Operator space fragmentation in perturbed Floquet-Clifford circuits
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AbstractFloquet quantum circuits are able to realise a wide range of non-equilibrium quantum states, exhibiting quantum chaos, topological order and localisation. In this work, we investigate the stability of operator localisation and emergence of chaos in random Floquet-Clifford circuits subjected to unitary perturbations which drive them away from the Clifford limit. We construct a nearest-neighbour Clifford circuit with a brickwork pattern and study the effect of including disordered non-Clifford gates. The perturbations are uniformly sampled from single-qubit unitaries with probability $p$ on each qubit. We show that the interacting model exhibits strong localisation of operators for $0 \le{p} \lt{1}$ that is characterised by the fragmentation of operator space into disjoint sectors due to the appearance of wall configurations. Such walls give rise to emergent local integrals of motion for the circuit that we construct exactly. We analytically establish the stability of localisation against generic perturbations and calculate the average length of operator spreading tunable by $p$. Although our circuit is not separable across any bi-partition, we further show that the operator localisation leads to an entanglement bottleneck, where initially unentangled states remain weakly entangled across typical fragment boundaries. Finally, we study the spectral form factor (SFF) to characterise the chaotic properties of the operator fragments and spectral fluctuations as a probe of non-ergodicity. In the $p = 1$ model, the emergence of a fragmentation time scale is found before random matrix theory sets in after which the SFF can be approximated by that of the circular unitary ensemble. Our work provides an explicit description of quantum phases in operator dynamics and circuit ergodicity which can be realised on current NISQ devices.Featured image: Circuit model exhibiting fragmented (non-ergodic) circuit evolution.Popular summaryMany-body quantum dynamics concerns the time-evolution of large interacting ensembles. Ergodicity-breaking from localisation is a fundamental problem within this field referring to the long-time preservation of locally encoded information. Understanding the conditions under which many-body systems exhibit localisation remains an open problem. In particular, significant theoretical effort has been dedicated to understanding the role of interaction strength, symmetry and disorder on the stability of localised quantum evolution which is analytically challenging in Hamiltonian models. This paper concerns brickwork quantum circuits as a toy model to study robust operator localisation in time-periodic disordered systems. Quantum circuits have emerged as minimally structured dynamical many-body systems due to their high degree of analytical tractability and the rich complexity of dynamical phases they can realise. They are also prime candidates for benchmarking contemporary quantum computers for simulating complex quantum evolution. Our work studies a robust ergodicity-breaking mechanism in brickwork Clifford unitaries where the emergence of chaotic evolution is obstructed even when the circuit is perturbed with local random gates. We analytically study local constraints emerging in the disordered circuit ensemble using Clifford group techniques from quantum computing theory. Our analytical predictions are then verified by numerically simulating the dynamics of entanglement entropy and many-body spectral correlations to understand observable features of the ergodicity-breaking dynamics and its stability. Our work sits at the intersection of quantum information theory and many-body physics with our analytical insights offering a mathematically salient description of stable ergodicity-breaking in the large system and long-time limit.► BibTeX data@article{Kovacs2026operatorspace, doi = {10.22331/q-2026-05-18-2107}, url = {https://doi.org/10.22331/q-2026-05-18-2107}, title = {Operator space fragmentation in perturbed {F}loquet-{C}lifford circuits}, author = {Kov{\'{a}}cs, Marcell D. and Turner, Christopher J. and Masanes, Lluis and Pal, Arijeet}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2107}, month = may, year = {2026} }► References [1] Daniel A Roberts and Beni Yoshida. ``Chaos and complexity by design''. Journal of High Energy Physics 2017, 1–64 (2017). https://doi.org/10.1007/JHEP04(2017)121 [2] J. Eisert. ``Entangling power and quantum circuit complexity''. Phys. Rev. Lett. 127, 020501 (2021). https://doi.org/10.1103/PhysRevLett.127.020501 [3] Wen Wei Ho and Soonwon Choi. ``Exact emergent quantum state designs from quantum chaotic dynamics''. Phys. Rev. Lett. 128, 060601 (2022). https://doi.org/10.1103/PhysRevLett.128.060601 [4] Jonas Haferkamp, Philippe Faist, Naga BT Kothakonda, Jens Eisert, and Nicole Yunger Halpern. ``Linear growth of quantum circuit complexity''. Nature Physics 18, 528–532 (2022). https://doi.org/10.1038/s41567-022-01539-6 [5] Matthew P.A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay. ``Random quantum circuits''. Annu. Rev. Condens. Matter Phys. 14, 335–379 (2023). https://doi.org/10.1146/annurev-conmatphys-031720-030658 [6] Dominik Hahn and Luis Colmenarez. ``Absence of localization in weakly interacting Floquet circuits''. Phys. Rev. B 109, 094207 (2024). https://doi.org/10.1103/PhysRevB.109.094207 [7] Chao-Ming Jian, Yi-Zhuang You, Romain Vasseur, and Andreas W. W. Ludwig. ``Measurement-induced criticality in random quantum circuits''. Phys. Rev. B 101, 104302 (2020). https://doi.org/10.1103/PhysRevB.101.104302 [8] 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 [9] 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 [10] Yaodong Li, Xiao Chen, and Matthew P. A. Fisher. ``Quantum Zeno effect and the