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arXiv:2209.10162 · Quantum

Infinite quantum signal processing

Yulong Dong, Lin Lin, Hongkang Ni, Jiasu Wang

Published Dec 10, 2024• 21 citations
arXivHTMLPDF

Abstract

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 → ∞ .

DOI
10.22331/q-2024-12-10-1558

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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 → ∞ .
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) 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 ${10}^{\ensuremath{-}12}$.
Abstract We provide a complete solution to the problem of infinite quantum signal processing (QSP) for the class of Szegő functions, which are functions that satisfy a logarithmic integrability condition and include almost any function that allows for a QSP representation. We do so by introducing a new algorithm called the Riemann–Hilbert–Weiss algorithm, which can compute any individual phase factor independent of all other phase factors. Our algorithm is also the first provably stable numerical algorithm for computing phase factors of any arbitrary Szegő function. The proof of stability involves solving a Riemann–Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory.
Quantum signal processing (QSP) provides a simple and efficient framework for implementing polynomial transformations using quantum circuits. Its classical design stage leads to a constrained minimax approximation problem: find a polynomial of prescribed parity that approximates a target function uniformly on a fitting set while remaining bounded in magnitude by one on the domain [0,1][0,1][0,1], which can be viewed as a semi-infinite constraint. Discretization converts the problem into a linear program, but feasibility at a set of finitely many sampled points does not ensure feasibility on the whole domain, especially when an optimal approximant reaches the boundary of the feasible set. We investigate two approaches to address this difficulty. A Remez exchange method combined with active-set constraint enforcement is efficient on many tested instances, but its stability depends on the target and problem geometry. We then introduce nonlinear Fourier retraction, which uses QSP completion and phase synthesis to turn a nearly feasible polynomial into phase factors for a feasible QSP polynomial without increasing the degree. Across representative problems, retraction largely preserves approximation accuracy and remains effective on instances where the Remez heuristic is unstable. The resulting workflow connects classical minimax approximation and semi-infinite optimization with nonlinear Fourier analysis, and is implemented in the qsppack software package.
Quantum signal processing (QSP) offers a framework for implementing polynomial transformations of operators that are block-encoded within unitary matrices. It has been widely adopted as a grand unification framework for quantum algorithms. In this work, we demonstrate that the coefficients of the induced polynomials P(x) and Q(x) of a QSP sequence in the Chebyshev basis can be uniquely determined by the coefficients of a Laurent polynomial. Based on this correspondence, we develop an efficient method for computing QSP matrix multiplications by evaluating this Laurent polynomial to obtain the QSP sequence. Finally, by applying this framework to the fixed-point iterative method for phase factor recovery, we achieve highly efficient computation of phase factors.
Quantum signal processing (QSP) and quantum singular value transformation (QSVT) are powerful techniques for the development of quantum procedures. They allow to derive circuits preparing desired polynomial transformations. Recent research [Alexis et al. 2024] showed that Non-Linear Fourier Analysis (NLFA) can be employed to numerically compute a QSP protocol, with provable stability. In this work we extend their result, showing that GQSP and the Non-Linear Fourier Transform over SU(2)SU(2)SU(2) are the same object. This statement - proven by a simple argument - has a bunch of consequences: first, the Riemann-Hilbert-Weiss algorithm can be turned, with little modifications and no penalty in complexity, into a unified, provably stable algorithm for the computation of phase factors in any QSP variant, including GQSP. Secondly, we derive a uniqueness result for the existence of GQSP phase factors based on the bijectivity of the Non-Linear Fourier Transform. Furthermore, NLFA provides a complete theory of infinite generalized quantum signal processing, which characterizes the class of functions approximable by GQSP protocols.