{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T04:04:33Z","timestamp":1777953873410,"version":"3.51.4"},"reference-count":80,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,19]],"date-time":"2021-08-19T00:00:00Z","timestamp":1629331200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2014000"],"award-info":[{"award-number":["2014000"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We show that the most important measures of quantum chaos, such as frame potentials, scrambling, Loschmidt echo and out-of-time-order correlators (OTOCs), can be described by the unified framework of the isospectral twirling, namely the Haar average of a k-fold unitary channel. We show that such measures can then always be cast in the form of an expectation value of the isospectral twirling. In literature, quantum chaos is investigated sometimes through the spectrum and some other times through the eigenvectors of the Hamiltonian generating the dynamics. We show that thanks to this technique, we can interpolate smoothly between integrable Hamiltonians and quantum chaotic Hamiltonians. The isospectral twirling of Hamiltonians with eigenvector stabilizer states does not possess chaotic features, unlike those Hamiltonians whose eigenvectors are taken from the Haar measure. As an example, OTOCs obtained with Clifford resources decay to higher values compared with universal resources. By doping Hamiltonians with non-Clifford resources, we show a crossover in the OTOC behavior between a class of integrable models and quantum chaos. Moreover, exploiting random matrix theory, we show that these measures of quantum chaos clearly distinguish the finite time behavior of probes to quantum chaos corresponding to chaotic spectra given by the Gaussian Unitary Ensemble (GUE) from the integrable spectra given by Poisson distribution and the Gaussian Diagonal Ensemble (GDE).<\/jats:p>","DOI":"10.3390\/e23081073","type":"journal-article","created":{"date-parts":[[2021,8,19]],"date-time":"2021-08-19T09:58:06Z","timestamp":1629367086000},"page":"1073","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Isospectral Twirling and Quantum Chaos"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0334-7419","authenticated-orcid":false,"given":"Lorenzo","family":"Leone","sequence":"first","affiliation":[{"name":"Physics Department, University of Massachusetts Boston, Boston, MA 02125, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3569-085X","authenticated-orcid":false,"given":"Salvatore F. E.","family":"Oliviero","sequence":"additional","affiliation":[{"name":"Physics Department, University of Massachusetts Boston, Boston, MA 02125, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0662-719X","authenticated-orcid":false,"given":"Alioscia","family":"Hamma","sequence":"additional","affiliation":[{"name":"Physics Department, University of Massachusetts Boston, Boston, MA 02125, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,19]]},"reference":[{"key":"ref_1","unstructured":"Lloyd, S. (1988). Black Holes, Demons and the Loss of Coherence: How Complex Systems Get Information, and What They Do with It. [Ph.D. Thesis, Rockefeller University]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1038\/nature06838","article-title":"Thermalization and its mechanism for generic isolated quantum systems","volume":"452","author":"Rigol","year":"2008","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"036206","DOI":"10.1103\/PhysRevE.81.036206","article-title":"Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization","volume":"81","author":"Santos","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"754","DOI":"10.1038\/nphys444","article-title":"Entanglement and the foundations of statistical mechanics","volume":"2","author":"Popescu","year":"2006","journal-title":"Nat. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1103\/PhysRevE.50.888","article-title":"Chaos and quantum thermalization","volume":"50","author":"Srednicki","year":"1994","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"160404","DOI":"10.1103\/PhysRevLett.99.160404","article-title":"Typicality for Generalized Microcanonical Ensembles","volume":"99","author":"Reimann","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1038\/nphys3215","article-title":"Quantum many-body systems out of equilibrium","volume":"11","author":"Eisert","year":"2015","journal-title":"Nat. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1103\/RevModPhys.83.863","article-title":"Colloquium: Nonequilibrium dynamics of closed interacting quantum systems","volume":"83","author":"Polkovnikov","year":"2011","journal-title":"Rev. Mod. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"113039","DOI":"10.1088\/1367-2630\/aaed68","article-title":"Relaxation, chaos, and thermalization in a three-mode model of a Bose\u2013Einstein condensate","volume":"20","author":"Bonneau","year":"2018","journal-title":"New J. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1007\/s10955-016-1511-2","article-title":"Typicality of Thermal Equilibrium and Thermalization in Isolated Macroscopic Quantum Systems","volume":"163","author":"Tasaki","year":"2016","journal-title":"J. Stat. