The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation

TL;DR

This paper advances quantum machine learning by developing block-encoding techniques for matrices, leading to improved algorithms for linear systems, least squares, and electrical network estimations with exponential precision gains.

Contribution

It introduces new block-encoding tools and algorithms that significantly improve quantum linear system solving and regression methods, especially for non-sparse matrices.

Findings

Exponential improvement in precision dependence for quantum linear system solvers.

New quantum algorithms for weighted and generalized least squares problems.

Enhanced quantum algorithms for electrical network estimations.

Abstract

We apply the framework of block-encodings, introduced by Low and Chuang (under the name standard-form), to the study of quantum machine learning algorithms and derive general results that are applicable to a variety of input models, including sparse matrix oracles and matrices stored in a data structure. We develop several tools within the block-encoding framework, such as singular value estimation of a block-encoded matrix, and quantum linear system solvers using block-encodings. The presented results give new techniques for Hamiltonian simulation of non-sparse matrices, which could be relevant for certain quantum chemistry applications, and which in turn imply an exponential improvement in the dependence on precision in quantum linear systems solvers for non-sparse matrices. In addition, we develop a technique of variable-time amplitude estimation, based on Ambainis' variable-time amplitude amplification technique, which we are also able to apply within the framework. As applications, we design the following algorithms: (1) a quantum algorithm for the quantum weighted least squares problem, exhibiting a 6-th power improvement in the dependence on the condition number and an exponential improvement in the dependence on the precision over the previous best algorithm of Kerenidis and Prakash; (2) the first quantum algorithm for the quantum generalized least squares problem; and (3) quantum algorithms for estimating electrical-network quantities, including effective resistance and dissipated power, improving upon previous work.