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October 8, 2025Open Access

Qubit-Efficient Quantum Algorithm for Linear Differential Equations

Key Points

The proposed algorithm effectively solves linear ordinary differential equations using only one ancilla qubit, enhancing qubit efficiency.
It achieves locality preservation, needing time evolution for $(k+1)$-local Hamiltonians when the coefficient matrix is $k$-local.
The runtime guarantee provides a theoretical framework for the algorithm's performance, showcasing a new approach to ODEs.
Linkages to Lindbladian simulation and applications in the interacting Hatano-Nelson model highlight the algorithm's practical relevance.

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.

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