Short remarks on shallow unitary circuits
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Abstract(i) We point out that every local unitary circuit of depth smaller than the linear system size is easily distinguished from a global Haar random unitary if there is a conserved quantity that is a sum of local operators. This is always the case with a continuous onsite symmetry or with a local energy conservation law. (ii) We explain a simple algorithm for a formulation of the shallow unitary circuit learning problem and relate it to an open question on strictly locality-preserving unitaries (quantum cellular automata). (iii) We show that any translation-invariant quantum cellular automaton in $D$-dimensional lattice of volume $V$ can be implemented using only $O(V)$ local gates in a staircase fashion using invertible subalgebra pumping.► BibTeX data@article{Haah2025shortremarksshallow, doi = {10.22331/q-2025-12-11-1940}, url = {https://doi.org/10.22331/q-2025-12-11-1940}, title = {Short remarks on shallow unitary circuits}, author = {Haah, Jeongwan}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1940}, month = dec, year = {2025} }► References [1] Pablo Arrighi, Vincent Nesme, and Reinhard Werner. Unitarity plus causality implies localizability. Journal of Computer and System Sciences, 77:372–378, 2011. arXiv:0711.3975, doi:10.1016/j.jcss.2010.05.004. https://doi.org/10.1016/j.jcss.2010.05.004 arXiv:0711.3975 [2] Xie Chen, Arpit Dua, Michael Hermele, David T. Stephen, Nathanan Tantivasadakarn, Robijn Vanhove, and Jing-Yu Zhao. Sequential quantum circuits as maps between gapped phases. Phys. Rev. B, 109(7):075116, 2024. arXiv:2307.01267, doi:10.1103/physrevb.109.075116. https://doi.org/10.1103/physrevb.109.075116 arXiv:2307.01267 [3] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Phys. Rev. B, 82(15):155138, 2010. arXiv:1004.3835, doi:10.1103/physrevb.82.155138. https://doi.org/10.1103/physrevb.82.155138 arXiv:1004.3835 [4] M. Freedman and M. B. Hastings. Classification of quantum cellular automata. Commun. Math. Phys., 376(2):1171–1222, 2020. arXiv:1902.10285, doi:10.1007/s00220-020-03735-y. https://doi.org/10.1007/s00220-020-03735-y arXiv:1902.10285 [5] Michael Freedman, Jeongwan Haah, and Matthew B. Hastings. The group structure of quantum cellular automata. Commun. Math. Phys. 389, 1277-1302 (2022), 389(3):1277–1302, 2019. arXiv:1910.07998, doi:10.1007/s00220-022-04316-x. https://doi.org/10.1007/s00220-022-04316-x arXiv:1910.07998 [6] Lukasz Fidkowski, Hoi Chun Po, Andrew C. Potter, and Ashvin Vishwanath. Interacting invariants for floquet phases of fermions in two dimensions. Phys. Rev. B, 99(8):085115, February 2019. arXiv:1703.07360, doi:10.1103/physrevb.99.085115. https://doi.org/10.1103/physrevb.99.085115 arXiv:1703.07360 [7] D. Gross, V. Nesme, H. Vogts, and R. F. Werner. Index theory of one dimensional quantum walks and cellular automata. Commun. Math. Phys., 310(2):419–454, 2012. arXiv:0910.3675, doi:10.1007/s00220-012-1423-1. https://doi.org/10.1007/s00220-012-1423-1 arXiv:0910.3675 [8] Jeongwan Haah. Clifford quantum cellular automata: Trivial group in 2d and Witt group in 3d. J. Math. Phys., 62(9):092202, 2021. arXiv:1907.02075, doi:10.1063/5.0022185. https://doi.org/10.1063/5.0022185 arXiv:1907.02075 [9] Jeongwan Haah. Invertible subalgebras. Commun. Math. Phys., 403(2):661–698, 2023. arXiv:2211.02086, doi:10.1007/s00220-023-04806-6. https://doi.org/10.1007/s00220-023-04806-6 arXiv:2211.02086 [10] Jeongwan Haah. Topological phases of unitary dynamics: Classification in Clifford category. Commun. Math. Phys., 406(76), 2025. arXiv:2205.09141, doi:10.1007/s00220-025-05239-z. https://doi.org/10.1007/s00220-025-05239-z arXiv:2205.09141 [11] Sumner N. Hearth, Michael O. Flynn, Anushya Chandran, and Chris R. Laumann. Unitary k-designs from random number-conserving quantum circuits. Phys. Rev. X, 15(021022), 2025. arXiv:2306.01035, doi:10.1103/PhysRevX.15.021022. https://doi.org/10.1103/PhysRevX.15.021022 arXiv:2306.01035 [12] Jeongwan Haah, Robin Kothari, and Ewin Tang. Learning quantum Hamiltonians from high-temperature Gibbs states and real-time evolutions. Nature Physics, 20(6):1027–1031, 2024. arXiv:2108.04842, doi:10.1038/s41567-023-02376-x. https://doi.org/10.1038/s41567-023-02376-x arXiv:2108.04842 [13] Hsin-Yuan Huang, Yunchao Liu, Michael Broughton, Isaac Kim, Anurag Anshu, Zeph Landau, and Jarrod R. McClean. Learning shallow quantum circuits. