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Non-unitary time evolution via the Chebyshev expansion method

Áron Holló1,2, Dániel Varjas3,4,5, Cosma Fulga3,4, László Oroszlány1,2, and Viktor Könye3,4,6

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Phys. Rev. E 113, 035301 – Published 4 March, 2026

DOI: https://doi.org/10.1103/nbxh-j277

Abstract

The Chebyshev expansion method is a well-established technique for computing the time evolution of quantum states, particularly in Hermitian systems with a bounded spectrum. Here, we show that the applicability of the Chebyshev expansion method extends well beyond this constraint: It remains valid across the entire complex plane and is thus suitable for arbitrary non-Hermitian matrices. We identify numerical rounding errors as the primary source of errors encountered when applying the method outside the conventional spectral bounds, and they are not caused by fundamental limitations. By carefully selecting the spectral radius and the time step, we show how these errors can be effectively suppressed, enabling accurate time evolution calculations in non-Hermitian systems. We derive an analytic upper bound for the rounding error, which serves as a practical guideline for selecting time steps in numerical simulations. As an application, we illustrate the performance of the method by computing the time evolution of wave packets in the Hatano-Nelson model.

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Phys. Rev. E 113, 035301– Published 4 March, 2026
  • Received 15 October 2025
  • Accepted 28 January 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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