many-body entanglement transition''. Phys. Rev. B 98, 205136 (2018). https://doi.org/10.1103/PhysRevB.98.205136 [11] Oliver Lunt, Marcin Szyniszewski, and Arijeet Pal. ``Measurement-induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional Clifford circuits''. Phys. Rev. B 104, 155111 (2021). https://doi.org/10.1103/PhysRevB.104.155111 [12] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Rev. Mod. Phys. 81, 865–942 (2009). https://doi.org/10.1103/RevModPhys.81.865 [13] Thomas Jennewein, Christoph Simon, Gregor Weihs, Harald Weinfurter, and Anton Zeilinger. ``Quantum cryptography with entangled photons''. Phys. Rev. Lett. 84, 4729–4732 (2000). https://doi.org/10.1103/PhysRevLett.84.4729 [14] Juan Yin, Yu-Huai Li, Sheng-Kai Liao, Meng Yang, Yuan Cao, Liang Zhang, Ji-Gang Ren, Wen-Qi Cai, Wei-Yue Liu, Shuang-Lin Li, et al. ``Entanglement-based secure quantum cryptography over 1,120 kilometres''. Nature 582, 501–505 (2020). https://doi.org/10.1038/s41586-020-2401-y [15] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. ``Mixed-state entanglement and quantum error correction''. Phys. Rev. A 54, 3824–3851 (1996). https://doi.org/10.1103/PhysRevA.54.3824 [16] Barbara M. Terhal. ``Quantum error correction for quantum memories''. Rev. Mod. Phys. 87, 307–346 (2015). https://doi.org/10.1103/RevModPhys.87.307 [17] Adam M Kaufman, M Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko, Philipp M Preiss, and Markus Greiner. ``Quantum thermalization through entanglement in an isolated many-body system''. Science 353, 794–800 (2016). https://doi.org/10.1126/science.aaf6725 [18] Rahul Nandkishore and David A Huse. ``Many-body localization and thermalization in quantum statistical mechanics''. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015). https://doi.org/10.1146/annurev-conmatphys-031214-014726 [19] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn. ``Colloquium: Many-body localization, thermalization, and entanglement''. Rev. Mod. Phys. 91, 021001 (2019). https://doi.org/10.1103/RevModPhys.91.021001 [20] Bruno Bertini, Pavel Kos, and Tomaž Prosen. ``Entanglement Spreading in a Minimal Model of Maximal Many-Body Quantum Chaos''. Phys. Rev. X 9, 021033 (2019). https://doi.org/10.1103/PhysRevX.9.021033 [21] Bruno Bertini, Pavel Kos, and Tomaž Prosen. ``Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits''. Commun. Math. Phys. 387, 597–620 (2021). https://doi.org/10.1007/s00220-021-04139-2 [22] Pieter W. Claeys and Austen Lamacraft. ``Maximum velocity quantum circuits''. Phys. Rev. Res. 2, 033032 (2020). https://doi.org/10.1103/PhysRevResearch.2.033032 [23] Aram W. Harrow and Richard A. Low. ``Random Quantum Circuits are Approximate 2-designs''. Commun. Math. Phys. 291, 257–302 (2009). https://doi.org/10.1007/s00220-009-0873-6 [24] Fernando G. S. L. Brandão, Aram W. Harrow, and Michal Horodecki. ``Efficient Quantum Pseudorandomness''. Phys. Rev. Lett. 116, 170502 (2016). https://doi.org/10.1103/PhysRevLett.116.170502 [25] D Gross, K Audenaert, and J Eisert. ``Evenly distributed unitaries: on the structure of unitary designs''. J. Math. Phys. 48, 052104 (2007). https://doi.org/10.1063/1.2716992 [26] Jonas Haferkamp. ``Random quantum circuits are approximate unitary $t$-designs in depth $O\left(nt^{5+o(1)}\right)$''. Quantum 6, 795 (2022). https://doi.org/10.22331/q-2022-09-08-795 [27] Huangjun Zhu. ``Multiqubit Clifford groups are unitary 3-designs''. Phys. Rev. A 96, 062336 (2017). https://doi.org/10.1103/PhysRevA.96.062336 [28] Amos Chan, Andrea De Luca, and J. T. Chalker. ``Solution of a minimal model for many-body quantum chaos''. Phys. Rev. X 8, 041019 (2018). https://doi.org/10.1103/PhysRevX.8.041019 [29] C. W. von Keyserlingk, Tibor Rakovszky, Frank Pollmann, and S. L. Sondhi. ``Operator hydrodynamics, otocs, and entanglement growth in systems without conservation laws''. Phys. Rev. X 8, 021013 (2018). https://doi.org/10.1103/PhysRevX.8.021013 [30] I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov. ``Interacting electrons in disordered wires: Anderson localization and low-t transport''. Phys. Rev. Lett. 95, 206603 (2005). https://doi.org/10.1103/PHYSREVLETT.95.206603 [31] D. M. Basko, I. L. Aleiner, and B. L. Altshuler. ``Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states''. Ann. Phys. (NY) 321, 1126–1205 (2006). https://doi.org/10.1016/j.aop.2005.11.014 [32] Arijeet Pal and David A. Huse. ``The many-body localization phase transition''. Phys. Rev. B 82, 174411 (2010). https://doi.org/10.1103/PhysRevB.82.174411 [33] David J. Luitz, Nicolas Laflorencie, and Fabien Alet. ``Many-body localization edge in the random-field heisenberg chain''. Phys. Rev. B 91, 081103(R) (2015). https://doi.org/10.1103/PHYSREVB.91.081103 [34] Anushya Chandran and C. R. Laumann. ``Semiclassical limit for the many-body localization transition''. Phys. Rev. B 92, 024301 (2015). https://doi.org/10.1103/PhysRevB.92.024301 [35] Christoph Sünderhauf, David Pérez-García, David A. Huse, Norbert Schuch, and J. Ignacio Cirac. ``Localization with random time-periodic quantum circuits''. Phys. Rev. B 98, 134204 (2018). https://doi.org/10.1103/PhysRevB.98.134204 [36] Amos Chan, Andrea De Luca, and J. T. Chalker. ``Spectral lyapunov exponents in chaotic and localized many-body quantum systems''. Phys. Rev. Res. 3, 023118 (2021). https://doi.org/10.1103/PhysRevResearch.3.023118 [37] Tom Farshi, Daniele Toniolo, Carlos E. González-Guillén, Álvaro M. Alhambra, and Lluis Masanes. ``Mixing and localization in random time-periodic quantum circuits of Clifford unitaries''. J. Math. Phys. 63, 032201 (2022). https://doi.org/10.1063/5.0054863 [38] Tom Farshi, Jonas Richter, Daniele Toniolo, Arijeet Pal, and Lluis Masanes. ``Absence of Localization in Two-Dimensional Clifford Circuits''. PRX Quantum 4, 030302 (2023). https://doi.org/10.1103/PRXQuantum.4.030302 [39] Maksym Serbyn, Z. Papić, and Dmitry