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024), pages 1343–1351, 2024. arXiv:2401.10095, doi:10.1145/3618260.3649722. https://doi.org/10.1145/3618260.3649722 arXiv:2401.10095 [14] Hyun-Soo Kim, Isaac H. Kim, and Daniel Ranard. Learning state preparation circuits for quantum phases of matter. October 2024. arXiv:2410.23544, doi:10.48550/ARXIV.2410.23544. https://doi.org/10.48550/ARXIV.2410.23544 arXiv:2410.23544 [15] Vedika Khemani, Ashvin Vishwanath, and D. A. Huse. Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws. Phys. Rev. X, 8(3):031057, September 2018. arXiv:1710.09835, doi:10.1103/physrevx.8.031057. https://doi.org/10.1103/physrevx.8.031057 arXiv:1710.09835 [16] Zeph Landau and Yunchao Liu. Learning quantum states prepared by shallow circuits in polynomial time. 2024. arXiv:2410.23618, doi:10.48550/ARXIV.2410.23618. https://doi.org/10.48550/ARXIV.2410.23618 arXiv:2410.23618 [17] Nicholas LaRacuente and Felix Leditzky. Approximate unitary $k$-designs from shallow, low-communication circuits. July 2024. arXiv:2407.07876, doi:10.48550/ARXIV.2407.07876. https://doi.org/10.48550/ARXIV.2407.07876 arXiv:2407.07876 [18] Iman Marvian. Restrictions on realizable unitary operations imposed by symmetry and locality. Nat. Phys., 18(3):283–289, 2022. arXiv:2003.05524, doi:10.1038/s41567-021-01464-0. https://doi.org/10.1038/s41567-021-01464-0 arXiv:2003.05524 [19] Ruochen Ma, Yabo Li, and Meng Cheng. Quantum cellular automata on symmetric subalgebras. November 2024. arXiv:2411.19280, doi:10.48550/ARXIV.2411.19280. https://doi.org/10.48550/ARXIV.2411.19280 arXiv:2411.19280 [20] Tibor Rakovszky, Frank Pollmann, and C. W. von Keyserlingk. Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation. Phys. Rev. X, 8(3):031058, September 2018. arXiv:1710.09827, doi:10.1103/physrevx.8.031058. https://doi.org/10.1103/physrevx.8.031058 arXiv:1710.09827 [21] Daniel Ranard, Michael Walter, and Freek Witteveen. A converse to lieb-robinson bounds in one dimension using index theory.
Annales Henri Poincaré, 23(11):3905–3979, July 2022. arXiv:2012.00741, doi:10.1007/s00023-022-01193-x. https://doi.org/10.1007/s00023-022-01193-x arXiv:2012.00741 [22] Thomas Schuster, Jonas Haferkamp, and Hsin-Yuan Huang. Random unitaries in extremely low depth. 2024. arXiv:2407.07754, doi:10.48550/ARXIV.2407.07754. https://doi.org/10.48550/ARXIV.2407.07754 arXiv:2407.07754 [23] B. Schumacher and R. F. Werner. Reversible quantum cellular automata. May 2004. arXiv:quant-ph/0405174, doi:10.48550/ARXIV.QUANT-PH/0405174. https://doi.org/10.48550/ARXIV.QUANT-PH/0405174 arXiv:quant-ph/0405174 [24] Emanuele Tirrito, Xhek Turkeshi, and Piotr Sierant. Anticoncentration and magic spreading under ergodic quantum dynamics. December 2024. arXiv:2412.10229, doi:10.48550/ARXIV.2412.10229. https://doi.org/10.48550/ARXIV.2412.10229 arXiv:2412.10229 [25] Yusen Wu, Yukun Zhang, Chuan Wang, and Xiao Yuan. Hamiltonian dynamics learning: A scalable approach to quantum process characterization. 