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"010403","DOI":"10.1103\/PhysRevLett.115.010403","article-title":"Generalization of von Neumann\u2019s Approach to Thermalization","volume":"115","author":"Reimann","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1007\/JHEP02(2016)004","article-title":"Chaos in quantum channels","volume":"2016","author":"Hosur","year":"2016","journal-title":"J. High Energy Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/JHEP12(2016)145","article-title":"Conditional mutual information of bipartite unitaries and scrambling","volume":"2016","author":"Ding","year":"2016","journal-title":"J. High Energy Phys."},{"key":"ref_14","unstructured":"Brown, W.G., and Fawzi, O. (2013). Scrambling speed of random quantum circuits. arXiv."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/JHEP07(2018)041","article-title":"Entanglement, quantum randomness, and complexity beyond scrambling","volume":"2018","author":"Liu","year":"2018","journal-title":"J. High Energy Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"130502","DOI":"10.1103\/PhysRevLett.120.130502","article-title":"Generalized Entanglement Entropies of Quantum Designs","volume":"120","author":"Liu","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Styliaris, G., Anand, N., and Zanardi, P. (2020). Information Scrambling over Bipartitions: Equilibration, Entropy Production, and Typicality. arXiv.","DOI":"10.1103\/PhysRevLett.126.030601"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1088\/1126-6708\/2007\/09\/120","article-title":"Black holes as mirrors: Quantum information in random subsystems","volume":"2007","author":"Hayden","year":"2007","journal-title":"J. High Energy Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1007\/JHEP05(2015)132","article-title":"Stringy effects in scrambling","volume":"2015","author":"Shenker","year":"2015","journal-title":"J. High Energy Phys."},{"key":"ref_20","unstructured":"Kitaev, A. (2021, August 15). Hidden Correlations in the Hawking Radiation and Thermal Noise. Talk Given at the Fundamental Physics Prize Symposium, 2014. Available online: https:\/\/online.kitp.ucsb.edu\/online\/joint98\/kitaev\/."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1007\/JHEP05(2017)118","article-title":"Black holes and random matrices","volume":"2017","author":"Cotler","year":"2017","journal-title":"J. High Energy Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"020408","DOI":"10.1103\/PhysRevB.96.020408","article-title":"Entanglement complexity in quantum many-body dynamics, thermalization, and localization","volume":"96","author":"Yang","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"240501","DOI":"10.1103\/PhysRevLett.112.240501","article-title":"Emergent Irreversibility and Entanglement Spectrum Statistics","volume":"112","author":"Chamon","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/s00220-009-0873-6","article-title":"Random Quantum Circuits are Approximate 2-designs","volume":"291","author":"Harrow","year":"2009","journal-title":"Commun. Math. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1007\/JHEP07(2018)124","article-title":"Onset of random matrix behavior in scrambling systems","volume":"2018","author":"Gharibyan","year":"2018","journal-title":"J. High Energy Phys."},{"key":"ref_26","unstructured":"Brown, W.G. (2010). Random Quantum Dynamics: From Random Quantum Circuits to Quantum Chaos. [Ph.D. Thesis, Dartmouth College]."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"250501","DOI":"10.1103\/PhysRevLett.104.250501","article-title":"Convergence Rates for Arbitrary Statistical Moments of Random Quantum Circuits","volume":"104","author":"Brown","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","first-page":"021014","article-title":"Operator Spreading in Random Unitary Circuits","volume":"8","author":"Nahum","year":"2018","journal-title":"Phys. Rev. X"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"086015","DOI":"10.1103\/PhysRevD.97.086015","article-title":"Second law of quantum complexity","volume":"97","author":"Brown","year":"2018","journal-title":"Phys. Rev. D"},{"key":"ref_30","first-page":"861","article-title":"The geometry of quantum computation","volume":"8","author":"Dowling","year":"2008","journal-title":"Quantum Inf. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"094206","DOI":"10.1103\/PhysRevB.95.094206","article-title":"Operator entanglement entropy of the time evolution operator in chaotic systems","volume":"95","author":"Zhou","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"025201","DOI":"10.1103\/PhysRevE.79.025201","article-title":"How complex is quantum motion?","volume":"79","author":"Benenti","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1007\/JHEP08(2016)106","article-title":"A bound on chaos","volume":"2016","author":"Maldacena","year":"2016","journal-title":"J. High Energy Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1007\/JHEP04(2013)022","article-title":"Towards the fast scrambling conjecture","volume":"2013","author":"Lashkari","year":"2013","journal-title":"J. High Energy Phys."