Quantum Multi-Resolution Measurement with application to Quantum Linear Solver

TL;DR

The paper introduces a quantum multi-resolution measurement method that significantly reduces the measurement cost for quantum solutions, demonstrated through a quantum linear solver application.

Contribution

It proposes a hybrid quantum-classical multi-resolution measurement approach that lowers measurement complexity from O(n/ε^2) to O(n log(1/ε)), with an application to quantum linear systems.

Findings

QMRM achieves solution accuracy with fewer measurements.

Numerical experiments validate the efficiency of QMRM.

Application to quantum linear solver demonstrates practical benefits.

Abstract

Quantum computation consists of a quantum state corresponding to a solution, and measurements with some observables. To obtain a solution with an accuracy $\epsilon$, measurements $O(n/\epsilon^2)$ are required, where $n$ is the size of a problem. The cost of these measurements requires a large computing time for an accurate solution. In this paper, we propose a quantum multi-resolution measurement (QMRM), which is a hybrid quantum-classical algorithm that gives a solution with an accuracy $\epsilon$ in $O(n\log(1/\epsilon))$ measurements using a pair of functions. The QMRM computational cost with an accuracy $\epsilon$ is smaller than $O(n/\epsilon^2)$. We also propose an algorithm entitled QMRM-QLS (quantum linear solver) for solving a linear system of equations using the Harrow-Hassidim-Lloyd (HHL) algorithm as one of the examples. We perform some numerical experiments that QMRM gives solutions to with an accuracy $\epsilon$ in $O(n\log(1/\epsilon))$ measurements.