Probes of Full Eigenstate Thermalization in Ergodicity-Breaking Quantum Circuits
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AbstractThe eigenstate thermalization hypothesis (ETH) is the leading interpretation in our current understanding of quantum thermalization. Recent results uncovered strong connections between quantum correlations in thermalizing systems and the structure of free probability theory, leading to the notion of full ETH. However, most studies have been performed for ergodic systems and it is still unclear whether or how full ETH manifests in ergodicity-breaking models. We fill this gap by studying standard probes of full ETH in ergodicity-breaking quantum circuits, presenting numerical and analytical results for interacting integrable systems. These probes can display distinct behavior and undergo a different scaling than the ones observed in ergodic systems. For the analytical results we consider an interacting integrable dual-unitary model and present the exact eigenstates, allowing us to analytically express common probes for full ETH. We discuss the underlying mechanisms responsible for these differences and show how the presence of solitons dictates the behavior of ETH-related quantities in the dual-unitary model. We show numerical evidence that this behavior is sufficiently generic away from dual-unitarity when restricted to the appropriate symmetry sectors.Featured image: According to the eigenstate thermalization hypothesis (ETH), eigenstates of quantum many-body systems appear thermal whenever local observables are probed. For such ergodic systems, the dynamics of out-of-time order correlators can be decomposed in the sum of thermal cumulants — an ETH-based analogue of free cumulants. This decomposition is based on the combinatorics of non-crossing partitions and correctly captures the long-time behavior of OTOCs. In this work, we investigate the behavior of such probes and their underlying structure in ergodicity-breaking quantum circuits.► BibTeX data@article{Alves2025probesoffull, doi = {10.22331/q-2025-12-15-1949}, url = {https://doi.org/10.22331/q-2025-12-15-1949}, title = {Probes of {F}ull {E}igenstate {T}hermalization in {E}rgodicity-{B}reaking {Q}uantum {C}ircuits}, author = {Alves, Gabriel O. and Fritzsch, Felix and Claeys, Pieter W.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1949}, month = dec, year = {2025} }► References [1] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (Wiley, New York, 1985). [2] H. Tasaki, Phys. Rev. Lett. 80, 1373 (1998). https://doi.org/10.1103/PhysRevLett.80.1373 [3] A. Nazir and G. Schaller, Thermodynamics in the Quantum Regime, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, Vol. 195 (Springer International Publishing, Cham, 2018) 1805.08307. https://doi.org/10.1007/978-3-319-99046-0 arXiv:1805.08307 [4] J. M. Deutsch, Phys. Rev. A 43, 2046 (1991). https://doi.org/10.1103/physreva.43.2046 [5] M. Srednicki, Phys. Rev. E 50, 888 (1994). https://doi.org/10.1103/PhysRevE.50.888 [6] L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016). https://doi.org/10.1080/00018732.2016.1198134 [7] J. M. Deutsch, Rep. Prog. Phys. 81, 082001 (2018). https://doi.org/10.1088/1361-6633/aac9f1 [8] L. Foini and J. Kurchan, Phys. Rev. E 99, 042139 (2019). https://doi.org/10.1103/PhysRevE.99.042139 [9] S. Pappalardi, L. Foini, and J. Kurchan, Phys. Rev. Lett. 129, 170603 (2022). https://doi.org/10.1103/PhysRevLett.129.170603 [10] D. Voiculescu, in Operator Algebras and Their Connections with Topology and Ergodic Theory, edited by H. Araki, C. C. Moore, Ş.