Fast computational deep thermalization

TL;DR

This paper introduces the fastest quantum circuit dynamics that produce states mimicking Haar randomness with low entanglement, cryptographic properties, and scalability, revealing a new form of computational deep thermalization.

Contribution

It constructs the fastest known dynamics exhibiting computational deep thermalization with low entanglement and pseudorandom properties at infinite temperature.

Findings

States are indistinguishable from Haar random to bounded observers

States maintain pseudorandomness under local measurements

Preparation of states is resource-efficient and scalable

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.