A. Abanin. ``Local Conservation Laws and the Structure of the Many-Body Localized States''. Phys. Rev. Lett. 111, 127201 (2013). https://doi.org/10.1103/PhysRevLett.111.127201 [40] David A. Huse, Rahul Nandkishore, and Vadim Oganesyan. ``Phenomenology of fully many-body-localized systems''. Phys. Rev. B 90, 174202 (2014). https://doi.org/10.1103/PhysRevB.90.174202 [41] John Z. Imbrie. ``On Many-Body Localization for Quantum Spin Chains''. J. Stat. Phys. 163, 998–1048 (2016). https://doi.org/10.1007/s10955-016-1508-x [42] Wojciech De Roeck and François Huveneers. ``Stability and instability towards delocalization in many-body localization systems''. Phys. Rev. B 95, 155129 (2017). https://doi.org/10.1103/PhysRevB.95.155129 [43] David J. Luitz, François Huveneers, and Wojciech De Roeck. ``How a small quantum bath can thermalize long localized chains''. Phys. Rev. Lett. 119, 150602 (2017). https://doi.org/10.1103/PhysRevLett.119.150602 [44] Dries Sels. ``Bath-induced delocalization in interacting disordered spin chains''. Phys. Rev. B 106, L020202 (2022). https://doi.org/10.1103/PhysRevB.106.L020202 [45] P. J. D. Crowley and A. Chandran. ``Mean-field theory of failed thermalizing avalanches''. Phys. Rev. B 106, 184208 (2022). https://doi.org/10.1103/PhysRevB.106.184208 [46] Alan Morningstar, Luis Colmenarez, Vedika Khemani, David J. Luitz, and David A. Huse. ``Avalanches and many-body resonances in many-body localized systems''. Phys. Rev. B 105, 174205 (2022). https://doi.org/10.1103/PhysRevB.105.174205 [47] Hyunsoo Ha, Alan Morningstar, and David A. Huse. ``Many-body resonances in the avalanche instability of many-body localization''. Phys. Rev. Lett. 130, 250405 (2023). https://doi.org/10.1103/PhysRevLett.130.250405 [48] Jared Jeyaretnam, Christopher J. Turner, and Arijeet Pal. ``Renormalization view on resonance proliferation between many-body localized phases''. Phys. Rev. B 108, 094205 (2023). https://doi.org/10.1103/PhysRevB.108.094205 [49] Thorsten B. Wahl, Arijeet Pal, and Steven H. Simon. ``Efficient representation of fully many-body localized systems using tensor networks''. Phys. Rev. X 7, 021018 (2017). https://doi.org/10.1103/PhysRevX.7.021018 [50] Elmer V. H. Doggen, Frank Schindler, Konstantin S. Tikhonov, Alexander D. Mirlin, Titus Neupert, Dmitry G. Polyakov, and Igor V. Gornyi. ``Many-body localization and delocalization in large quantum chains''. Phys. Rev. B 98, 174202 (2018). https://doi.org/10.1103/PhysRevB.98.174202 [51] Thorsten B Wahl, Arijeet Pal, and Steven H Simon. ``Signatures of the many-body localized regime in two dimensions''. Nature Physics 15, 164–169 (2019). https://doi.org/10.1038/s41567-018-0339-x [52] Elmer VH Doggen, Igor V Gornyi, Alexander D Mirlin, and Dmitry G Polyakov. ``Many-body localization in large systems: Matrix-product-state approach''. Annals of Physics 435, 168437 (2021). https://doi.org/10.1016/j.aop.2021.168437 [53] Jeongwan Haah. ``Topological phases of unitary dynamics: Classification in Clifford category''. Commun. Math. Phys. 406, 76 (2025). https://doi.org/10.1007/s00220-025-05239-z [54] Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli. ``Topological order and criticality in (2+ 1) d monitored random quantum circuits''. Phys. Rev. Lett. 127, 235701 (2021). https://doi.org/10.1103/PhysRevLett.127.235701 [55] Grace M. Sommers, David A. Huse, and Michael J. Gullans. ``Crystalline quantum circuits''. PRX Quantum 4, 030313 (2023). https://doi.org/10.1103/PRXQuantum.4.030313 [56] Iosifina Angelidi, Marcin Szyniszewski, and Arijeet Pal. ``Stabilization of symmetry-protected long-range entanglement in stochastic quantum circuits''. Quantum 8, 1430 (2004). https://doi.org/10.22331/q-2024-08-02-1430 [57] Jonas Richter, Oliver Lunt, and Arijeet Pal. ``Transport and entanglement growth in long-range random clifford circuits''. Phys. Rev. Res. 5, L012031 (2023). https://doi.org/10.1103/PhysRevResearch.5.L012031 [58] J Tolar. ``On Clifford groups in quantum computing''. J. Phys.: Conf. Ser. 1071, 012022 (2018). https://doi.org/10.1088/1742-6596/1071/1/012022 [59] Scott Aaronson and Daniel Gottesman. ``Improved simulation of stabilizer circuits''. Phys. Rev. A 70, 052328 (2004). https://doi.org/10.1103/PhysRevA.70.052328 [60] Yaodong Li, Romain Vasseur, Matthew P. A. Fisher, and Andreas W. W. Ludwig. ``Statistical mechanics model for clifford random tensor networks and monitored quantum circuits''. Phys. Rev. B 109, 174307 (2024). https://doi.org/10.1103/PhysRevB.109.174307 [61] Nastasia Makki, Nicolai Lang, and Hans Peter Büchler. ``Absorbing state phase transition with clifford circuits''. Phys. Rev. Res. 6, 013278 (2024). https://doi.org/10.1103/PhysRevResearch.6.013278 [62] Sergey Bravyi and David Gosset. ``Improved classical simulation of quantum circuits dominated by clifford gates''. Phys. Rev. Lett. 116, 250501 (2016). https://doi.org/10.1103/PhysRevLett.116.250501 [63] Pradeep Niroula, Christopher David White, Qingfeng Wang, Sonika Johri, Daiwei Zhu, Christopher Monroe, Crystal Noel, and Michael J Gullans. ``Phase transition in magic with random quantum circuits''. Nature Phys. 20, 1786 (2024). https://doi.org/10.1038/s41567-024-02637-3 [64] Mircea Bejan, Campbell McLauchlan, and Benjamin Béri. ``Dynamical magic transitions in monitored Clifford+T circuits''. PRX Quantum 5, 030332 (2024). https://doi.org/10.1103/PRXQuantum.5.030332 [65] Poetri Sonya Tarabunga, Emanuele Tirrito, Titas Chanda, and Marcello Dalmonte. ``Many-body magic via pauli-markov chains—from criticality to gauge theories''. PRX Quantum 4, 040317 (2023). https://doi.org/10.1103/PRXQuantum.4.040317 [66] Shiyu Zhou, Zhi-Cheng Yang, Alioscia Hamma, and Claudio Chamon. ``Single T gate in a Clifford circuit drives transition to universal entanglement spectrum statistics''. SciPost Phys. 9, 087 (2020). https://doi.org/10.21468/SciPostPhys.9.6.087 [67] Nicholas Hunter-Jones. ``Unitary designs from statistical mechanics in random quantum circuits'' (2019) arXiv:1905.12053. arXiv:1905.12053 [68] Saúl Pilatowsky-Cameo, Ceren B. Dag, Wen Wei Ho, and Soonwon Choi. ``Complete Hilbert-Space Ergodicity in Quantum Dynamics of Generalized Fibonacci Drives''. Phys. Rev. Lett. 131, 250401 (2023). https://doi.org/10.1103/PhysRevLett.131.250401 [69] Saúl Pilatowsky-Cameo, Iman Marvian, Soonwon Choi, and Wen Wei Ho. ``Hilbert-Space Ergodicity in Driven Quantum Systems: Obstructions and Designs'' (2024). arXiv:2402.06720. https://doi.org/10.1103/PhysRevX.14.041059 arXiv:2402.06720 [70] Louk Rademaker and Miguel Ortuño. ``Explicit local integrals of motion for the many-body localized state''. Phys. Rev. Lett. 116, 010404 (2016). https://doi.org/10.1103/PhysRevLett.116.010404 [71] Abishek K. Kulshreshtha, Arijeet Pal, Thorsten B. Wahl, and Steven H. Simon. ``Behavior of l-bits near the many-body localization transition''. Phys. Rev. B 98, 184201 (2018). https://doi.org/10.1103/PhysRevB.98.184201 [72] S. J. Thomson and M. Schiró. ``Time evolution of many-body localized systems with the flow equation approach''. Phys. Rev. B 97, 060201(R) (2018). https://doi.org/10.1103/PhysRevB.97.060201 [73] M. Goihl, M. Gluza, C. Krumnow, and J. Eisert. ``Construction of exact constants of motion and effective models for many-body localized systems''. Phys. Rev. B 97, 134202 (2018). https://doi.org/10.1103/PhysRevB.97.134202 [74] Sergey Bravyi, Dan Browne, Padraic Calpin, Earl Campbell, David Gosset, and Mark Howard. ``Simulation of quantum circuits by low-rank stabilizer decompositions''. Quantum 3, 181 (2019). https://doi.org/10.22331/q-2019-09-02-181 [75] E. Brézin and S. Hikami. ``Spectral form factor in a random matrix theory''. Phys. Rev. E 55, 4067–4083 (1997). https://doi.org/10.1103/PhysRevE.55.4067 [76] Fritz Haake, Hans-Jürgen Sommers, and Joachim Weber. ``Fluctuations and ergodicity of the form factor of quantum propagators and random unitary matrices''. J. Phys. A: Math. Gen 32, 6903–6913 (1999). https://doi.org/10.1088/0305-4470/32/40/301 [77] Amos Chan, Andrea De Luca, and J. T. Chalker. ``Spectral Statistics in Spatially Extended Chaotic Quantum Many-Body Systems''. Phys. Rev. Lett. 121, 060601 (2018). https://doi.org/10.1103/PhysRevLett.121.060601 [78] Fabian H. L. Essler and Lorenzo Piroli. ``Integrability of one-dimensional lindbladians from operator-space fragmentation''. Phys. Rev. E 102, 062210 (2020). https://doi.org/10.1103/PhysRevE.102.062210 [79] Yahui Li, Pablo Sala, and Frank Pollmann. ``Hilbert space fragmentation in open quantum systems''. Phys. Rev. Res. 5, 043239 (2023). https://doi.org/10.1103/PhysRevResearch.5.043239 [80] Dawid Paszko, Dominic C. Rose, Marzena H. Szymańska, and Arijeet Pal. ``Edge modes and symmetry-protected topological states in open quantum systems''. PRX Quantum 5, 030304 (2024). https://doi.org/10.1103/PRXQuantum.5.030304 [81] Sanjay Moudgalya, B Andrei Bernevig, and Nicolas Regnault. ``Quantum many-body scars and hilbert space fragmentation: a review of exact results''. Reports on Progress in Physics 85, 086501 (2022). https://doi.org/10.1088/1361-6633/ac73a0 [82] Daniel Grier and Luke Schaeffer. ``Multiqubit Clifford groups are unitary 3-designs''. Quantum 6, 734 (2022). https://doi.org/10.22331/q-2022-06-13-734 [83] Jens Siewert. ``On orthogonal bases in the Hilbert-Schmidt space of matrices''. J. Phys. Commun. 6, 055014 (2022). https://doi.org/10.1088/2399-6528/ac6f43 [84] Antonio Anna Mele. ``Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial''. Quantum 8, 1340 (2024). https://doi.org/10.22331/q-2024-05-08-1340 [85] Bruno Bertini, Pavel Kos, and TomažProsen. ``Operator Entanglement in Local Quantum Circuits I: Chaotic Dual-Unitary Circuits''. SciPost Phys. 8, 067 (2020). https://doi.org/10.21468/SciPostPhys.8.4.067 [86] Bruno Bertini, Pavel Kos, and TomažProsen. ``Exact spectral form factor in a minimal model of many-body quantum chaos''. Phys. Rev. Lett. 121, 264101 (2018). https://doi.org/10.1103/PhysRevLett.121.264101 [87] Robert Zwanzig. ``Nonequilibrium statistical mechanics''.
Oxford University Press. (2001). https://doi.org/10.1093/oso/9780195140187.001.0001 [88] Tom Holden-Dye, Lluis Masanes, and Arijeet Pal. ``Fundamental charges for dual-unitary circuits''. Quantum 9, 1615 (2025). https://doi.org/10.22331/q-2025-01-30-1615 [89] Allan Steel. ``Conquering inseparability: Primary decomposition and multivariate factorization over algebraic function fields of positive characteristic''. Journal of Symbolic Computation 40, 1053–1075 (2005). https://doi.org/10.1016/j.jsc.2005.03.002 [90] P. W. Anderson. ``Absence of Diffusion in Certain Random Lattices''. Phys. Rev. 109, 1492–1505 (1958). https://doi.org/10.1103/PhysRev.109.1492 [91] Sanjay Moudgalya and Olexei I. Motrunich. ``Hilbert space fragmentation and commutant algebras''. Phys. Rev. X 12, 011050 (2022). https://doi.org/10.1103/PhysRevX.12.011050 [92] Eugene P. Wigner. ``Random Matrices in Physics''. SIAM Review 9, 1–23 (1967). arXiv:https://doi.org/10.1137/1009001. https://doi.org/10.1137/1009001 arXiv:https://doi.org/10.1137/1009001 [93] Madan Lal Mehta. ``Random matrices''. Academic Press, New York. (1990). 2nd edition. url: https://doi.org/10.1016/C2009-0-22297-5. https://doi.org/10.1016/C2009-0-22297-5 [94] Daniel A. Roberts and Beni Yoshida. ``Chaos and complexity by design''. J.
