Quantum differential equation solvers: limitations and fast-forwarding

TL;DR

This paper investigates the limitations of quantum algorithms for linear ODEs, establishing lower bounds and proposing methods for exponential speed-ups in specific cases, especially for inhomogeneous equations.

Contribution

It introduces a framework for proving lower bounds on quantum algorithms and develops novel fast-forwarding algorithms that avoid traditional discretization and high-dimensional linear system solutions.

Findings

Quantum algorithms face fundamental overheads due to non-quantumness.

Homogeneous ODEs without non-quantumness are equivalent to quantum dynamics.

Exponential speed-ups are achievable for certain inhomogeneous ODEs with efficient eigensystem implementations.

Abstract

We study the limitations and fast-forwarding of quantum algorithms for linear ordinary differential equation (ODE) systems with a particular focus on non-quantum dynamics, where the coefficient matrix in the ODE is not anti-Hermitian or the ODE is inhomogeneous. On the one hand, for generic linear ODEs, by proving worst-case lower bounds, we show that quantum algorithms suffer from computational overheads due to two types of ``non-quantumness'': real part gap and non-normality of the coefficient matrix. We then show that homogeneous ODEs in the absence of both types of ``non-quantumness'' are equivalent to quantum dynamics, and reach the conclusion that quantum algorithms for quantum dynamics work best. To obtain these lower bounds, we propose a general framework for proving lower bounds on quantum algorithms that are amplifiers, meaning that they amplify the difference between a pair of input quantum states. On the other hand, we show how to fast-forward quantum algorithms for solving special classes of ODEs which leads to improved efficiency. More specifically, we obtain exponential improvements in both $T$ and the spectral norm of the coefficient matrix for inhomogeneous ODEs with efficiently implementable eigensystems, including various spatially discretized linear evolutionary partial differential equations. We give fast-forwarding algorithms that are conceptually different from existing ones in the sense that they neither require time discretization nor solving high-dimensional linear systems.