Qubit-Efficient Quantum Algorithm for Linear Differential Equations

TL;DR

This paper introduces a quantum algorithm for solving linear differential equations that is efficient in qubit usage and hardware-friendly, suitable for near-term quantum devices, with applications to non-Hermitian models.

Contribution

The paper presents a qubit-efficient, locality-preserving quantum algorithm for linear ODEs with provable runtime guarantees, suitable for near-term quantum hardware.

Findings

Uses only one ancilla qubit

Maintains locality when the coefficient matrix is local

Applicable to the Hatano-Nelson model

Abstract

As quantum hardware rapidly advances toward the early fault-tolerant era, a key challenge is to develop quantum algorithms that are not only theoretically sound but also hardware-friendly on near-term devices. In this work, we propose a quantum algorithm for solving linear ordinary differential equations (ODEs) with a provable runtime guarantee. Our algorithm uses only a single ancilla qubit, and is locality preserving, i.e., when the coefficient matrix of the ODE is $k$-local, the algorithm only needs to implement the time evolution of $(k+1)$-local Hamiltonians. We also discuss the connection between our proposed algorithm and Lindbladian simulation as well as its application to the interacting Hatano-Nelson model, a widely studied non-Hermitian model with rich phenomenology.