High Energy Phys. 2017, 121 (2017). https://doi.org/10.1007/JHEP04(2017)121 [95] Alexei Y. Kitaev. ``Fault-tolerant quantum computation by anyons''. Annals of Physics 303, 2–30 (2003). https://doi.org/10.1016/S0003-4916(02)00018-0Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-18 07:52:57: Could not fetch cited-by data for 10.22331/q-2026-05-18-2107 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-05-18 07:52:58: Cannot retrieve data from ADS due to rate limitations.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractFloquet quantum circuits are able to realise a wide range of non-equilibrium quantum states, exhibiting quantum chaos, topological order and localisation. In this work, we investigate the stability of operator localisation and emergence of chaos in random Floquet-Clifford circuits subjected to unitary perturbations which drive them away from the Clifford limit. We construct a nearest-neighbour Clifford circuit with a brickwork pattern and study the effect of including disordered non-Clifford gates. The perturbations are uniformly sampled from single-qubit unitaries with probability $p$ on each qubit. We show that the interacting model exhibits strong localisation of operators for $0 \le{p} \lt{1}$ that is characterised by the fragmentation of operator space into disjoint sectors due to the appearance of wall configurations. Such walls give rise to emergent local integrals of motion for the circuit that we construct exactly. We analytically establish the stability of localisation against generic perturbations and calculate the average length of operator spreading tunable by $p$. Although our circuit is not separable across any bi-partition, we further show that the operator localisation leads to an entanglement bottleneck, where initially unentangled states remain weakly entangled across typical fragment boundaries. Finally, we study the spectral form factor (SFF) to characterise the chaotic properties of the operator fragments and spectral fluctuations as a probe of non-ergodicity. In the $p = 1$ model, the emergence of a fragmentation time scale is found before random matrix theory sets in after which the SFF can be approximated by that of the circular unitary ensemble. Our work provides an explicit description of quantum phases in operator dynamics and circuit ergodicity which can be realised on current NISQ devices.Featured image: Circuit model exhibiting fragmented (non-ergodic) circuit evolution.Popular summaryMany-body quantum dynamics concerns the time-evolution of large interacting ensembles. Ergodicity-breaking from localisation is a fundamental problem within this field referring to the long-time preservation of locally encoded information. Understanding the conditions under which many-body systems exhibit localisation remains an open problem. In particular, significant theoretical effort has been dedicated to understanding the role of interaction strength, symmetry and disorder on the stability of localised quantum evolution which is analytically challenging in Hamiltonian models. This paper concerns brickwork quantum circuits as a toy model to study robust operator localisation in time-periodic disordered systems. Quantum circuits have emerged as minimally structured dynamical many-body systems due to their high degree of analytical tractability and the rich complexity of dynamical phases they can realise. They are also prime candidates for benchmarking contemporary quantum computers for simulating complex quantum evolution. Our work studies a robust ergodicity-breaking mechanism in brickwork Clifford unitaries where the emergence of chaotic evolution is obstructed even when the circuit is perturbed with local random gates. We analytically study local constraints emerging in the disordered circuit ensemble using Clifford group techniques from quantum computing theory. Our analytical predictions are then verified by numerically simulating the dynamics of entanglement entropy and many-body spectral correlations to understand observable features of the ergodicity-breaking dynamics and its stability. Our work sits at the intersection of quantum information theory and many-body physics with our analytical insights offering a mathematically salient description of stable ergodicity-breaking in the large system and long-time limit.► BibTeX data@article{Kovacs2026operatorspace, doi = {10.22331/q-2026-05-18-2107}, url = {https://doi.org/10.22331/q-2026-05-18-2107}, title = {Operator space fragmentation in perturbed {F}loquet-{C}lifford circuits}, author = {Kov{\'{a}}cs, Marcell D. and Turner, Christopher J. and Masanes, Lluis and Pal, Arijeet}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2107}, month = may, year = {2026} }► References [1] Daniel A Roberts and Beni Yoshida. ``Chaos and complexity by design''. Journal of High Energy Physics 2017, 1–64 (2017). https://doi.org/10.1007/JHEP04(2017)121 [2] J. Eisert. ``Entangling power and quantum circuit complexity''. Phys. Rev. Lett. 127, 020501 (2021). https://doi.org/10.1103/PhysRevLett.127.020501 [3] Wen Wei Ho and Soonwon Choi. ``Exact emergent quantum state designs from quantum chaotic dynamics''. Phys. Rev. Lett. 128, 060601 (2022). https://doi.org/10.1103/PhysRevLett.128.060601 [4] Jonas Haferkamp, Philippe Faist, Naga BT Kothakonda, Jens Eisert, and Nicole Yunger Halpern. ``Linear growth of quantum circuit complexity''. Nature Physics 18, 528–532 (2022). https://doi.org/10.1038/s41567-022-01539-6 [5] Matthew P.A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay. ``Random quantum circuits''. Annu. Rev. Condens. Matter Phys. 14, 335–379 (2023). https://doi.org/10.1146/annurev-conmatphys-031720-030658 [6] Dominik Hahn and Luis Colmenarez. ``Absence of localization in weakly interacting Floquet circuits''. Phys. Rev. B 109, 094207 (2024). https://doi.org/10.1103/PhysRevB.109.094207 [7] Chao-Ming Jian, Yi-Zhuang You, Romain Vasseur, and Andreas W. W. Ludwig. ``Measurement-induced criticality in random quantum circuits''. Phys. Rev. B 101, 104302 (2020). https://doi.org/10.1103/PhysRevB.101.104302 [8] 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 [9] 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 [10] Yaodong Li, Xiao Chen, and Matthew P. A. Fisher. ``Quantum Zeno effect and the many-body entanglement transition''. Phys. Rev. B 98, 205136 (2018). https://doi.org/10.1103/PhysRevB.98.205136 [11] Oliver Lunt, Marcin Szyniszewski, and Arijeet Pal. ``Measurement-induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional Clifford circuits''. Phys. Rev. B 104, 155111 (2021). https://doi.org/10.1103/PhysRevB.104.155111 [12] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Rev. Mod. Phys. 81, 865–942 (2009). https://doi.org/10.1103/RevModPhys.81.865 [13] Thomas Jennewein, Christoph Simon, Gregor Weihs, Harald Weinfurter, and Anton Zeilinger. ``Quantum cryptography with entangled photons''. Phys. Rev. Lett. 84, 4729–4732 (2000). https://doi.org/10.1103/PhysRevLett.84.4729 [14] Juan Yin, Yu-Huai Li, Sheng-Kai Liao, Meng Yang, Yuan Cao, Liang Zhang, Ji-Gang Ren, Wen-Qi Cai, Wei-Yue Liu, Shuang-Lin Li, et al. ``Entanglement-based secure quantum cryptography over 1,120 kilometres''. Nature 582, 501–505 (2020). https://doi.org/10.1038/s41586-020-2401-y [15] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. ``Mixed-state entanglement and quantum error correction''. Phys. Rev. A 54, 3824–3851 (1996). https://doi.org/10.1103/PhysRevA.54.3824 [16] Barbara M. Terhal. ``Quantum error correction for quantum memories''. Rev. Mod. Phys. 87, 307–346 (2015). https://doi.org/10.1103/RevModPhys.87.307 [17] Adam M Kaufman, M Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko, Philipp M Preiss, and Markus Greiner. ``Quantum thermalization through entanglement in an isolated many-body system''. Science 353, 794–800 (2016). https://doi.org/10.1126/science.aaf6725 [18] Rahul Nandkishore and David A Huse. ``Many-body localization and thermalization in quantum statistical mechanics''. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015). https://doi.org/10.1146/annurev-conmatphys-031214-014726 [19] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn. ``Colloquium: Many-body localization, thermalization, and entanglement''. Rev. Mod. Phys. 91, 021001 (2019). https://doi.org/10.1103/RevModPhys.91.021001 [20] Bruno Bertini, Pavel Kos, and Tomaž Prosen. ``Entanglement Spreading in a Minimal Model of Maximal Many-Body Quantum Chaos''. Phys. Rev. X 9, 021033 (2019). https://doi.org/10.1103/PhysRevX.9.021033 [21] Bruno Bertini, Pavel Kos, and Tomaž Prosen. ``Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits''. Commun. Math. Phys. 387, 597–620 (2021). https://doi.org/10.1007/s00220-021-04139-2 [22] Pieter W. Claeys and Austen Lamacraft. ``Maximum velocity quantum circuits''. Phys. Rev. Res. 2, 033032 (2020). https://doi.org/10.1103/PhysRevResearch.2.033032 [23] Aram W. Harrow and Richard A. Low. ``Random Quantum Circuits are Approximate 2-designs''. Commun. Math. Phys. 291, 257–302 (2009). https://doi.org/10.1007/s00220-009-0873-6 [24] Fernando G. S. L. Brandão, Aram W. Harrow, and Michal Horodecki. ``Efficient Quantum Pseudorandomness''. Phys. Rev. Lett. 116, 170502 (2016). https://doi.org/10.1103/PhysRevLett.116.170502 [25] D Gross, K Audenaert, and J Eisert. ``Evenly distributed unitaries: on the structure of unitary designs''. J. Math. Phys. 48, 052104 (2007). https://doi.org/10.1063/1.2716992 [26] Jonas Haferkamp. ``Random quantum circuits are approximate unitary $t$-designs in depth $O\left(nt^{5+o(1)}\right)$''. Quantum 6, 795 (2022). https://doi.org/10.22331/q-2022-09-08-795 [27] Huangjun Zhu. ``Multiqubit Clifford groups are unitary 3-designs''. Phys. Rev. A 96, 062336 (2017). https://doi.org/10.1103/PhysRevA.96.062336 [28] Amos Chan, Andrea De Luca, and J. T. Chalker. ``Solution of a minimal model for many-body quantum chaos''. Phys. Rev. X 8, 041019 (2018). https://doi.org/10.1103/PhysRevX.8.041019 [29] C. W. von Keyserlingk, Tibor Rakovszky, Frank Pollmann, and S. L. Sondhi. ``Operator hydrodynamics, otocs, and entanglement growth in systems without conservation laws''. Phys. Rev. X 8, 021013 (2018). https://doi.org/10.1103/PhysRevX.8.021013 [30] I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov. ``Interacting electrons in disordered wires: Anderson localization and low-t transport''. Phys. Rev. Lett. 95, 206603 (2005). https://doi.org/10.1103/PHYSREVLETT.95.206603 [31] D. M. Basko, I. L. Aleiner, and B. L. Altshuler. ``Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states''. Ann. Phys. (NY) 321, 1126–1205 (2006). https://doi.org/10.1016/j.aop.2005.11.014 [32] Arijeet Pal and David A. Huse. ``The many-body localization phase transition''. Phys. Rev. B 82, 174411 (2010). https://doi.org/10.1103/PhysRevB.82.174411 [33] David J. Luitz, Nicolas Laflorencie, and Fabien Alet. ``Many-body localization edge in the random-field heisenberg chain''. Phys. Rev. B 91, 081103(R) (2015). https://doi.org/10.1103/PHYSREVB.91.081103 [34] Anushya Chandran and C. R. Laumann. ``Semiclassical limit for the many-body localization transition''. Phys. Rev. B 92, 024301 (2015). https://doi.org/10.1103/PhysRevB.92.024301 [35] Christoph Sünderhauf, David Pérez-García, David A. Huse, Norbert Schuch, and J. Ignacio Cirac. ``Localization with random time-periodic quantum circuits''. Phys. Rev. B 98, 134204 (2018). https://doi.org/10.1103/PhysRevB.98.134204 [36] Amos Chan, Andrea De Luca, and J. T. Chalker. ``Spectral lyapunov exponents in chaotic and localized many-body quantum systems''. Phys. Rev. Res. 3, 023118 (2021). https://doi.org/10.1103/PhysRevResearch.3.023118 [37] Tom Farshi, Daniele Toniolo, Carlos E. González-Guillén, Álvaro M. Alhambra, and Lluis Masanes. ``Mixing and localization in random time-periodic quantum circuits of Clifford unitaries''. J. Math. Phys. 63, 032201 (2022). https://doi.org/10.1063/5.0054863 [38] Tom Farshi, Jonas Richter, Daniele Toniolo, Arijeet Pal, and Lluis Masanes. ``Absence of Localization in Two-Dimensional Clifford Circuits''. PRX Quantum 4, 030302 (2023). https://doi.org/10.1103/PRXQuantum.4.030302 [39] Maksym Serbyn, Z. Papić, and Dmitry A. Abanin. ``Local Conservation Laws and the Structure of