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  • Open Access

Probabilistic quantum algorithm for Lyapunov equations and matrix inversion

Marcello Benedetti*, Ansis Rosmanis†, and Matthias Rosenkranz‡

  • Quantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom

  • *Contact author: marcello.benedetti@quantinuum.com
  • †Contact author: ansis.rosmanis@quantinuum.com
  • ‡Contact author: matthias.rosenkranz@quantinuum.com
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Phys. Rev. Applied 25, 044040 – Published 16 April, 2026

DOI: https://doi.org/10.1103/mprk-9vx3

Abstract

We present a probabilistic quantum algorithm for preparing mixed states which, in expectation, are proportional to the solutions of Lyapunov equations—linear matrix equations ubiquitous in the analysis of classical and quantum dynamical systems. Building on previous results by Zhang et al., arXiv:2304.04526, at each step the algorithm can (i) return the current state, (ii) apply a trace nonincreasing completely positive map, or (iii) restart. We introduce a deterministic stopping rule, which leads to an efficient algorithm with a bounded expected number of calls to oracles representing the two input matrices of the Lyapunov equations. We also consider preparing a mixed state that approximates the normalized inverse of a positive definite matrix A. In its most general form, the algorithm generates mixed states, which approximate matrix-valued weighted sums and integrals. It can be shown that block encodings and states yield two incomparable computational resources even when they represent the same piece of data. While block encodings of functions have received much attention in the literature, our work takes a step toward the less explored problem of encoding functions into mixed states.

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Phys. Rev. Applied 25, 044040– Published 16 April, 2026
  • Received 29 August 2025
  • Revised 11 February 2026
  • Accepted 9 March 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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