Quantum communication complexity of linear regression

TL;DR

This paper demonstrates that quantum computers can achieve significant polynomial and exponential reductions in communication complexity for linear regression and Hamiltonian simulation, surpassing classical methods under certain conditions.

Contribution

It introduces new quantum protocols for linear regression and Hamiltonian simulation, establishing lower bounds and proposing an efficient quantum singular value transformation technique.

Findings

Quantum protocols outperform classical in communication complexity for certain linear algebra problems.

Quantum lower bounds match the performance of proposed protocols, indicating near-optimality.

The work advances quantum algorithms for fundamental linear algebra tasks.

Abstract

Quantum computers may achieve speedups over their classical counterparts for solving linear algebra problems. However, in some cases -- such as for low-rank matrices -- dequantized algorithms demonstrate that there cannot be an exponential quantum speedup. In this work, we show that quantum computers have provable polynomial and exponential speedups in terms of communication complexity for some fundamental linear algebra problems \update{if there is no restriction on the rank}. We mainly focus on solving linear regression and Hamiltonian simulation. In the quantum case, the task is to prepare the quantum state of the result. To allow for a fair comparison, in the classical case, the task is to sample from the result. We investigate these two problems in two-party and multiparty models, propose near-optimal quantum protocols and prove quantum/classical lower bounds. In this process, we propose an efficient quantum protocol for quantum singular value transformation, which is a powerful technique for designing quantum algorithms. This will be helpful in developing efficient quantum protocols for many other problems.