-V. Stratila, and D.-V. Voiculescu (Springer, Berlin, Heidelberg, 1985) pp. 556–588. https://doi.org/10.1007/BFb0074909 [11] D. Voiculescu, J. Funct. Anal. 66, 323 (1986). https://doi.org/10.1016/0022-1236(86)90062-5 [12] D. Voiculescu, J. Operat. Theor. 18, 223 (1987). [13] J. Novak and M. LaCroix, Three lectures on free probability (2012), arXiv:1205.2097. https://doi.org/10.48550/arXiv.1205.2097 arXiv:1205.2097 [14] J. A. Mingo and R. Speicher, Free Probability and Random Matrices, Fields Institute Monographs, Vol. 35 (Springer New York, New York, NY, 2017). https://doi.org/10.1007/978-1-4939-6942-5 [15] S. Pappalardi, F. Fritzsch, and T. Prosen, Phys. Rev. Lett. 134, 140404 (2025). https://doi.org/10.1103/PhysRevLett.134.140404 [16] S. Pappalardi, L. Foini, and J. Kurchan, Quantum 8, 1227 (2024). https://doi.org/10.22331/q-2024-01-11-1227 [17] S. Jindal and P. Hosur, J. High Energ. Phys. 2024 (66), 66. https://doi.org/10.1007/JHEP09(2024)066 [18] F. Fritzsch, T. Prosen, and S. Pappalardi, Phys. Rev. B 111, 054303 (2025). https://doi.org/10.1103/PhysRevB.111.054303 [19] E. Vallini and S. Pappalardi, Long-time Freeness in the Kicked Top (2024), arXiv:2411.12050. https://doi.org/10.48550/arXiv.2411.12050 arXiv:2411.12050 [20] H. J. Chen and J. Kudler-Flam, Phys. Rev. B 111, 014311 (2025). https://doi.org/10.1103/PhysRevB.111.014311 [21] H. A. Camargo, Y. Fu, V. Jahnke, K. Pal, and K.-Y. Kim, Quantum Signatures of Chaos from Free Probability (2025), arxiv:2503.20338. https://doi.org/10.48550/arXiv.2503.20338 arXiv:arxiv:2503.20338 [22] M. Fava, J. Kurchan, and S. Pappalardi, Phys. Rev. X 15, 011031 (2025). https://doi.org/10.1103/PhysRevX.15.011031 [23] S. Gopalakrishnan and A. Lamacraft, Phys. Rev. B 100, 064309 (2019). https://doi.org/10.1103/PhysRevB.100.064309 [24] B. Bertini, P. Kos, and T. Prosen, Phys. Rev. Lett. 123, 210601 (2019). https://doi.org/10.1103/PhysRevLett.123.210601 [25] B. Bertini, P. Kos, and T. Prosen, SciPost Phys. 8, 068 (2020). https://doi.org/10.21468/SciPostPhys.8.4.068 [26] T. Gombor and B. Pozsgay, SciPost Phys. 12, 102 (2022). https://doi.org/10.21468/SciPostPhys.12.3.102 [27] T. Holden-Dye, L. Masanes, and A. Pal, Quantum 9, 1615 (2025). https://doi.org/10.22331/q-2025-01-30-1615 [28] G. M. Sommers, D. A. Huse, and M. J. Gullans, PRX Quantum 4, 030313 (2023). https://doi.org/10.1103/PRXQuantum.4.030313 [29] P. W. Claeys and A. Lamacraft, J. Phys. A: Math. Theor. 57, 405301 (2024). https://doi.org/10.1088/1751-8121/ad776a [30] A. Foligno, P. Calabrese, and B. Bertini, PRX Quantum 6, 010324 (2025). https://doi.org/10.1103/PRXQuantum.6.010324 [31] N. Dowling, P. Kos, and K. Modi, Phys. Rev. Lett. 131, 180403 (2023). https://doi.org/10.1103/PhysRevLett.131.180403 [32] M. A. Rampp and P. W. Claeys, Quantum 8, 1434 (2024). https://doi.org/10.22331/q-2024-08-08-1434 [33] G. Giudice, G. Giudici, M. Sonner, J. Thoenniss, A. Lerose, D. A. Abanin, and L. Piroli, Phys. Rev. Lett. 128, 220401 (2022). https://doi.org/10.1103/PhysRevLett.128.220401 [34] J. A. Montañà López and P. Kos, J. Phys. A: Math. Theor. 