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Bridging Classical and Quantum Information Scrambling with the Operator Entanglement Spectrum

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Researchers identified the operator entanglement spectrum (OES) as a definitive marker distinguishing classical automaton circuits from fully quantum dynamics, resolving a key ambiguity in chaotic system modeling. Automaton circuits—classical reversible gates—exhibit OES statistics matching Bernoulli random matrices (binary 0/1 entries), unlike quantum circuits governed by Gaussian random matrix eigenvalues, revealing fundamental differences in information scrambling. Adding even a constant number of superposition-generating gates (e.g., Hadamard or Rₓ) to automaton circuits drives their OES toward universal quantum chaos, mirroring prior Clifford-T gate transitions. The study bridges classical and quantum chaos, showing minimal quantum resources suffice to achieve full universality, with implications for quantum advantage in cryptography and thermalization studies. This work establishes OES as a robust tool for classifying dynamical universality, quantifying operator delocalization, and assessing computational hardness in hybrid classical-quantum systems.

Bridging Classical and Quantum Information Scrambling with the Operator Entanglement Spectrum

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AbstractUniversal 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 summaryIdentifying 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} }► References [1] Xiao Mi, Pedram Roushan, Chris Quintana, Salvatore Mandra, Jeffrey Marshall, Charles Neill, Frank Arute, Kunal Arya, Juan Atalaya, Ryan Babbush, et al. ``Information scrambling in quantum circuits''. 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AbstractUniversal 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 summaryIdentifying 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} }► References [1] Xiao Mi, Pedram Roushan, Chris Quintana, Salvatore Mandra, Jeffrey Marshall, Charles Neill, Frank Arute, Kunal Arya, Juan Atalaya, Ryan Babbush, et al. ``Information scrambling in quantum circuits''. Science 374, 1479–1483 (2021). https://doi.org/10.1126/science.abg5029 [2] Matthew PA Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay. ``Random quantum circuits''. Annual Review of Condensed Matter Physics 14, 335–379 (2023). https://doi.org/10.1146/annurev-conmatphys-031720-030658 [3] Jordan Cotler, Nicholas Hunter-Jones, Junyu Liu, and Beni Yoshida. ``Chaos, complexity, and random matrices''. Journal of High Energy Physics 2017, 1–60 (2017). https://doi.org/10.1007/JHEP11(2017)048 [4] Scott D. Geraedts, Rahul Nandkishore, and Nicolas Regnault. ``Many-body localization and thermalization: Insights from the entanglement spectrum''. Phys. Rev. B 93, 174202 (2016). https://doi.org/10.1103/PhysRevB.93.174202 [5] Zhi-Cheng Yang, Alioscia Hamma, Salvatore M. Giampaolo, Eduardo R. Mucciolo, and Claudio Chamon. ``Entanglement complexity in quantum many-body dynamics, thermalization, and localization''. Phys. Rev. B 96, 020408 (2017). https://doi.org/10.1103/PhysRevB.96.020408 [6] Claudio Chamon, Alioscia Hamma, and Eduardo R. Mucciolo. ``Emergent irreversibility and entanglement spectrum statistics''. Phys. Rev. Lett. 112, 240501 (2014). https://doi.org/10.1103/PhysRevLett.112.240501 [7] Po-Yao Chang, Xiao Chen, Sarang Gopalakrishnan, and J. H. Pixley. ``Evolution of entanglement spectra under generic quantum dynamics''. Phys. Rev. Lett. 123, 190602 (2019). https://doi.org/10.1103/PhysRevLett.123.190602 [8] Daniel Shaffer, Claudio Chamon, Alioscia Hamma, and Eduardo R Mucciolo. ``Irreversibility and entanglement spectrum statistics in quantum circuits''. Journal of Statistical Mechanics: Theory and Experiment 2014, P12007 (2014). https://doi.org/10.1088/1742-5468/2014/12/P12007 [9] T. Rakovszky, S. Gopalakrishnan, S. A. Parameswaran, and F. Pollmann. ``Signatures of information scrambling in the dynamics of the entanglement spectrum''. Phys. Rev. B 100, 125115 (2019). https://doi.org/10.1103/PhysRevB.100.125115 [10] Xiao Chen and Andreas W. W. Ludwig. ``Universal spectral correlations in the chaotic wave function and the development of quantum chaos''. Phys. Rev. B 98, 064309 (2018). https://doi.org/10.1103/PhysRevB.98.064309 [11] S D Geraedts, N Regnault, and R M Nandkishore. ``Characterizing the many-body localization transition using the entanglement spectrum''. New Journal of Physics 19, 113021 (2017). https://doi.org/10.1088/1367-2630/aa93a5 [12] Shiyu Zhou, Zhicheng Yang, Alioscia Hamma, and Claudio Chamon. ``Single T gate in a Clifford circuit drives transition to universal entanglement spectrum statistics''. SciPost Physics 9, 087 (2020). https://doi.org/10.21468/SciPostPhys.9.6.087 [13] Adam Nahum, Sagar Vijay, and Jeongwan Haah. ``Operator spreading in random unitary circuits''. Phys. Rev. X 8, 021014 (2018). https://doi.org/10.1103/PhysRevX.8.021014 [14] Pavan Hosur, Xiao-Liang Qi, Daniel A Roberts, and Beni Yoshida. ``Chaos in quantum channels''. Journal of High Energy Physics 2016, 1–49 (2016). https://doi.org/10.1007/JHEP02(2016)004 [15] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. ``Quantum entanglement growth under random unitary dynamics''. Phys. Rev. X 7, 031016 (2017). https://doi.org/10.1103/PhysRevX.7.031016 [16] Xiaoguang Wang and Paolo Zanardi. ``Quantum entanglement of unitary operators on bipartite systems''. Phys. Rev. A 66, 044303 (2002). https://doi.org/10.1103/PhysRevA.66.044303 [17] Paolo Zanardi. ``Entanglement of quantum evolutions''. Phys. Rev. A 63, 040304 (2001). https://doi.org/10.1103/PhysRevA.63.040304 [18] Shenglong Xu and Brian Swingle. ``Accessing scrambling using matrix product operators''. Nat. Phys. 16, 199–204 (2019). https://doi.org/10.1038/s41567-019-0712-4 [19] Georgios Styliaris, Namit Anand, and Paolo Zanardi. ``Information scrambling over bipartitions: Equilibration, entropy production, and typicality''. Phys. Rev. Lett. 126, 030601 (2021). https://doi.org/10.1103/PhysRevLett.126.030601 [20] Shenglong Xu and Brian Swingle. ``Scrambling dynamics and out-of-time-ordered correlators in quantum many-body systems''. PRX Quantum 5, 010201 (2024). https://doi.org/10.1103/PRXQuantum.5.010201 [21] Iztok Pižorn and Tomaž Prosen. ``Operator space entanglement entropy in $xy$ spin chains''. Phys. Rev. B 79, 184416 (2009). https://doi.org/10.1103/PhysRevB.79.184416 [22] Tomaž Prosen. ``Chaos and complexity of quantum motion''. Journal of Physics A: Mathematical and Theoretical 40, 7881–7918 (2007). https://doi.org/10.1088/1751-8113/40/28/s02 [23] Tomaž Prosen and Iztok Pižorn. ``Operator space entanglement entropy in a transverse Ising chain''. Phys. Rev. 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Bridging Classical and Quantum Information Scrambling with the Operator Entanglement Spectrum

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