factorization method avoids direct root finding but relies on Prony's method to recover a polynomial from weighted power sums of its roots (arXiv:2202.02671). We present a modification of this approach that eliminates Prony's method by using the logarithmic derivative of an associated polynomial. Because the logarithmic derivative has the same residue at every root after accounting for multiplicity, its Fourier coefficients directly yield the unweighted power sums of the roots inside the unit disk. The Newton--Girard identities then recover the coefficients of the corresponding polynomial without computing its roots. The resulting algorithm has O(dlog⁡2d)O(d\log^2 d)O(dlog2d) computational complexity and simplifies the factorization procedure of the original method. In numerical experiments on Hamiltonian simulation, eigenstate filtering, matrix inversion, and the Fermi--Dirac operator, our implementation computes phase factors in seconds with maximum errors on the order of 10−1210^{-12}10−12.
Quantum signal processing (QSP) serves as the asymptotically optimal technique for Hamiltonian simulation on a quantum computer. By approximating the time evolution operator via the Jacobi-Anger expansion, the Hamiltonian simulation problem reduces to a problem in polynomial approximation theory: find a sufficient degree-ddd polynomial series to approximate e−iτxe^{-iτx}e−iτx on [−1,1][-1,1][−1,1] within error εεε. While d∈O~(τ)d\in\tilde{\mathcal{O}}(τ)d∈O~(τ) is known to be asymptotically optimal, there exists a gap between state-of-the-art bounds and the optimal constant multiplicative factor, which is approximately equal to 1. Here, we close this gap almost entirely, to the point where possible future improvements will not be of practical significance. Our improvement resides in a careful treatment of the Bessel tail in the Jacobi-Anger series using Kapteyn's and Watson's inequalities, thereby reducing the overhead estimates for all Hamiltonian simulation tasks on quantum computers by a factor of ≈e/2\approx e/2≈e/2.
Quantum signal processing (QSP) represents a real scalar polynomial of degree d using a product of unitary matrices of size 2 × 2 , parameterized by ( d + 1 ) real numbers called the phase factors. This innovative representation of polynomials has a wide range of applications in quantum computation. When the polynomial of interest is obtained by truncating an infinite polynomial series, a natural question is whether the phase factors have a well defined limit as the degree d → ∞ . While the phase factors are generally not unique, we find that there exists a consistent choice of parameterization so that the limit is well defined in the ℓ 1 space. This generalization of QSP, called the infinite quantum signal processing, can be used to represent a large class of non-polynomial functions. Our analysis reveals a surprising connection between the regularity of the target function and the decay properties of the phase factors. Our analysis also inspires a very simple and efficient algorithm to approximately compute the phase factors in the ℓ 1 space. The algorithm uses only double precision arithmetic operations, and provably converges when the ℓ 1 norm of the Chebyshev coefficients of the target function is upper bounded by a constant that is independent of d . This is also the first numerically stable algorithm for finding phase factors with provable performance guarantees in the limit d → ∞ .
Abstract. This paper addresses the problem of solving nonlinear systems in the context of symmetric quantum signal processing (QSP), a powerful technique for implementing matrix functions on quantum computers. Symmetric QSP focuses on representing target polynomials as products of matrices in SU(2) that possess symmetry properties. We present a novel Newton’s method tailored for efficiently solving the nonlinear system involved in determining the phase factors within the symmetric QSP framework. Our method demonstrates rapid and robust convergence in all parameter regimes, including the challenging scenario with ill-conditioned Jacobian matrices, using standard double precision arithmetic operations. For instance, solving symmetric QSP for a highly oscillatory target function [Formula: see text] (polynomial degree [Formula: see text]) takes 6 iterations to converge to machine precision when [Formula: see text], and the number of iterations only increases to 18 iterations when [Formula: see text] with a highly ill-conditioned Jacobian matrix. Leveraging the matrix product state structure of symmetric QSP, the computation of the Jacobian matrix incurs a computational cost comparable to a single function evaluation. Moreover, we introduce a reformulation of symmetric QSP using real-number arithmetics, further enhancing the method’s efficiency. Extensive numerical tests validate the effectiveness and robustness of our approach, which has been implemented in the QSPPACK software package.