Design nearly optimal quantum algorithm for linear differential equations via Lindbladians

TL;DR

This paper introduces a nearly optimal quantum algorithm for solving linear differential equations by utilizing Lindbladians and non-diagonal density matrix encoding, outperforming previous methods and enabling applications like Gibbs state preparation.

Contribution

The paper presents a novel quantum algorithm for linear ODEs using Lindbladians and non-diagonal density matrix encoding, achieving near-optimal performance and broad applications.

Findings

Outperforms all existing quantum ODE algorithms

Achieves near-optimal dependence on parameters

Enables applications like Gibbs state preparation

Abstract

Solving linear ordinary differential equations (ODE) is one of the most promising applications for quantum computers to demonstrate exponential advantages. The challenge of designing a quantum ODE algorithm is how to embed non-unitary dynamics into intrinsically unitary quantum circuits. In this work, we propose a new quantum algorithm for solving ODEs by harnessing open quantum systems. Specifically, we propose a novel technique called non-diagonal density matrix encoding, which leverages the inherent non-unitary dynamics of Lindbladians to encode general linear ODEs into the non-diagonal blocks of density matrices. This framework enables us to design quantum algorithms with both theoretical simplicity and high performance. Combined with the state-of-the-art quantum Lindbladian simulation algorithms, our algorithm can outperform all existing quantum ODE algorithms and achieve near-optimal dependence on all parameters under a plausible input model. We also give applications of our algorithm including the Gibbs state preparations and the partition function estimations.