Statistical Features of Chaos in Quantum Evolutions

Sudhir R. Jain

Asim Sharma, Avah Banerjee

We investigate the emergence of quantum chaos and unitary T-design behavior in derandomized Clifford+T circuits using causal cover architectures. Motivated by the need for deterministic constructions that can exhibit chaotic behavior across diverse quantum hardware platforms, we explore deterministic Clifford circuit architectures (random Clifford circuits with causal cover, bitonic sorting networks, and permutation-based routing circuits) to drive quantum circuits toward Wigner-Dyson (WD) entanglement spectrum statistics and OTOC decay.Our experiments demonstrate that causal connectivity, not circuit depth or randomness, is a critical feature that drives circuits to chaos. We show that initializing with n T-states and adding a second T-layer after a causally covered Clifford evolution yields consistent OTOC decay and WD statistics. This also enables deeper understanding of the circuit structures that generate complex entanglement behavior. Notably, our work suggests polylogarithmic-depth deterministic circuits suffice to approximate chaotic behavior, highlighting that causal connectivity is sufficient for operator spreading to induce Wigner-Dyson entanglement statistics and OTOC decay.

Marcin Płodzień

Information scrambling, the process by which quantum information spreads and becomes effectively inaccessible, is central to modern quantum statistical physics and quantum chaos. These lecture notes provide an introduction to information scrambling from both static and dynamical perspectives. The spectral properties of reduced density matrices arising from Haar-random states are developed through the geometry of the unitary group and the universal results of random matrix theory. This geometric framework yields universal, model-independent predictions for entanglement and spectral statistics, capturing generic features of quantum chaos without reference to microscopic details. Dynamical diagnostics such as the spectral form factor and out-of-time-ordered correlators further reveal the onset of chaos in time-dependent evolution. The notes are aimed at advanced undergraduate and graduate students in physics, mathematics, and computer science who are interested in the connections between quantum chaos, information dynamics, and quantum computing. Concepts such as entanglement growth, Haar randomness, random-matrix statistics, and unitary t-designs are introduced through their realization in random quantum circuits and circuit complexity. The same mathematical framework also underpins modern quantum-device benchmarking, where approximate unitary designs and random circuits translate geometric ideas into quantitative tools for assessing fidelity, noise, and scrambling efficiency in real quantum processors.

L. E. Ballentine, J. P. Zibin

Sensitivity of the time evolution to small changes in the state is a characteristic feature of classical chaos. It has been believed that state sensitivity could not exist in quantum mechanics because of the unitary invariance of the Hilbert-space overlap of states. We argue that this Hilbert-space criterion is irrelevant and show that both quantum states and classical statistical states exhibit a similar kind of state sensitivity. This is demonstrated by the degree to which the initial state can be recovered in computational motion reversal: forward evolution for a time T, perturbation of the state, and backward time evolution. Some differences between classical and quantum state sensitivity remain, and these seem to be insensitive to decoherence. \textcopyright{} 1996 The American Physical Society.

P. Lebœuf

Sandro Wimberger