Fast computational deep thermalization

TL;DR

This work shows that fast computational deep thermalization is attainable with polylogarithmic-depth quantum circuits that generate states with low entanglement yet Haar-like statistics for polynomially many copies. By building two families of states—Schmidt-patched states and large-brick phase states—the authors prove Haar-indistinguishability for both global and projected ensembles, while maintaining entanglement at or near a sublinear, quasi-area-law scale. They establish rigorous bounds on the $t$-th moments and demonstrate that projected ensembles across most cuts behave like Haar random states, enabling deep thermalization to arise from computational pseudorandomness rather than chaotic or highly entropic dynamics. The work further discusses robustness beyond BQP, suggesting the possibility of pseudorandomness persisting against stronger adversaries such as PDQP and outlining open questions for finite-temperature deep thermalization and practical quantum simulations. Together, these results provide a scalable, cryptography-inspired route to deeply thermalized quantum states with potential implications for benchmarking, state certification, and quantum simulations of many-body 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.

Figures (3)

• Figure 1: Two-layered brickwork circuit for preparing a large-brick phase state. The circuit applies $m$ bricks to the equal superposition state of $n$ qubits. Each brick is of size $2b$ and applies a phase gate in the computational basis, constructed out of pseudorandom functions $f_i, g_i \in \{0,1\}^{2b}\mapsto\{0,1\}$, such that each $x_i\in \{0,1\}^b$. In this figure, $m=3$ and the final state is of $n=2mb=6b$ qubits.
• Figure S1: Quantum circuit for preparing a Schmidt patch subset state
• Figure S2: Example of the notions introduced in Definition \ref{['def:bipartitecomponents']}. A subset of $[\sqrt{N}]\times [\sqrt{N}]$ can be naturally viewed as a bipartite graph; in the above example, the subset is $T=\{(1,1), (1,2), (2,2), (2,3), (7,5), (7,6)\}$. According to Definition \ref{['def:bipartitecomponents']}, the bipartite components are $X_1=\{1,2\}, Y_1=\{1,2,3\}$, and $X_2=\{7\},Y_2=\{5,6\}$. The number of unique half-strings would be $u_T=8$.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2: Large-brick phase states are pseudorandom and pseudoentangled
• Definition S3
• Definition S4
• Definition S5: Subset-phase state aaronson_et_al:LIPIcs.ITCS.2024.2
• Lemma S6: Subset distributions for Haar-like subset-phase states
• Lemma S7: Subset distributions for Haar-like subset state ensembles
• Lemma S8: Proximity to Haar
• Definition S9: Doubly-unique $t$-subsets
• Remark