The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation
Shantanav Chakraborty, András Gilyén, Stacey Jeffery
TL;DR
This work unifies quantum linear-algebra tasks under the block-encoding framework, enabling input models ranging from sparse access to quantum data structures to be encoded as unitaries and manipulated efficiently. By introducing variable-time amplitude amplification/estimation, the authors achieve exponential or polylogarithmic improvements in precision dependence and improved condition-number scaling for quantum linear system solvers and least-squares problems. The paper provides concrete quantum algorithms for weighted and generalized least squares, along with quantum-estimation methods for electrical-network quantities, achieving advantages in both sparse and data-structure input models. Overall, the approach yields faster, more versatile quantum regression techniques and network-analysis tools with broad applicability across quantum machine learning and quantum simulation domains. The results advance practical quantum algorithms by enabling efficient handling of non-sparse inputs and by reducing sensitivity to precision and conditioning in key problems.$
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.