Measurement Schemes for Quantum Linear Equation Solvers

TL;DR

This paper introduces a measurement scheme for quantum linear solvers tailored for CFD problems, reducing measurement overhead by focusing on significant amplitudes, and demonstrates its effectiveness through simulation.

Contribution

It proposes a novel measurement scheme combining QSP-based amplitude estimation with QSVT for CFD, optimizing resource use and accuracy.

Findings

Focusing on large amplitudes reduces measurement costs.

Simulations show acceptable error levels with fewer measurements.

Resource estimation guides practical implementation of quantum CFD solvers.

Abstract

Solving Computational Fluid Dynamics (CFD) problems requires the inversion of a linear system of equations, which can be done using a quantum algorithm for matrix inversion arxiv:1806.01838. However, the number of shots required to measure the output of the system can be prohibitive and remove any advantage obtained by quantum computing. In this work we propose a scheme for measuring the output of QSVT matrix inversion algorithms specifically for the CFD use case. We use a Quantum Signal Processing (QSP) based amplitude estimation algorithm arxiv:2207.08628 and show how it can be combined with the QSVT matrix inversion algorithm. We perform a detailed resource estimation of the amount of computational resources required for a single iteration of amplitude estimation, and compare the costs of amplitude estimation with the cost of not doing amplitude estimation and measuring the whole wavefunction. We also propose a measurement scheme to reduce the number of amplitudes measured in the CFD example by focusing on large amplitudes only. We simulate the whole CFD loop, finding that thus measuring only a small number of the total amplitudes in the output vector still results in an acceptable level of overall error.