Error and Resource Estimates of Variational Quantum Algorithms for Solving Differential Equations Based on Runge-Kutta Methods

TL;DR

This paper provides a detailed error and resource analysis for variational quantum algorithms that use Runge-Kutta methods to solve differential equations, aiming to optimize their efficiency on near-term quantum devices.

Contribution

It offers the first rigorous error and resource estimates for variational Runge-Kutta quantum algorithms, including shot noise and truncation errors, guiding their practical implementation.

Findings

Order 4 methods are most resource-efficient for 1D ODEs.

Order 2 methods are optimal for option pricing PDEs.

Analysis excludes hardware noise and representation errors.

Abstract

A focus of recent research in quantum computing has been on developing quantum algorithms for differential equations solving using variational methods on near-term quantum devices. A promising approach involves variational algorithms, which combine classical Runge-Kutta methods with quantum computations. However, a rigorous error analysis, essential for assessing real-world feasibility, has so far been lacking. In this paper, we provide an extensive analysis of error sources and determine the resource requirements needed to achieve specific target errors. In particular, we derive analytical error and resource estimates for scenarios with and without shot noise, examining shot noise in quantum measurements and truncation errors in Runge-Kutta methods. Our analysis does not take into account representation errors and hardware noise, as these are specific to the instance and the used device. We evaluate the implications of our results by applying them to two scenarios: classically solving a $1$D ordinary differential equation and solving an option pricing linear partial differential equation with the variational algorithm, showing that the most resource-efficient methods are of order 4 and 2, respectively. This work provides a framework for optimizing quantum resources when applying Runge-Kutta methods, enhancing their efficiency and accuracy in both solving differential equations and simulating quantum systems.