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Approximate quantum error correction, eigenstate thermalization and the chaos bound

Shozab Qasim, Jason Pollack

TL;DR

This work unifies quantum information and many-body physics by deriving a quantitative link between eigenstate thermalization (ETH), information scrambling, and approximate quantum error correction (AQECC). Building on the ETH matrix structure and the chaos bound, it shows that the AQECC code error is tightly controlled by the Lyapunov exponent via $\varepsilon_{\mathrm{code}} \lesssim 2^{d+2k} \exp\left(-\frac{S}{4} - \frac{\pi}{2\lambda}|\omega|\right)$, tying scrambling dynamics to information preservation. It further derives dynamical and static fluctuation bounds and fluctuation–dissipation relations from ETH, illuminating how chaotic dynamics constrain observable fluctuations in thermalizing systems. The results suggest deep connections to holography, random tensor networks, and algebraic quantum error correction, and open avenues for a hydrodynamic description of AQECC in chaotic media.

Abstract

Quantum error correction, thermalization, and quantum chaos are fundamental aspects of quantum many-body physics that have each developed largely independently, despite their deep conceptual overlap. In this work, we establish a precise link between all three in systems that satisfy the eigenstate thermalization hypothesis (ETH) and exhibit a well-defined hierarchy of time scales between dissipation and scrambling. Building on the ETH matrix ansatz and the structure of the out-of-time-order correlator (OTOC), we show that the chaos bound directly constrains the error of an approximate quantum error-correcting code. This establishes a quantitative relation between information scrambling, thermalization, and correctability. Furthermore, we derive bounds on dynamical fluctuations around the infinite-time average and on fluctuation-dissipation relations, expressed in terms of both the code error and the Lyapunov exponent. Our results reveal how the limits of quantum chaos constrain information preservation in thermalizing quantum systems.

Approximate quantum error correction, eigenstate thermalization and the chaos bound

TL;DR

This work unifies quantum information and many-body physics by deriving a quantitative link between eigenstate thermalization (ETH), information scrambling, and approximate quantum error correction (AQECC). Building on the ETH matrix structure and the chaos bound, it shows that the AQECC code error is tightly controlled by the Lyapunov exponent via , tying scrambling dynamics to information preservation. It further derives dynamical and static fluctuation bounds and fluctuation–dissipation relations from ETH, illuminating how chaotic dynamics constrain observable fluctuations in thermalizing systems. The results suggest deep connections to holography, random tensor networks, and algebraic quantum error correction, and open avenues for a hydrodynamic description of AQECC in chaotic media.

Abstract

Quantum error correction, thermalization, and quantum chaos are fundamental aspects of quantum many-body physics that have each developed largely independently, despite their deep conceptual overlap. In this work, we establish a precise link between all three in systems that satisfy the eigenstate thermalization hypothesis (ETH) and exhibit a well-defined hierarchy of time scales between dissipation and scrambling. Building on the ETH matrix ansatz and the structure of the out-of-time-order correlator (OTOC), we show that the chaos bound directly constrains the error of an approximate quantum error-correcting code. This establishes a quantitative relation between information scrambling, thermalization, and correctability. Furthermore, we derive bounds on dynamical fluctuations around the infinite-time average and on fluctuation-dissipation relations, expressed in terms of both the code error and the Lyapunov exponent. Our results reveal how the limits of quantum chaos constrain information preservation in thermalizing quantum systems.

Paper Structure

This paper contains 12 sections, 4 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Relation to previous results
  3. Organization
  4. Preliminaries
  5. Eigenstate Thermalization Hypothesis (ETH) and fluctuations
  6. Bound on chaos from ETH
  7. Approximate Quantum Error-Correcting Codes (AQECC)
  8. Results
  9. Quantum error correction and the chaos bound
  10. Dynamical and static fluctuations
  11. Fluctuation-dissipation relations
  12. Discussion

Key Result

Proposition 1

(Brand_o_2019, Proposition 4) A code defined by the projector $P$ is said to be $\epsilon$-correctable under a noise channel $\Lambda$ iff where $\lambda_{ij}$ are the components of a density operator, and $d(\mathcal{N} + \mathcal{B}, \mathcal{N}) \leq \epsilon$, where $\mathcal{N}(\rho)=\sum_{i,j} \lambda_{ij} \Tr(\rho)\ket{i}\bra{j}$ and $(\mathcal{N}+\mathcal{B}(\rho))=\mathcal{N}(\rho) + \su

Theorems & Definitions (7)

  • Proposition 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof