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Experimental Signatures of Hilbert-Space Ergodicity: Universal Bitstring Distributions and Applications in Noise Learning

Adam L. Shaw1,*, Daniel K. Mark2,*, Joonhee Choi3, Ran Finkelstein1, Pascal Scholl1, Soonwon Choi2,†, and Manuel Endres1,‡

  • *These authors contributed equally to this work.
  • †Contact author: soonwon@mit.edu
  • ‡Contact author: mendres@caltech.edu
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Phys. Rev. X 15, 031001 – Published 1 July, 2025

DOI: https://doi.org/10.1103/h6xy-zpx4

Abstract

Systems reaching thermal equilibrium are ubiquitous. For classical systems, this phenomenon is typically understood statistically through ergodicity in phase space, but translating this to quantum systems is a long-standing problem of interest. Recently, a strong notion of quantum ergodicity has been proposed, namely, that isolated, global quantum states uniformly explore their available state space, dubbed Hilbert-space ergodicity. Here, we observe signatures of this process with an experimental Rydberg quantum simulator and various numerical models, before generalizing to the case of a local quantum system interacting with its environment. For a closed system, where the environment is a complementary subsystem, we predict and observe a smooth quantum-to-classical transition in that observables progress from large, quantum fluctuations to small, Gaussian fluctuations as the bath size grows. This transition exhibits universal properties on a quantitative level among a wide range of systems, including those at finite temperature, those with itinerant particles, and random circuits. For an open system, where the environment is uncontrolled, we predict the statistics of observables under largely arbitrary noise channels including those with correlated errors, allowing us to discriminate between candidate error models both for continuous Hamiltonian time evolution and for digital random circuits. This allows for computationally efficient experimental noise learning and, more broadly, is a new avenue for quantitatively classifying the behavior of noisy quantum systems. Ultimately, our results clarify the role of ergodicity in quantum dynamics, with fundamental and practical consequences.

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Popular Summary

A major question in quantum physics is how systems made of many particles reach thermal equilibrium, given that quantum mechanics itself is reversible. Traditional approaches get around this by focusing only on part of the system and treating the rest as a heat bath. In this study, we take a different route by measuring the entire system using an analog quantum simulator, allowing us to directly examine how quantum systems behave as they evolve over time.

We test the idea of “Hilbert-space ergodicity,” which suggests that a quantum state, when evolving chaotically, spreads out evenly across all the states it can reach. Interestingly, while local parts of the system appear to reach thermal equilibrium with tiny fluctuations, global measurements of the full system still show large, ongoing fluctuations. By gradually moving from measuring local to global properties, we observe a smooth change in behavior. These fluctuations turn out to be universal—they do not depend on the specific system used. We also use this understanding to study how environmental noise affects the system. Small deviations from ideal behavior help us identify and model the types of noise in our setup.

Our findings reveal new insights into how quantum systems evolve and how noise affects them. This helps not only with understanding the foundations of quantum mechanics but also with improving the accuracy of quantum experiments. We anticipate that these methods could help calibrate quantum devices and study the fundamental properties of open quantum systems.

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Phys. Rev. X 15, 031001– Published 1 July, 2025
  • Received 21 September 2024
  • Revised 28 April 2025
  • Accepted 29 May 2025
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