Quantum Error Correction, Eigenstate Thermalization, and Chaos Bound Demonstrate Link Between Scrambling and Correctability

The interplay between error correction, thermalisation, and chaos represents a long-standing puzzle in physics, with each area developing as a largely separate field of study despite underlying connections. Shozab Qasim from Freie Universitat Berlin and Jason Pollack from Syracuse University now demonstrate a precise link between these three concepts, revealing how they are fundamentally intertwined in systems that satisfy the eigenstate thermalisation hypothesis. Their work establishes that the established chaos bound directly limits the error rate of approximate quantum error-correcting codes, providing a quantitative relationship between information scrambling, thermalisation, and the ability to preserve information. This achievement reveals how the inherent limits of chaotic behaviour constrain information preservation within thermalising systems, offering new insights into the fundamental boundaries of quantum information processing.

This work demonstrates that these seemingly independent aspects of physics are fundamentally interconnected, revealing a unified microscopic structure governing their behavior. The research builds upon the ETH framework, connecting it to the ability to protect quantum information from noise using approximate quantum error correction (AQEC). Researchers demonstrate that the limits of chaos directly constrain the performance of these approximate error-correcting codes, establishing a quantifiable relationship between how quickly information spreads, how systems reach equilibrium, and how well information can be preserved.

A key concept is scrambling, which describes how information disperses rapidly within a chaotic quantum system. The team shows that the rate of scrambling, quantified by the Lyapunov exponent, directly influences the accuracy of approximate error correction, meaning more chaotic systems present greater challenges for preserving quantum information. Furthermore, the research connects these findings to holographic duality and random tensor networks, suggesting that the ability to protect quantum information is an intrinsic property of chaotic systems that satisfy ETH. The team suggests a potential connection to hydrodynamics, hinting that the laws of thermodynamics might emerge from the underlying chaotic dynamics of quantum systems. Understanding this connection could have implications for designing more robust quantum computers and for understanding how information is preserved in black holes. This research opens up new avenues for exploring algebraic quantum error correction and extending the analysis to different error models.

Error, Chaos, and Thermalization are Unified

Scientists have established a precise link between error correction, thermalization, and quantum chaos in systems that adhere to the eigenstate thermalization hypothesis (ETH). This work demonstrates that these seemingly independent aspects of physics are fundamentally interconnected, revealing a unified microscopic structure governing their behavior. Researchers began with the ETH framework and its equivalence to approximate quantum error correction (AQEC), then showed that the error associated with the approximate error correction is directly controlled by the Lyapunov exponent and the chaos bound, establishing a quantifiable relationship between the code error, system size, and the rate of chaos. The research reveals that dynamical fluctuations, static fluctuations, and fluctuation-dissipation relations can all be expressed in terms of the Lyapunov exponent and the chaos bound, demonstrating how the structure of quantum chaos constrains information preservation within chaotic eigenstates. The team emphasizes that this analysis holds for systems exhibiting a well-defined hierarchy of timescales, where the scrambling time significantly exceeds the dissipation time, a condition often found in large or finely tuned systems. This detailed connection between the code error and fundamental quantities provides a new tool for understanding and quantifying the limits of information preservation in complex quantum systems.

Chaos, Thermalization, and Quantum Error Correction

This research establishes a fundamental connection between error correction, thermalization, and chaos in quantum physics, demonstrating that these seemingly disparate concepts are deeply intertwined within systems adhering to the eigenstate thermalization hypothesis. By building upon the ETH matrix ansatz and analysing the out-of-time-order correlator, scientists have shown that the limits of chaos directly constrain the performance of approximate error-correcting codes, establishing a quantifiable relationship between how quickly information spreads, how systems reach equilibrium, and how well information can be preserved. Furthermore, the team derived bounds on both dynamical fluctuations around average behaviour and on fluctuation-dissipation relations, expressing these in terms of both code error and the Lyapunov exponent, a measure of chaotic behaviour. These findings reveal how the inherent limits of chaos restrict the preservation of information within thermalizing systems, offering new insights into the fundamental limits of information processing in complex quantum systems. The authors acknowledge that their results apply specifically to systems satisfying the ETH, and future work could explore the extent to which these connections hold in more general scenarios, potentially extending our understanding of non-equilibrium dynamics and the emergence of statistical mechanics in isolated quantum systems. This research provides a crucial step towards understanding the fundamental limits of information processing in complex quantum systems and opens up new avenues for designing more robust quantum technologies.