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Parallel-in-time quantum simulation via Page and Wootters quantum time

N. L. Diaz1,2,3,*, Paolo Braccia4,5, Martin Larocca4,6, J. M. Matera2, R. Rossignoli2,7, and M. Cerezo1

  • *Contact author: nldiaz.unlp@gmail.com
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Phys. Rev. Research 7, 033294 – Published 29 September, 2025

DOI: https://doi.org/10.1103/wpnf-4nnn

Abstract

In the past few decades, researchers have created a veritable zoo of quantum algorithms by drawing inspiration from classical computing, information theory, and even from physical phenomena. Here, we present quantum algorithms for parallel-in-time simulations that are inspired by the Page and Wootters formalism. In this framework, and thus in our algorithms, the classical time variable of quantum mechanics is promoted to the quantum realm by introducing a Hilbert space of “clock” qubits that are then entangled with the “system” qubits. We show that our algorithms can compute temporal properties over N different times of many-body systems by only using log(N) clock qubits. As such, we achieve an exponential trade-off between time and spatial complexities. In addition, we rigorously prove that the entanglement created between the system qubits and the clock qubits has operational meaning, as it encodes valuable information about the system’s dynamics. We also provide a circuit depth estimation of all the protocols, showing a running time advantage in computation times over traditional sequential-in-time algorithms. In particular, for the case when the dynamics are determined by the Aubry-Andre model, we present a hybrid method for which our algorithms have a depth that only scales as O(log(N)n). As a by-product, we can relate the previous schemes to the problem of equilibration of an isolated quantum system, thus indicating that our framework enables a new dimension for studying dynamical properties of many-body systems.

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Phys. Rev. Research 7, 033294– Published 29 September, 2025
  • Received 30 December 2023
  • Accepted 29 August 2025
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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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