},{"key":"ref_35","first-page":"031048","article-title":"Locality, Quantum Fluctuations, and Scrambling","volume":"9","author":"Xu","year":"2019","journal-title":"Phys. Rev. X"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Anand, N., Styliaris, G., Kumari, M., and Zanardi, P. (2020). Quantum coherence as a signature of chaos. arXiv.","DOI":"10.1103\/PhysRevResearch.3.023214"},{"key":"ref_37","first-page":"1200","article-title":"Quasiclassical method in the theory of superconductivity","volume":"28","author":"Larkin","year":"1969","journal-title":"J. Exp. Theor. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"144304","DOI":"10.1103\/PhysRevB.97.144304","article-title":"Out-of-time-ordered correlators in a quantum Ising chain","volume":"97","author":"Lin","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"127","DOI":"10.22331\/q-2019-03-04-127","article-title":"Work Statistics, Loschmidt Echo and Information Scrambling in Chaotic Quantum Systems","volume":"3","author":"Chenu","year":"2019","journal-title":"Quantum"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"035005","DOI":"10.1088\/2058-9565\/ab8ebb","article-title":"Quantum scrambling and the growth of mutual information","volume":"5","author":"Touil","year":"2020","journal-title":"Quantum Sci. Technol."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"032114","DOI":"10.1103\/PhysRevA.85.032114","article-title":"Universality and robustness of revivals in the transverse field XY model","volume":"85","author":"Hamma","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"040302","DOI":"10.1103\/PhysRevA.94.040302","article-title":"Measuring the scrambling of quantum information","volume":"94","author":"Swingle","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_43","first-page":"021013","article-title":"Operator Hydrodynamics, OTOCs, and Entanglement Growth in Systems without Conservation Laws","volume":"8","author":"Rakovszky","year":"2018","journal-title":"Phys. Rev. X"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"988","DOI":"10.1038\/s41567-018-0295-5","article-title":"Unscrambling the physics of out-of-time-order correlators","volume":"14","author":"Swingle","year":"2018","journal-title":"Nat. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"010601","DOI":"10.1103\/PhysRevLett.123.010601","article-title":"Finite-Size Scaling of Out-of-Time-Ordered Correlators at Late Times","volume":"123","author":"Huang","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"055308","DOI":"10.1088\/1751-8113\/41\/5\/055308","article-title":"Optimizing quantum process tomography with unitary 2-designs","volume":"41","author":"Scott","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/JHEP04(2017)121","article-title":"Chaos and complexity by design","volume":"2017","author":"Roberts","year":"2017","journal-title":"J. High Energy Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"790","DOI":"10.1017\/S0305004100027237","article-title":"On the statistical distribution of the widths and spacings of nuclear resonance levels","volume":"47","author":"Wigner","year":"1951","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Haake, F., Gnutzmann, S., and Ku\u015b, M. (2018). Quantum Signatures of Chaos, Springer International Publishing.","DOI":"10.1007\/978-3-319-97580-1"},{"key":"ref_50","unstructured":"Mehta, M.L. (1991). Random Matrices, Elsevier."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Tao, T. (2012). Topics in Random Matrix Theory, American Mathematical Society.","DOI":"10.1090\/gsm\/132"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"054202","DOI":"10.1103\/PhysRevB.102.054202","article-title":"Higher-order level spacings in random matrix theory based on Wigner\u2019s conjecture","volume":"102","author":"Rao","year":"2020","journal-title":"Phys. Rev. B"},{"key":"ref_53","unstructured":"Chen, X., and Zhou, T. (2018). Operator scrambling and quantum chaos. arXiv."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1007\/JHEP11(2017)048","article-title":"Chaos, complexity, and random matrices","volume":"2017","author":"Cotler","year":"2017","journal-title":"J. High Energy Phys."},{"key":"ref_55","unstructured":"Hunter-Jones, N.R. (2018). Chaos and Randomness in Strongly-Interacting Quantum Systems. [Ph.D. Thesis, California Institute of Technology]."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"185009","DOI":"10.1088\/0264-9381\/31\/18\/185009","article-title":"Black holes, entanglement and random matrices","volume":"31","author":"Balasubramanian","year":"2014","journal-title":"Class. Quantum Gravity"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"085103","DOI":"10.1088\/1572-9494\/ab8a28","article-title":"Out-of-time-order correlators in the one-dimensional XY model","volume":"72","author":"Bao","year":"2020","journal-title":"Commun. Theor. Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"032106","DOI":"10.1103\/PhysRevE.94.032106","article-title":"Integrable matrix theory: Level statistics","volume":"94","author":"Scaramazza","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Riser, R., and Kanzieper, E. (2020). Power spectrum and form factor in random diagonal matrices and integrable billiards. arXiv.","DOI":"10.1016\/j.aop.2020.168393"},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Prakash, A., Pixley, J.H., and Kulkarni, M. (2020). The universal spectral form factor for many-body localization. arXiv.","DOI":"10.1103\/PhysRevResearch.3.L012019"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"168065","DOI":"10.1016\/j.aop.2019.168065","article-title":"Nonperturbative theory of power spectrum in complex systems","volume":"413","author":"Riser","year":"2020","journal-title":"Ann. Phys."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"031101","DOI":"10.1103\/PhysRevE.86.031101","article-title":"Convergence to equilibrium under a random Hamiltonian","volume":"86","author":"Horodecki","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"023095","DOI":"10.1103\/PhysRevResearch.2.023095","article-title":"Random quantum batteries","volume":"2","author":"Caravelli","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1155\/S107379280320917X","article-title":"Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability","volume":"2003","author":"Collins","year":"2003","journal-title":"Int. Math. Res. Not."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"170502","DOI":"10.1103\/PhysRevLett.121.170502","article-title":"Recovering Quantum Gates from Few Average Gate Fidelities","volume":"121","author":"Roth","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"453","DOI":"10.22331\/q-2021-05-04-453","article-title":"Quantum Chaos is Quantum","volume":"5","author":"Leone","year":"2021","journal-title":"Quantum"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"76","DOI":"10.21468\/SciPostPhys.10.3.076","article-title":"Random Matrix Theory of the Isospectral twirling","volume":"10","author":"Oliviero","year":"2021","journal-title":"SciPost Phys."},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Gemmer, J., Michel, M., and Mahler, G. (2009). Quantum Thermodynamics, Springer.","DOI":"10.1007\/978-3-540-70510-9"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"R5185","DOI":"10.1103\/PhysRevE.51.R5185","article-title":"Normalization sum rule and spontaneous breaking of U(N) invariance in random matrix ensembles","volume":"51","author":"Canali","year":"1995","journal-title":"Phys. Rev. E"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"R3291","DOI":"10.1103\/PhysRevE.61.R3291","article-title":"Spontaneous symmetry breaking in U(N) invariant ensembles with a soft confinement potential","volume":"61","author":"Pato","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"086026","DOI":"10.1103\/PhysRevD.98.086026","article-title":"Spectral form factors and late time quantum chaos","volume":"98","author":"Liu","year":"2018","journal-title":"Phys. Rev. D"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"160603","DOI":"10.1103\/PhysRevLett.124.160603","article-title":"Information Scrambling and Loschmidt Echo","volume":"124","author":"Yan","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_73","unstructured":"Bhattacharyya, A., Chemissany, W., Haque, S.S., and Yan, B. (2019). Towards the Web of Quantum Chaos Diagnostics. arXiv."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"1610","DOI":"10.1103\/PhysRevA.30.1610","article-title":"Stability of quantum motion in chaotic and regular systems","volume":"30","author":"Peres","year":"1984","journal-title":"Phys. Rev. A"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"214101","DOI":"10.1103\/PhysRevLett.89.214101","article-title":"Border between Regular and Chaotic Quantum Dynamics","volume":"89","author":"Weinstein","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"1334","DOI":"10.1016\/j.physleta.2009.02.022","article-title":"Long-time fidelity and chaos for a kicked nonlinear oscillator system","volume":"373","author":"Kalaga","year":"2009","journal-title":"Phys. Lett. A"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0024-3795(75)90075-0","article-title":"Completely positive linear maps on complex matrices","volume":"10","author":"Choi","year":"1975","journal-title":"Linear Algebra Its Appl."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"040304","DOI":"10.1103\/PhysRevA.63.040304","article-title":"Entanglement of quantum evolutions","volume":"63","author":"Zanardi","year":"2001","journal-title":"Phys. Rev. A"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"87","DOI":"10.21468\/SciPostPhys.9.6.087","article-title":"Single T gate in a Clifford circuit drives transition to universal entanglement spectrum statistics","volume":"9","author":"Zhou","year":"2020","journal-title":"SciPost Phys."},{"key":"ref_80","doi-asserted-by":"crossref","unstructured":"Watrous, J. (2018). The Theory of Quantum Information, Cambridge University Press.","DOI":"10.1017\/9781316848142"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1073\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:47:13Z","timestamp":1760165233000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1073"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,19]]},"references-count":80,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["e23081073"],"URL":"https:\/\/doi.org\/10.3390\/e23081073","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,19]]}}}