57, 475301 (2024). https://doi.org/10.1088/1751-8121/ad85b0 [35] A. Chan, A. De Luca, and J. T. Chalker, Phys. Rev. Lett. 122, 220601 (2019). https://doi.org/10.1103/PhysRevLett.122.220601 [36] D. Hahn, D. J. Luitz, and J. T. Chalker, Phys. Rev. X 14, 031029 (2024). https://doi.org/10.1103/PhysRevX.14.031029 [37] R. Speicher, Math. Ann. 298, 611 (1994). https://doi.org/10.1007/BF01459754 [38] M. Vanicat, L. Zadnik, and T. Prosen, Phys. Rev. Lett. 121, 030606 (2018). https://doi.org/10.1103/PhysRevLett.121.030606 [39] M. Ljubotina, L. Zadnik, and T. Prosen, Phys. Rev. Lett. 122, 150605 (2019). https://doi.org/10.1103/PhysRevLett.122.150605 [40] A. J. Friedman, A. Chan, A. De Luca, and J. T. Chalker, Phys. Rev. Lett. 123, 210603 (2019). https://doi.org/10.1103/PhysRevLett.123.210603 [41] Ž. Krajnik, E. Ilievski, and T. Prosen, SciPost Phys. 9, 038 (2020). https://doi.org/10.21468/SciPostPhys.9.3.038 [42] P. W. Claeys, J. Herzog-Arbeitman, and A. Lamacraft, SciPost Phys. 12, 007 (2022). https://doi.org/10.21468/SciPostPhys.12.1.007 [43] R. Suzuki, K. Mitarai, and K. Fujii, Quantum 6, 631 (2022). https://doi.org/10.22331/q-2022-01-24-631 [44] T. Gombor and B. Pozsgay, SciPost Phys. 16, 114 (2024). https://doi.org/10.21468/SciPostPhys.16.4.114 [45] E. Ilievski, E. Quinn, J. De Nardis, and M. Brockmann, J. Stat. Mech. 2016, 063101 (2016). https://doi.org/10.1088/1742-5468/2016/06/063101 [46] S. Gopalakrishnan and B. Zakirov, Quantum Sci. Technol. 3, 044004 (2018). https://doi.org/10.1088/2058-9565/aad759 [47] A. Maillard, L. Foini, A. L. Castellanos, F. Krzakala, M. Mézard, and L. Zdeborová, J. Stat. Mech. , 113301 (2019). https://doi.org/10.1088/1742-5468/ab4bbb [48] T. LeBlond, K. Mallayya, L. Vidmar, and M. Rigol, Phys. Rev. E 100, 062134 (2019). https://doi.org/10.1103/PhysRevE.100.062134 [49] T. LeBlond and M. Rigol, Phys. Rev. E 102, 062113 (2020). https://doi.org/10.1103/PhysRevE.102.062113 [50] F. H. L. Essler and A. J. J. M. de Klerk, Phys. Rev. X 14, 031048 (2024). https://doi.org/10.1103/PhysRevX.14.031048 [51] M. Pandey, P. W. Claeys, D. K. Campbell, A. Polkovnikov, and D. Sels, Phys. Rev. X 10, 041017 (2020). https://doi.org/10.1103/PhysRevX.10.041017 [52] M. Brenes, J. Goold, and M. Rigol, Phys. Rev. B 102, 075127 (2020). https://doi.org/10.1103/PhysRevB.102.075127 [53] P. W. Claeys and A. Lamacraft, Phys. Rev. Lett. 126, 100603 (2021). https://doi.org/10.1103/PhysRevLett.126.100603 [54] M. A. Rampp, R. Moessner, and P. W. Claeys, Phys. Rev. Lett. 130, 130402 (2023). https://doi.org/10.1103/PhysRevLett.130.130402 [55] G. O. Alves, F. Fritzsch, and P. W. Claeys, Alves-gabriel/dual_unitary_eth: First release, Zenodo (2025). https://doi.org/10.5281/ZENODO.15576372 [56] J. B. Carrell, in Groups, Matrices, and Vector Spaces: A Group Theoretic Approach to Linear Algebra, edited by J. B. Carrell (Springer, New York, NY, 2017) pp. 337–382. https://doi.org/10.1007/978-0-387-79428-0_11 [57] B. Collins and P. Śniady, Commun. Math. Phys. 264, 773 (2006). https://doi.org/10.1007/s00220-006-1554-3 [58] H. Fredricksen and J. Maiorana, Universitext. 23, 207 (1978). https://doi.org/10.1016/0012-365X(78)90002-X [59] H. Fredricksen and I. J. Kessler, Universitext. 61, 181 (1986). https://doi.org/10.1016/0012-365X(86)90089-0 [60] J. Sawada, SIAM J. Comput. 31, 259 (2001). https://doi.org/10.1137/S0097539700377037 [61] F. Ruskey, C. Savage, and T.