2025. arXiv:2503.24171, doi:10.48550/ARXIV.2503.24171. https://doi.org/10.48550/ARXIV.2503.24171 arXiv:2503.24171Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-11 15:59:16: Could not fetch cited-by data for 10.22331/q-2025-12-11-1940 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-11 15:59:16: 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. Abstract(i) We point out that every local unitary circuit of depth smaller than the linear system size is easily distinguished from a global Haar random unitary if there is a conserved quantity that is a sum of local operators. This is always the case with a continuous onsite symmetry or with a local energy conservation law. (ii) We explain a simple algorithm for a formulation of the shallow unitary circuit learning problem and relate it to an open question on strictly locality-preserving unitaries (quantum cellular automata). (iii) We show that any translation-invariant quantum cellular automaton in $D$-dimensional lattice of volume $V$ can be implemented using only $O(V)$ local gates in a staircase fashion using invertible subalgebra pumping.► BibTeX data@article{Haah2025shortremarksshallow, doi = {10.22331/q-2025-12-11-1940}, url = {https://doi.org/10.22331/q-2025-12-11-1940}, title = {Short remarks on shallow unitary circuits}, author = {Haah, Jeongwan}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1940}, month = dec, year = {2025} }► References [1] Pablo Arrighi, Vincent Nesme, and Reinhard Werner. Unitarity plus causality implies localizability. Journal of Computer and System Sciences, 77:372–378, 2011. arXiv:0711.3975, doi:10.1016/j.jcss.2010.05.004. https://doi.org/10.1016/j.jcss.2010.05.004 arXiv:0711.3975 [2] Xie Chen, Arpit Dua, Michael Hermele, David T. Stephen, Nathanan Tantivasadakarn, Robijn Vanhove, and Jing-Yu Zhao. Sequential quantum circuits as maps between gapped phases. Phys. Rev. B, 109(7):075116, 2024. arXiv:2307.01267, doi:10.1103/physrevb.109.075116. https://doi.org/10.1103/physrevb.109.075116 arXiv:2307.01267 [3] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Phys. Rev. B, 82(15):155138, 2010. arXiv:1004.3835, doi:10.1103/physrevb.82.155138. https://doi.org/10.1103/physrevb.82.155138 arXiv:1004.3835 [4] M. Freedman and M. B. Hastings. Classification of quantum cellular automata. Commun. Math. Phys., 376(2):1171–1222, 2020. arXiv:1902.10285, doi:10.1007/s00220-020-03735-y. https://doi.org/10.1007/s00220-020-03735-y arXiv:1902.10285 [5] Michael Freedman, Jeongwan Haah, and Matthew B. Hastings. The group structure of quantum cellular automata. Commun. Math. Phys. 389, 1277-1302 (2022), 389(3):1277–1302, 2019. arXiv:1910.07998, doi:10.1007/s00220-022-04316-x. https://doi.org/10.1007/s00220-022-04316-x arXiv:1910.07998 [6] Lukasz Fidkowski, Hoi Chun Po, Andrew C. Potter, and Ashvin Vishwanath. Interacting invariants for floquet phases of fermions in two dimensions. Phys. Rev. B, 99(8):085115, February 2019. arXiv:1703.07360, doi:10.1103/physrevb.99.085115. https://doi.org/10.1103/physrevb.99.085115 arXiv:1703.07360 [7] D. Gross, V. Nesme, H. Vogts, and R. F. Werner. Index theory of one dimensional quantum walks and cellular automata. Commun. Math. Phys., 310(2):419–454, 2012. arXiv:0910.3675, doi:10.1007/s00220-012-1423-1. https://doi.org/10.1007/s00220-012-1423-1 arXiv:0910.3675 [8] Jeongwan Haah. Clifford quantum cellular automata: Trivial group in 2d and Witt group in 3d. J. Math. Phys., 62(9):092202, 2021. arXiv:1907.02075, doi:10.1063/5.0022185. https://doi.org/10.1063/5.0022185 arXiv:1907.02075 [9] Jeongwan Haah. Invertible subalgebras. Commun. Math. Phys., 403(2):661–698, 2023. arXiv:2211.02086, doi:10.1007/s00220-023-04806-6. https://doi.org/10.1007/s00220-023-04806-6 arXiv:2211.02086 [10] Jeongwan Haah. Topological phases of unitary dynamics: Classification in Clifford category. Commun. Math. Phys., 406(76), 2025. arXiv:2205.09141, doi:10.1007/s00220-025-05239-z. https://doi.org/10.1007/s00220-025-05239-z arXiv:2205.09141 [11] Sumner N. Hearth, Michael O. Flynn, Anushya Chandran, and Chris R. Laumann. Unitary k-designs from random number-conserving quantum circuits. Phys. Rev. X, 15(021022), 2025. arXiv:2306.01035, doi:10.1103/PhysRevX.15.021022. https://doi.org/10.1103/PhysRevX.15.021022 arXiv:2306.01035 [12] Jeongwan Haah, Robin Kothari, and Ewin Tang. Learning quantum Hamiltonians from high-temperature Gibbs states and real-time evolutions. Nature Physics, 20(6):1027–1031, 2024. arXiv:2108.04842, doi:10.1038/s41567-023-02376-x. https://doi.org/10.1038/s41567-023-02376-x arXiv:2108.04842 [13] Hsin-Yuan Huang, Yunchao Liu, Michael Broughton, Isaac Kim, Anurag Anshu, Zeph Landau, and Jarrod R. McClean. Learning shallow quantum circuits. