An Improved Classical Singular Value Transformation for Quantum Machine Learning

The field of quantum machine learning (QML) produces many proposals for attaining quantum speedups for tasks in machine learning and data analysis. Such speedups can only manifest if classical algorithms for these tasks perform significantly slower than quantum ones. We study quantum-classical gaps in QML through the quantum singular value transformation (QSVT) framework. QSVT, introduced by Gilyen, Su, Low and Wiebe [GSLW19], unifies all major types of quantum speedup [MRTC21]; in particular, a wide variety of QML proposals are applications of QSVT on low-rank classical data. We challenge these proposals by providing a classical algorithm that matches the performance of QSVT in this regime up to a small polynomial overhead. We show that, given a matrix A is an element of C-mxn, a vector b is an element of C-n, a bounded degree-d polynomial p, and linear-time pre-processing, we can output a description of a vector v such that parallel to v - p(A)b parallel to <= epsilon parallel to b parallel to in (O) over tilde (d(11)parallel to A parallel to(4)(F) /(epsilon(2) parallel to A parallel to(4))) time. This improves upon the best known classical algorithm [CGLLTW22], which requires (O) over tilde (d(22)parallel to A parallel to(6)(F)/ (epsilon(6)parallel to A parallel to(6))) time, and narrows the gap with QSVT, which, after linear-time pre-processing to load input into a quantum-accessible memory, can estimate the magnitude of an entry p(A)b to epsilon parallel to b parallel to error in (O) over tilde(parallel to A parallel to(F)/(epsilon parallel to A parallel to)) time. Instantiating our algorithm with different polynomials, we improve on prior classical algorithms for quantum-inspired regression [CGLLTW22; GST22], recommendation systems [Tan19; CGLLTW22], and Hamiltonian simulation [CGLLTW22]. Our key insight is to combine the Clenshaw recurrence, an iterative method for computing matrix polynomials, with sketching techniques to simulate QSVT classically. We introduce several new classical techniques in this work, including (a) a non-oblivious matrix sketch for approximately preserving bi-linear forms, (b) a new stability analysis for the Clenshaw recurrence, and (c) a new technique to bound arithmetic progressions of the coefficients appearing in the Chebyshev series expansion of bounded functions, each of which may be of independent interest.