Min Yih Wang, J. Algorithm. 13, 414 (1992). https://doi.org/10.1016/0196-6774(92)90047-G [62] R. Speicher, Lecture Notes on "Free Probability Theory" (2019), arxiv:1908.08125. https://doi.org/10.48550/arXiv.1908.08125 arXiv:1908.08125Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-15 15:08:14: Could not fetch cited-by data for 10.22331/q-2025-12-15-1949 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-15 15:08:15: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractThe eigenstate thermalization hypothesis (ETH) is the leading interpretation in our current understanding of quantum thermalization. Recent results uncovered strong connections between quantum correlations in thermalizing systems and the structure of free probability theory, leading to the notion of full ETH. However, most studies have been performed for ergodic systems and it is still unclear whether or how full ETH manifests in ergodicity-breaking models. We fill this gap by studying standard probes of full ETH in ergodicity-breaking quantum circuits, presenting numerical and analytical results for interacting integrable systems. These probes can display distinct behavior and undergo a different scaling than the ones observed in ergodic systems. For the analytical results we consider an interacting integrable dual-unitary model and present the exact eigenstates, allowing us to analytically express common probes for full ETH. We discuss the underlying mechanisms responsible for these differences and show how the presence of solitons dictates the behavior of ETH-related quantities in the dual-unitary model. We show numerical evidence that this behavior is sufficiently generic away from dual-unitarity when restricted to the appropriate symmetry sectors.Featured image: According to the eigenstate thermalization hypothesis (ETH), eigenstates of quantum many-body systems appear thermal whenever local observables are probed. For such ergodic systems, the dynamics of out-of-time order correlators can be decomposed in the sum of thermal cumulants — an ETH-based analogue of free cumulants. This decomposition is based on the combinatorics of non-crossing partitions and correctly captures the long-time behavior of OTOCs. In this work, we investigate the behavior of such probes and their underlying structure in ergodicity-breaking quantum circuits.► BibTeX data@article{Alves2025probesoffull, doi = {10.22331/q-2025-12-15-1949}, url = {https://doi.org/10.22331/q-2025-12-15-1949}, title = {Probes of {F}ull {E}igenstate {T}hermalization in {E}rgodicity-{B}reaking {Q}uantum {C}ircuits}, author = {Alves, Gabriel O. and Fritzsch, Felix and Claeys, Pieter W.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1949}, month = dec, year = {2025} }► References [1] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (Wiley, New York, 1985). [2] H. Tasaki, Phys. Rev. Lett. 80, 1373 (1998). https://doi.org/10.1103/PhysRevLett.80.1373 [3] A. Nazir and G. Schaller, Thermodynamics in the Quantum Regime, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, Vol. 195 (Springer International Publishing, Cham, 2018) 1805.08307. https://doi.org/10.1007/978-3-319-99046-0 arXiv:1805.08307 [4] J. M. Deutsch, Phys. Rev. A 43, 2046 (1991). https://doi.org/10.1103/physreva.43.2046 [5] M. Srednicki, Phys. Rev. E 50, 888 (1994). https://doi.org/10.1103/PhysRevE.50.888 [6] L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016). https://doi.org/10.1080/00018732.2016.1198134 [7] J. M. Deutsch, Rep. Prog. Phys. 81, 082001 (2018). https://doi.org/10.1088/1361-6633/aac9f1 [8] L. Foini and J. Kurchan, Phys. Rev. E 99, 042139 (2019). https://doi.org/10.1103/PhysRevE.99.042139 [9] S. Pappalardi, L. Foini, and J. Kurchan, Phys. Rev. Lett. 129, 170603 (2022). https://doi.org/10.1103/PhysRevLett.129.170603 [10] D. Voiculescu, in Operator Algebras and Their Connections with Topology and Ergodic Theory, edited by H. Araki, C. C. Moore, Ş.-V. Stratila, and D.