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024), pages 1343–1351, 2024. arXiv:2401.10095, doi:10.1145/3618260.3649722. https://doi.org/10.1145/3618260.3649722 arXiv:2401.10095 [14] Hyun-Soo Kim, Isaac H. Kim, and Daniel Ranard. Learning state preparation circuits for quantum phases of matter. October 2024. arXiv:2410.23544, doi:10.48550/ARXIV.2410.23544. https://doi.org/10.48550/ARXIV.2410.23544 arXiv:2410.23544 [15] Vedika Khemani, Ashvin Vishwanath, and D. A. Huse. Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws. Phys. Rev. X, 8(3):031057, September 2018. arXiv:1710.09835, doi:10.1103/physrevx.8.031057. https://doi.org/10.1103/physrevx.8.031057 arXiv:1710.09835 [16] Zeph Landau and Yunchao Liu. Learning quantum states prepared by shallow circuits in polynomial time. 2024. arXiv:2410.23618, doi:10.48550/ARXIV.2410.23618. https://doi.org/10.48550/ARXIV.2410.23618 arXiv:2410.23618 [17] Nicholas LaRacuente and Felix Leditzky. Approximate unitary $k$-designs from shallow, low-communication circuits. July 2024. arXiv:2407.07876, doi:10.48550/ARXIV.2407.07876. https://doi.org/10.48550/ARXIV.2407.07876 arXiv:2407.07876 [18] Iman Marvian. Restrictions on realizable unitary operations imposed by symmetry and locality. Nat. Phys., 18(3):283–289, 2022. arXiv:2003.05524, doi:10.1038/s41567-021-01464-0. https://doi.org/10.1038/s41567-021-01464-0 arXiv:2003.05524 [19] Ruochen Ma, Yabo Li, and Meng Cheng. Quantum cellular automata on symmetric subalgebras. November 2024. arXiv:2411.19280, doi:10.48550/ARXIV.2411.19280. https://doi.org/10.48550/ARXIV.2411.19280 arXiv:2411.19280 [20] Tibor Rakovszky, Frank Pollmann, and C. W. von Keyserlingk. Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation. Phys. Rev. X, 8(3):031058, September 2018. arXiv:1710.09827, doi:10.1103/physrevx.8.031058. https://doi.org/10.1103/physrevx.8.031058 arXiv:1710.09827 [21] Daniel Ranard, Michael Walter, and Freek Witteveen. A converse to lieb-robinson bounds in one dimension using index theory.
Annales Henri Poincaré, 23(11):3905–3979, July 2022. arXiv:2012.00741, doi:10.1007/s00023-022-01193-x. https://doi.org/10.1007/s00023-022-01193-x arXiv:2012.00741 [22] Thomas Schuster, Jonas Haferkamp, and Hsin-Yuan Huang. Random unitaries in extremely low depth. 2024. arXiv:2407.07754, doi:10.48550/ARXIV.2407.07754. https://doi.org/10.48550/ARXIV.2407.07754 arXiv:2407.07754 [23] B. Schumacher and R. F. Werner. Reversible quantum cellular automata. May 2004. arXiv:quant-ph/0405174, doi:10.48550/ARXIV.QUANT-PH/0405174. https://doi.org/10.48550/ARXIV.QUANT-PH/0405174 arXiv:quant-ph/0405174 [24] Emanuele Tirrito, Xhek Turkeshi, and Piotr Sierant. Anticoncentration and magic spreading under ergodic quantum dynamics. December 2024. arXiv:2412.10229, doi:10.48550/ARXIV.2412.10229. https://doi.org/10.48550/ARXIV.2412.10229 arXiv:2412.10229 [25] Yusen Wu, Yukun Zhang, Chuan Wang, and Xiao Yuan. Hamiltonian dynamics learning: A scalable approach to quantum process characterization. 2025. arXiv:2503.24171, doi:10.48550/ARXIV.2503.24171. https://doi.org/10.48550/ARXIV.2503.24171 arXiv:2503.24171Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-11 15:59:16: Could not fetch cited-by data for 10.22331/q-2025-12-11-1940 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-11 15:59:16: 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.