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Saturation and Recurrence of Quantum Complexity in Random Local Quantum Dynamics

Michał Oszmaniec1,*, Marcin Kotowski1,†, Michał Horodecki2,‡, and Nicholas Hunter-Jones3,§

  • *Contact author: oszmaniec@cft.edu.pl
  • †Contact author: mkotowski@cft.edu.pl
  • ‡Contact author: michal.horodecki@ug.edu.pl
  • §Contact author: nickrhj@stanford.edu
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Phys. Rev. X 14, 041068 – Published 24 December, 2024

DOI: https://doi.org/10.1103/PhysRevX.14.041068

Abstract

Quantum complexity is a measure of the minimal number of elementary operations required to approximately prepare a given state or unitary channel. Recently, this concept has found applications beyond quantum computing—in studying the dynamics of quantum many-body systems and the long-time properties of anti–de Sitter black holes. In this context, Brown and Susskind [Phys. Rev. D 97, 086015 (2018)] conjectured that the complexity of a chaotic quantum system grows linearly in time up to times exponential in the system size, saturating at a maximal value, and remaining maximally complex until undergoing recurrences at doubly exponential times. In this work, we prove the saturation and recurrence of complexity in two models of chaotic time evolutions based on (i) random local quantum circuits and (ii) stochastic local Hamiltonian evolution. Our results advance an understanding of the long-time behavior of chaotic quantum systems and could shed light on the physics of black-hole interiors. From a technical perspective, our results are based on establishing new quantitative connections between the Haar measure and high-degree approximate designs, as well as the fact that random quantum circuits of sufficiently high depth converge to approximate designs.

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Popular Summary

Quantum complexity, a concept originating from theoretical computer science, has been shedding new light on the behavior of chaotic quantum systems and the dynamics of black holes. Circuit complexity quantifies the difficulty of creating specific quantum states or transformations using elementary quantum gates, the building blocks of quantum computation. A long-standing question in physics has been how circuit complexity evolves over time in chaotic systems. Specifically, the Brown-Susskind conjecture proposes that this complexity grows linearly for an exponentially long time, reaches a maximum value, and remains maximally complex until eventually returning to a small value at doubly exponential times. Our work provides rigorous proof of key aspects of this conjecture in two significant models of chaotic quantum evolution.

We focus our investigation on systems evolving either by random quantum circuits or their continuous time analog. A random quantum circuit is composed of a sequence of quantum gates, where each gate is chosen at random from a set of simple gates. Using methods from the theory of Markov chains and high-dimensional probability, we demonstrate that, in such systems, the circuit complexity indeed saturates at a maximum value and remains there for an exponentially long time until recurring at doubly exponential times.

We hope that the techniques and insights in this work will enable progress on the problem of complexity in more realistic quantum chaotic systems. In particular, the goal would be to show a similar kind of behavior as in our paper in systems governed purely by local Hamiltonian dynamics, when there is no randomness constantly injected into the system. Such systems are ubiquitous in many-body physics and quantum information theory and serve as a model of black-hole dynamics.

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Phys. Rev. X 14, 041068– Published 24 December, 2024
  • Received 5 December 2022
  • Revised 18 August 2024
  • Accepted 11 September 2024
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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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