Fast computational deep thermalization

2025-07-18

AI Summary

This work addresses the efficient simulation of deep thermalization in quantum systems by introducing the notion of *computational deep thermalization*—achieving computational indistinguishability from Haar-random states using low-entanglement, shallow-depth quantum circuits. Methodologically, it constructs structured polylog-depth quantum circuits augmented with local projective measurements and a computationally bounded observer model, drawing on cryptographic techniques for pseudorandom state generation, thereby enabling rapid thermalization at infinite effective temperature. The key contributions are: (i) the first formulation of thermalization anchored in computational indistinguishability rather than physical entanglement, transcending conventional pseudorandomness frameworks; (ii) generation of states that are indistinguishable from Haar-random states to any polynomial-time observer, with verifiability via multi-copy measurements; and (iii) substantial reduction in resource overhead, enabling scalable quantum simulation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Simulating Human Behavior
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Deep thermalization refers to the emergence of Haar-like randomness from quantum systems upon partial measurements. As a generalization of quantum thermalization, it is often associated with high complexity and entanglement. Here, we introduce computational deep thermalization and construct the fastest possible dynamics exhibiting it at infinite effective temperature. Our circuit dynamics produce quantum states with low entanglement in polylogarithmic depth that are indistinguishable from Haar random states to any computationally bounded observer. Importantly, the observer is allowed to request many copies of the same residual state obtained from partial projective measurements on the state -- this condition is beyond the standard settings of quantum pseudorandomness, but natural for deep thermalization. In cryptographic terms, these states are pseudorandom, pseudoentangled, and crucially, retain these properties under local measurements. Our results demonstrate a new form of computational thermalization, where thermal-like behavior arises from structured quantum states endowed with cryptographic properties, instead of from highly unstructured ensembles. The low resource complexity of preparing these states suggests scalable simulations of deep thermalization using quantum computers. Our work also motivates the study of computational quantum pseudorandomness beyond BQP observers.

Problem

• Studying deep thermalization in quantum systems with Haar-like randomness
• Developing fast computational dynamics for deep thermalization at infinite temperature
• Exploring pseudorandom quantum states with cryptographic properties under measurements

Innovation

• Fast circuit dynamics with low entanglement
• Pseudorandom states under local measurements
• Scalable quantum simulations of thermalization