the Many-Body Localized States''. Phys. Rev. Lett. 111, 127201 (2013). https://doi.org/10.1103/PhysRevLett.111.127201 [40] David A. Huse, Rahul Nandkishore, and Vadim Oganesyan. ``Phenomenology of fully many-body-localized systems''. Phys. Rev. B 90, 174202 (2014). https://doi.org/10.1103/PhysRevB.90.174202 [41] John Z. Imbrie. ``On Many-Body Localization for Quantum Spin Chains''. J. Stat. Phys. 163, 998–1048 (2016). https://doi.org/10.1007/s10955-016-1508-x [42] Wojciech De Roeck and François Huveneers. ``Stability and instability towards delocalization in many-body localization systems''. Phys. Rev. B 95, 155129 (2017). https://doi.org/10.1103/PhysRevB.95.155129 [43] David J. Luitz, François Huveneers, and Wojciech De Roeck. ``How a small quantum bath can thermalize long localized chains''. Phys. Rev. Lett. 119, 150602 (2017). https://doi.org/10.1103/PhysRevLett.119.150602 [44] Dries Sels. ``Bath-induced delocalization in interacting disordered spin chains''. Phys. Rev. B 106, L020202 (2022). https://doi.org/10.1103/PhysRevB.106.L020202 [45] P. J. D. Crowley and A. Chandran. ``Mean-field theory of failed thermalizing avalanches''. Phys. Rev. B 106, 184208 (2022). https://doi.org/10.1103/PhysRevB.106.184208 [46] Alan Morningstar, Luis Colmenarez, Vedika Khemani, David J. Luitz, and David A. Huse. ``Avalanches and many-body resonances in many-body localized systems''. Phys. Rev. B 105, 174205 (2022). https://doi.org/10.1103/PhysRevB.105.174205 [47] Hyunsoo Ha, Alan Morningstar, and David A. Huse. ``Many-body resonances in the avalanche instability of many-body localization''. Phys. Rev. Lett. 130, 250405 (2023). https://doi.org/10.1103/PhysRevLett.130.250405 [48] Jared Jeyaretnam, Christopher J. Turner, and Arijeet Pal. ``Renormalization view on resonance proliferation between many-body localized phases''. Phys. Rev. B 108, 094205 (2023). https://doi.org/10.1103/PhysRevB.108.094205 [49] Thorsten B. Wahl, Arijeet Pal, and Steven H. Simon. ``Efficient representation of fully many-body localized systems using tensor networks''. Phys. Rev. X 7, 021018 (2017). https://doi.org/10.1103/PhysRevX.7.021018 [50] Elmer V. H. Doggen, Frank Schindler, Konstantin S. Tikhonov, Alexander D. Mirlin, Titus Neupert, Dmitry G. Polyakov, and Igor V. Gornyi. ``Many-body localization and delocalization in large quantum chains''. Phys. Rev. B 98, 174202 (2018). https://doi.org/10.1103/PhysRevB.98.174202 [51] Thorsten B Wahl, Arijeet Pal, and Steven H Simon. ``Signatures of the many-body localized regime in two dimensions''. Nature Physics 15, 164–169 (2019). https://doi.org/10.1038/s41567-018-0339-x [52] Elmer VH Doggen, Igor V Gornyi, Alexander D Mirlin, and Dmitry G Polyakov. ``Many-body localization in large systems: Matrix-product-state approach''. Annals of Physics 435, 168437 (2021). https://doi.org/10.1016/j.aop.2021.168437 [53] Jeongwan Haah. ``Topological phases of unitary dynamics: Classification in Clifford category''. Commun. Math. Phys. 406, 76 (2025). https://doi.org/10.1007/s00220-025-05239-z [54] Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli. ``Topological order and criticality in (2+ 1) d monitored random quantum circuits''. Phys. Rev. Lett. 127, 235701 (2021). https://doi.org/10.1103/PhysRevLett.127.235701 [55] Grace M. Sommers, David A. Huse, and Michael J. Gullans. ``Crystalline quantum circuits''. PRX Quantum 4, 030313 (2023). https://doi.org/10.1103/PRXQuantum.4.030313 [56] Iosifina Angelidi, Marcin Szyniszewski, and Arijeet Pal. ``Stabilization of symmetry-protected long-range entanglement in stochastic quantum circuits''. Quantum 8, 1430 (2004). https://doi.org/10.22331/q-2024-08-02-1430 [57] Jonas Richter, Oliver Lunt, and Arijeet Pal. ``Transport and entanglement growth in long-range random clifford circuits''. Phys. Rev. Res. 5, L012031 (2023). https://doi.org/10.1103/PhysRevResearch.5.L012031 [58] J Tolar. ``On Clifford groups in quantum computing''. J. Phys.: Conf. Ser. 1071, 012022 (2018). https://doi.org/10.1088/1742-6596/1071/1/012022 [59] Scott Aaronson and Daniel Gottesman. ``Improved simulation of stabilizer circuits''. Phys. Rev. A 70, 052328 (2004). https://doi.org/10.1103/PhysRevA.70.052328 [60] Yaodong Li, Romain Vasseur, Matthew P. A. Fisher, and Andreas W. W. Ludwig. ``Statistical mechanics model for clifford random tensor networks and monitored quantum circuits''. Phys. Rev. B 109, 174307 (2024). https://doi.org/10.1103/PhysRevB.109.174307 [61] Nastasia Makki, Nicolai Lang, and Hans Peter Büchler. ``Absorbing state phase transition with clifford circuits''. Phys. Rev. Res. 6, 013278 (2024). https://doi.org/10.1103/PhysRevResearch.6.013278 [62] Sergey Bravyi and David Gosset. ``Improved classical simulation of quantum circuits dominated by clifford gates''. Phys. Rev. Lett. 116, 250501 (2016). https://doi.org/10.1103/PhysRevLett.116.250501 [63] Pradeep Niroula, Christopher David White, Qingfeng Wang, Sonika Johri, Daiwei Zhu, Christopher Monroe, Crystal Noel, and Michael J Gullans. ``Phase transition in magic with random quantum circuits''. Nature Phys. 20, 1786 (2024). https://doi.org/10.1038/s41567-024-02637-3 [64] Mircea Bejan, Campbell McLauchlan, and Benjamin Béri. ``Dynamical magic transitions in monitored Clifford+T circuits''. PRX Quantum 5, 030332 (2024). https://doi.org/10.1103/PRXQuantum.5.030332 [65] Poetri Sonya Tarabunga, Emanuele Tirrito, Titas Chanda, and Marcello Dalmonte. ``Many-body magic via pauli-markov chains—from criticality to gauge theories''. PRX Quantum 4, 040317 (2023). https://doi.org/10.1103/PRXQuantum.4.040317 [66] Shiyu Zhou, Zhi-Cheng Yang, Alioscia Hamma, and Claudio Chamon. ``Single T gate in a Clifford circuit drives transition to universal entanglement spectrum statistics''. SciPost Phys. 9, 087 (2020). https://doi.org/10.21468/SciPostPhys.9.6.087 [67] Nicholas Hunter-Jones. ``Unitary designs from statistical mechanics in