-V. Voiculescu (Springer, Berlin, Heidelberg, 1985) pp. 556–588. https://doi.org/10.1007/BFb0074909 [11] D. Voiculescu, J. Funct. Anal. 66, 323 (1986). https://doi.org/10.1016/0022-1236(86)90062-5 [12] D. Voiculescu, J. Operat. Theor. 18, 223 (1987). [13] J. Novak and M. LaCroix, Three lectures on free probability (2012), arXiv:1205.2097. https://doi.org/10.48550/arXiv.1205.2097 arXiv:1205.2097 [14] J. A. Mingo and R. Speicher, Free Probability and Random Matrices, Fields Institute Monographs, Vol. 35 (Springer New York, New York, NY, 2017). https://doi.org/10.1007/978-1-4939-6942-5 [15] S. Pappalardi, F. Fritzsch, and T. Prosen, Phys. Rev. Lett. 134, 140404 (2025). https://doi.org/10.1103/PhysRevLett.134.140404 [16] S. Pappalardi, L. Foini, and J. Kurchan, Quantum 8, 1227 (2024). https://doi.org/10.22331/q-2024-01-11-1227 [17] S. Jindal and P. Hosur, J. High Energ. Phys. 2024 (66), 66. https://doi.org/10.1007/JHEP09(2024)066 [18] F. Fritzsch, T. Prosen, and S. Pappalardi, Phys. Rev. B 111, 054303 (2025). https://doi.org/10.1103/PhysRevB.111.054303 [19] E. Vallini and S. Pappalardi, Long-time Freeness in the Kicked Top (2024), arXiv:2411.12050. https://doi.org/10.48550/arXiv.2411.12050 arXiv:2411.12050 [20] H. J. Chen and J. Kudler-Flam, Phys. Rev. B 111, 014311 (2025). https://doi.org/10.1103/PhysRevB.111.014311 [21] H. A. Camargo, Y. Fu, V. Jahnke, K. Pal, and K.-Y. Kim, Quantum Signatures of Chaos from Free Probability (2025), arxiv:2503.20338. https://doi.org/10.48550/arXiv.2503.20338 arXiv:arxiv:2503.20338 [22] M. Fava, J. Kurchan, and S. Pappalardi, Phys. Rev. X 15, 011031 (2025). https://doi.org/10.1103/PhysRevX.15.011031 [23] S. Gopalakrishnan and A. Lamacraft, Phys. Rev. B 100, 064309 (2019). https://doi.org/10.1103/PhysRevB.100.064309 [24] B. Bertini, P. Kos, and T. Prosen, Phys. Rev. Lett. 123, 210601 (2019). https://doi.org/10.1103/PhysRevLett.123.210601 [25] B. Bertini, P. Kos, and T. Prosen, SciPost Phys. 8, 068 (2020). https://doi.org/10.21468/SciPostPhys.8.4.068 [26] T. Gombor and B. Pozsgay, SciPost Phys. 12, 102 (2022). https://doi.org/10.21468/SciPostPhys.12.3.102 [27] T. Holden-Dye, L. Masanes, and A. Pal, Quantum 9, 1615 (2025). https://doi.org/10.22331/q-2025-01-30-1615 [28] G. M. Sommers, D. A. Huse, and M. J. Gullans, PRX Quantum 4, 030313 (2023). https://doi.org/10.1103/PRXQuantum.4.030313 [29] P. W. Claeys and A. Lamacraft, J. Phys. A: Math. Theor. 57, 405301 (2024). https://doi.org/10.1088/1751-8121/ad776a [30] A. Foligno, P. Calabrese, and B. Bertini, PRX Quantum 6, 010324 (2025). https://doi.org/10.1103/PRXQuantum.6.010324 [31] N. Dowling, P. Kos, and K. Modi, Phys. Rev. Lett. 131, 180403 (2023). https://doi.org/10.1103/PhysRevLett.131.180403 [32] M. A. Rampp and P. W. Claeys, Quantum 8, 1434 (2024). https://doi.org/10.22331/q-2024-08-08-1434 [33] G. Giudice, G. Giudici, M. Sonner, J. Thoenniss, A. Lerose, D. A. Abanin, and L. Piroli, Phys. Rev. Lett. 128, 220401 (2022). https://doi.org/10.1103/PhysRevLett.128.220401 [34] J. A. Montañà López and P. Kos, J. Phys. A: Math. Theor. 