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  • Open Access

From Linear Differential Equations to Unitaries: A Moment-Matching Dilation Framework with Near-Optimal Quantum Algorithms

Xiantao Li*

  • *Contact author: Xiantao.Li@psu.edu
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PRX Quantum 7, 020350 – Published 4 June, 2026

DOI: https://doi.org/10.1103/xm61-ytf7

Abstract

We introduce a universal dilation framework to simulate arbitrary non-Hermitian dynamics x˙=L(t)x on unitary quantum processors. By imposing algebraic moment-matching conditions on an ancilla, our framework generates diverse families of dilation schemes that unify existing methods and broaden the design space for hardware-aware implementations. A tight-binding dilation that maps the ancilla to a 1D lattice with nearest-neighbor hopping is derived, overcoming the connectivity bottlenecks of dilation schemes. This construction handles general dynamics, including gain, without a priori rescaling and achieves near-optimal complexity, as demonstrated by benchmarks on dissipative wave propagations.

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Many important scientific models are governed by linear differential equations whose evolution is not unitary, for example, because of dissipation, gain and loss, or effective reduced dynamics. Quantum computers, however, naturally implement unitary evolution, which makes such problems difficult to simulate directly. In this work, we develop a general framework for converting broad classes of nonunitary linear dynamics into larger unitary dynamics that can be implemented on a quantum computer. Our framework unifies several recently proposed approaches and also generates new dilation families with different mathematical structures. We then study one particularly useful family in detail, showing that it leads to quantum algorithms with near-optimal complexity and a natural tight-binding interpretation. The results provide both a unifying perspective on recent methods and a new route toward efficient quantum simulation of nonunitary processes arising in scientific computing and physics.

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PRX Quantum 7, 020350– Published 4 June, 2026
  • Received 9 December 2025
  • Revised 25 March 2026
  • Accepted 12 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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