Quantum framework for parameterizing partial differential equations via diagonal block-encoding

TL;DR

This paper introduces a quantum algorithmic framework that uses diagonal block-encodings to efficiently simulate and optimize solutions to parameterized partial differential equations, demonstrated on the 2D wave equation.

Contribution

It develops a novel quantum encoding method for PDE coefficients that enables efficient simulation and optimization for a broad class of problems with diagonalizable parameter fields.

Findings

Successfully simulated 2D wave equation with Gaussian profile

Extended quantum simulation to parameter-dependent PDEs

Provided numerical illustration of the framework's effectiveness

Abstract

We study a quantum-algorithmic framework for parameterizing partial differential equations (PDEs). For a broad class of problems in which the discretized parameter field admits a diagonal representation, block-encodings of diagonal matrices, or diagonal block-encodings, can be used to represent spatially varying coefficients with structured, potentially complicated profiles. This encoding enables efficient quantum simulation of forward PDEs and extends naturally to parameter-dependent settings. Such simulations are a key primitive for quantum algorithms for PDE-constrained optimization, where the goal is to identify optimal design parameters. We illustrate the framework numerically through forward simulation and parameter design for the two-dimensional wave equation with a Gaussian parameter profile.