57, 475301 (2024). https://doi.org/10.1088/1751-8121/ad85b0 [35] A. Chan, A. De Luca, and J. T. Chalker, Phys. Rev. Lett. 122, 220601 (2019). https://doi.org/10.1103/PhysRevLett.122.220601 [36] D. Hahn, D. J. Luitz, and J. T. Chalker, Phys. Rev. X 14, 031029 (2024). https://doi.org/10.1103/PhysRevX.14.031029 [37] R. Speicher, Math. Ann. 298, 611 (1994). https://doi.org/10.1007/BF01459754 [38] M. Vanicat, L. Zadnik, and T. Prosen, Phys. Rev. Lett. 121, 030606 (2018). https://doi.org/10.1103/PhysRevLett.121.030606 [39] M. Ljubotina, L. Zadnik, and T. Prosen, Phys. Rev. Lett. 122, 150605 (2019). https://doi.org/10.1103/PhysRevLett.122.150605 [40] A. J. Friedman, A. Chan, A. De Luca, and J. T. Chalker, Phys. Rev. Lett. 123, 210603 (2019). https://doi.org/10.1103/PhysRevLett.123.210603 [41] Ž. Krajnik, E. Ilievski, and T. Prosen, SciPost Phys. 9, 038 (2020). https://doi.org/10.21468/SciPostPhys.9.3.038 [42] P. W. Claeys, J. Herzog-Arbeitman, and A. Lamacraft, SciPost Phys. 12, 007 (2022). https://doi.org/10.21468/SciPostPhys.12.1.007 [43] R. Suzuki, K. Mitarai, and K. Fujii, Quantum 6, 631 (2022). https://doi.org/10.22331/q-2022-01-24-631 [44] T. Gombor and B. Pozsgay, SciPost Phys. 16, 114 (2024). https://doi.org/10.21468/SciPostPhys.16.4.114 [45] E. Ilievski, E. Quinn, J. De Nardis, and M. Brockmann, J. Stat. Mech. 2016, 063101 (2016). https://doi.org/10.1088/1742-5468/2016/06/063101 [46] S. Gopalakrishnan and B. Zakirov, Quantum Sci. Technol. 3, 044004 (2018). https://doi.org/10.1088/2058-9565/aad759 [47] A. Maillard, L. Foini, A. L. Castellanos, F. Krzakala, M. Mézard, and L. Zdeborová, J. Stat. Mech. , 113301 (2019). https://doi.org/10.1088/1742-5468/ab4bbb [48] T. LeBlond, K. Mallayya, L. Vidmar, and M. Rigol, Phys. Rev. E 100, 062134 (2019). https://doi.org/10.1103/PhysRevE.100.062134 [49] T. LeBlond and M. Rigol, Phys. Rev. E 102, 062113 (2020). https://doi.org/10.1103/PhysRevE.102.062113 [50] F. H. L. Essler and A. J. J. M. de Klerk, Phys. Rev. X 14, 031048 (2024). https://doi.org/10.1103/PhysRevX.14.031048 [51] M. Pandey, P. W. Claeys, D. K. Campbell, A. Polkovnikov, and D. Sels, Phys. Rev. X 10, 041017 (2020). https://doi.org/10.1103/PhysRevX.10.041017 [52] M. Brenes, J. Goold, and M. Rigol, Phys. Rev. B 102, 075127 (2020). https://doi.org/10.1103/PhysRevB.102.075127 [53] P. W. Claeys and A. Lamacraft, Phys. Rev. Lett. 126, 100603 (2021). https://doi.org/10.1103/PhysRevLett.126.100603 [54] M. A. Rampp, R. Moessner, and P. W. Claeys, Phys. Rev. Lett. 130, 130402 (2023). https://doi.org/10.1103/PhysRevLett.130.130402 [55] G. O. Alves, F. Fritzsch, and P. W. Claeys, Alves-gabriel/dual_unitary_eth: First release, Zenodo (2025). https://doi.org/10.5281/ZENODO.15576372 [56] J. B. Carrell, in Groups, Matrices, and Vector Spaces: A Group Theoretic Approach to Linear Algebra, edited by J. B. Carrell (Springer, New York, NY, 2017) pp. 337–382. https://doi.org/10.1007/978-0-387-79428-0_11 [57] B. Collins and P. Śniady, Commun. Math. Phys. 264, 773 (2006). https://doi.org/10.1007/s00220-006-1554-3 [58] H. Fredricksen and J. Maiorana, Universitext. 23, 207 (1978). https://doi.org/10.1016/0012-365X(78)90002-X [59] H. Fredricksen and I. J. Kessler, Universitext. 61, 181 (1986). https://doi.org/10.1016/0012-365X(86)90089-0 [60] J. Sawada, SIAM J. Comput. 31, 259 (2001). https://doi.org/10.1137/S0097539700377037 [61] F. Ruskey, C. Savage, and T.
Min Yih Wang, J. Algorithm. 13, 414 (1992). https://doi.org/10.1016/0196-6774(92)90047-G [62] R. Speicher, Lecture Notes on "Free Probability Theory" (2019), arxiv:1908.08125. https://doi.org/10.48550/arXiv.1908.08125 arXiv:1908.08125Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-15 15:08:14: Could not fetch cited-by data for 10.22331/q-2025-12-15-1949 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-15 15:08:15: No response from ADS or unable to decode the received json data when getting the list of citing works.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.