An Improved Classical Singular Value Transformation for Quantum Machine Learning

Abstract

Quantum machine learning (QML) has shown great potential to produce large quantum speedups for computationally intensive linear algebra tasks. The quantum singular value transformation (QSVT), introduced by Gilyén, Su, Low and Wiebe [GSLW19], is a unifying framework to obtain QML algorithms. We provide a classical algorithm that matches the performance of QSVT on low-rank inputs, up to a small polynomial overhead. Under efficient quantum-accessible memory assumptions, given a bounded matrix A ∈ C m×n , a vector b ∈ C n , and a bounded degree-d polynomial p, QSVT can output a measurement from the state |p(A)b in O(d A F ) time after linear-time pre-processing. We show that, in the same setting, for any ε > 0, we can output a vectorF /ε 2 ) time after linear-time pre-processing. This improves upon the best known classical algorithm [CGL + 20a], which requires O(d 22 A 6 F /ε 6 ) time. Instantiating the aforementioned algorithm with different polynomials, we obtain fast quantum-inspired algorithms for regression, recommendation systems, and Hamiltonian simulation. We improve in numerous parameter settings on prior work, including those that use problem-specialized approaches, for quantum-inspired regression [CGL + 20b, CGL + 20a, SM21, GST22, CCH + 22] and quantum-inspired recommendation systems [Tan19, CGL + 20a, CCH + 22].Our key insight is to combine the Clenshaw recurrence, an iterative method for computing matrix polynomials, with sketching techniques to simulate QSVT classically. The tools we introduce in this work include (a) a non-oblivious matrix sketch for approximately preserving bi-linear forms, (b) a non-oblivious asymmetric approximate matrix product sketch based on 2 2 sampling, (c) a new stability analysis for the Clenshaw recurrence, and (d) 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.