Bridging Classical and Quantum Information Scrambling with the Operator Entanglement Spectrum

Ben T. McDonough; Claudio Chamon; Justin H. Wilson; Thomas Iadecola

doi:10.22331/q-2026-03-05-2012

Bridging Classical and Quantum Information Scrambling with the Operator Entanglement Spectrum

Ben T. McDonough^1,2, Claudio Chamon³, Justin H. Wilson^4,5, and Thomas Iadecola^6,2,7,8,9

¹Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA
²Ames National Laboratory, Ames, IA 50011, USA
³Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA
⁴Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA
⁵Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA
⁶Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA
⁷Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA
⁸Institute for Computational and Data Sciences, The Pennsylvania State University, University Park, PA 16802, USA
⁹Materials Research Institute, The Pennsylvania State University, University Park, PA 16802, USA

Published:	2026-03-05, volume 10, page 2012
Editor:	Daniel Malz
Eprint:	arXiv:2505.05575v3
Doi:	https://doi.org/10.22331/q-2026-03-05-2012
Citation:	Quantum 10, 2012 (2026).

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Abstract

Universal features of chaotic quantum dynamics underlie our understanding of thermalization in closed quantum systems and the complexity of quantum computations. Reversible automaton circuits, comprised of classical logic gates, have emerged as a tractable means to study such dynamics. Despite generating no entanglement in the computational basis, these circuits nevertheless capture many features expected from fully quantum evolutions. In this work, we demonstrate that the differences between automaton dynamics and fully quantum dynamics are revealed by the operator entanglement spectrum, much like the entanglement spectrum of a quantum state distinguishes between the dynamics of states under Clifford and Haar random circuits. While the operator entanglement spectrum under random unitary dynamics is governed by the eigenvalue statistics of random Gaussian matrices, we show evidence that under random automaton dynamics it is described by the statistics of Bernoulli random matrices, whose entries are random variables taking values $0$ or $1$. We study the crossover between automaton and generic unitary operator dynamics as the automaton circuit is doped with gates that introduce superpositions, namely Hadamard or $R_x = e^{-i\frac{\pi}{4}X}$ gates. We find that a constant number of superposition-generating gates is sufficient to drive the operator dynamics to the random-circuit universality class, similar to earlier results on Clifford circuits doped with $T$ gates. This establishes the operator entanglement spectrum as a useful tool for probing the chaoticity and universality class of quantum dynamics.

Popular summary

Identifying universal features of chaotic quantum evolution has been an active pursuit of physics in the last decade, with applications ranging from quantum cryptography to the quantum dynamics of black holes. In the quest to model these dynamics, automaton circuits—classical reversible binary gates viewed as unitary quantum operators—have been introduced in recent years as a computationally tractable model sharing many of the features of random unitary circuits. Despite their similarity, it has remained unclear what features distinguish these two classes of quantum dynamics. We identify a universal feature of chaotic quantum evolution, the operator entanglement spectrum (OES), and show that it establishes a definitive difference between generic unitary and reversible classical dynamics. Furthermore, automaton circuits represent universal classical logic and form a complete quantum gate set when a superposition-generating "Hadamard" gate is added. We find that a finite number of Hadamard gates drives the OES from the automaton circuit class exponentially close to the random unitary class, creating a bridge between classical and quantum information chaos. Since the OES quantifies delocalization in operator space, our characterization of the OES in automaton circuits may have applications to classifying the hardness of classical and quantum cryptographic protocols.

► BibTeX data

@article{McDonough2026bridgingclassical,
  doi = {10.22331/q-2026-03-05-2012},
  url = {https://doi.org/10.22331/q-2026-03-05-2012},
  title = {Bridging {C}lassical and {Q}uantum {I}nformation {S}crambling with the {O}perator {E}ntanglement {S}pectrum},
  author = {McDonough, Ben T. and Chamon, Claudio and Wilson, Justin H. and Iadecola, Thomas},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2012},
  month = mar,
  year = {2026}
}

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[3] Bruno Bertini, Katja Klobas, Pavel Kos, and Daniel Malz, "Random permutation circuits beyond qubits are quantum chaotic", Physical Review B 113 10, L100302 (2026).

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