random quantum circuits'' (2019) arXiv:1905.12053. arXiv:1905.12053 [68] Saúl Pilatowsky-Cameo, Ceren B. Dag, Wen Wei Ho, and Soonwon Choi. ``Complete Hilbert-Space Ergodicity in Quantum Dynamics of Generalized Fibonacci Drives''. Phys. Rev. Lett. 131, 250401 (2023). https://doi.org/10.1103/PhysRevLett.131.250401 [69] Saúl Pilatowsky-Cameo, Iman Marvian, Soonwon Choi, and Wen Wei Ho. ``Hilbert-Space Ergodicity in Driven Quantum Systems: Obstructions and Designs'' (2024). arXiv:2402.06720. https://doi.org/10.1103/PhysRevX.14.041059 arXiv:2402.06720 [70] Louk Rademaker and Miguel Ortuño. ``Explicit local integrals of motion for the many-body localized state''. Phys. Rev. Lett. 116, 010404 (2016). https://doi.org/10.1103/PhysRevLett.116.010404 [71] Abishek K. Kulshreshtha, Arijeet Pal, Thorsten B. Wahl, and Steven H. Simon. ``Behavior of l-bits near the many-body localization transition''. Phys. Rev. B 98, 184201 (2018). https://doi.org/10.1103/PhysRevB.98.184201 [72] S. J. Thomson and M. Schiró. ``Time evolution of many-body localized systems with the flow equation approach''. Phys. Rev. B 97, 060201(R) (2018). https://doi.org/10.1103/PhysRevB.97.060201 [73] M. Goihl, M. Gluza, C. Krumnow, and J. Eisert. ``Construction of exact constants of motion and effective models for many-body localized systems''. Phys. Rev. B 97, 134202 (2018). https://doi.org/10.1103/PhysRevB.97.134202 [74] Sergey Bravyi, Dan Browne, Padraic Calpin, Earl Campbell, David Gosset, and Mark Howard. ``Simulation of quantum circuits by low-rank stabilizer decompositions''. Quantum 3, 181 (2019). https://doi.org/10.22331/q-2019-09-02-181 [75] E. Brézin and S. Hikami. ``Spectral form factor in a random matrix theory''. Phys. Rev. E 55, 4067–4083 (1997). https://doi.org/10.1103/PhysRevE.55.4067 [76] Fritz Haake, Hans-Jürgen Sommers, and Joachim Weber. ``Fluctuations and ergodicity of the form factor of quantum propagators and random unitary matrices''. J. Phys. A: Math. Gen 32, 6903–6913 (1999). https://doi.org/10.1088/0305-4470/32/40/301 [77] Amos Chan, Andrea De Luca, and J. T. Chalker. ``Spectral Statistics in Spatially Extended Chaotic Quantum Many-Body Systems''. Phys. Rev. Lett. 121, 060601 (2018). https://doi.org/10.1103/PhysRevLett.121.060601 [78] Fabian H. L. Essler and Lorenzo Piroli. ``Integrability of one-dimensional lindbladians from operator-space fragmentation''. Phys. Rev. E 102, 062210 (2020). https://doi.org/10.1103/PhysRevE.102.062210 [79] Yahui Li, Pablo Sala, and Frank Pollmann. ``Hilbert space fragmentation in open quantum systems''. Phys. Rev. Res. 5, 043239 (2023). https://doi.org/10.1103/PhysRevResearch.5.043239 [80] Dawid Paszko, Dominic C. Rose, Marzena H. Szymańska, and Arijeet Pal. ``Edge modes and symmetry-protected topological states in open quantum systems''. PRX Quantum 5, 030304 (2024). https://doi.org/10.1103/PRXQuantum.5.030304 [81] Sanjay Moudgalya, B Andrei Bernevig, and Nicolas Regnault. ``Quantum many-body scars and hilbert space fragmentation: a review of exact results''. Reports on Progress in Physics 85, 086501 (2022). https://doi.org/10.1088/1361-6633/ac73a0 [82] Daniel Grier and Luke Schaeffer. ``Multiqubit Clifford groups are unitary 3-designs''. Quantum 6, 734 (2022). https://doi.org/10.22331/q-2022-06-13-734 [83] Jens Siewert. ``On orthogonal bases in the Hilbert-Schmidt space of matrices''. J. Phys. Commun. 6, 055014 (2022). https://doi.org/10.1088/2399-6528/ac6f43 [84] Antonio Anna Mele. ``Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial''. Quantum 8, 1340 (2024). https://doi.org/10.22331/q-2024-05-08-1340 [85] Bruno Bertini, Pavel Kos, and TomažProsen. ``Operator Entanglement in Local Quantum Circuits I: Chaotic Dual-Unitary Circuits''. SciPost Phys. 8, 067 (2020). https://doi.org/10.21468/SciPostPhys.8.4.067 [86] Bruno Bertini, Pavel Kos, and TomažProsen. ``Exact spectral form factor in a minimal model of many-body quantum chaos''. Phys. Rev. Lett. 121, 264101 (2018). https://doi.org/10.1103/PhysRevLett.121.264101 [87] Robert Zwanzig. ``Nonequilibrium statistical mechanics''.
Oxford University Press. (2001). https://doi.org/10.1093/oso/9780195140187.001.0001 [88] Tom Holden-Dye, Lluis Masanes, and Arijeet Pal. ``Fundamental charges for dual-unitary circuits''. Quantum 9, 1615 (2025). https://doi.org/10.22331/q-2025-01-30-1615 [89] Allan Steel. ``Conquering inseparability: Primary decomposition and multivariate factorization over algebraic function fields of positive characteristic''. Journal of Symbolic Computation 40, 1053–1075 (2005). https://doi.org/10.1016/j.jsc.2005.03.002 [90] P. W. Anderson. ``Absence of Diffusion in Certain Random Lattices''. Phys. Rev. 109, 1492–1505 (1958). https://doi.org/10.1103/PhysRev.109.1492 [91] Sanjay Moudgalya and Olexei I. Motrunich. ``Hilbert space fragmentation and commutant algebras''. Phys. Rev. X 12, 011050 (2022). https://doi.org/10.1103/PhysRevX.12.011050 [92] Eugene P. Wigner. ``Random Matrices in Physics''. SIAM Review 9, 1–23 (1967). arXiv:https://doi.org/10.1137/1009001. https://doi.org/10.1137/1009001 arXiv:https://doi.org/10.1137/1009001 [93] Madan Lal Mehta. ``Random matrices''. Academic Press, New York. (1990). 2nd edition. url: https://doi.org/10.1016/C2009-0-22297-5. https://doi.org/10.1016/C2009-0-22297-5 [94] Daniel A. Roberts and Beni Yoshida. ``Chaos and complexity by design''. J.
High Energy Phys. 2017, 121 (2017). https://doi.org/10.1007/JHEP04(2017)121 [95] Alexei Y. Kitaev. ``Fault-tolerant quantum computation by anyons''. Annals of Physics 303, 2–30 (2003). https://doi.org/10.1016/S0003-4916(02)00018-0Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-18 07:52:57: Could not fetch cited-by data for 10.22331/q-2026-05-18-2107 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-05-18 07:52:58: Cannot retrieve data from ADS due to rate limitations.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.