- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Clifford Quantum Cellular Automata from Topological Quantum Field Theories and Invertible Subalgebras

Meng Sun (孙萌)1,*, Bowen Yang (杨博闻)2,*, Zongyuan Wang (王宗远)1,3,*, Nathanan Tantivasadakarn4,3,5,†, and Yu-An Chen (陳昱安)1,‡

  • 1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
  • 2Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts 02138, USA
  • 3Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
  • 4C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York 11794, USA
  • 5Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA

  • *These authors contributed equally to this work.
  • †Contact author: nathanan.tantivasadakarn@stonybrook.edu
  • ‡Contact author: yuanchen@pku.edu.cn
PDF

PRX Quantum 7, 010362 – Published 31 March, 2026

DOI: https://doi.org/10.1103/4519-v15s

Abstract

We present a general framework for constructing quantum cellular automata (QCAs) from topological quantum field theories (TQFTs) and invertible subalgebras (ISAs) using the cup-product formalism. This approach explicitly realizes all Z2 and Zp Clifford QCAs (for prime p) in all admissible dimensions, in precise agreement with the classification predicted by algebraic L-theory. We determine the orders of these QCAs by explicitly showing that finite powers reduce to the identity up to finite-depth quantum circuits (FDQCs) and lattice translations. In particular, we demonstrate that the Z2 Clifford QCAs in (4l+1) spatial dimensions can be disentangled by non-Clifford FDQCs. Our construction applies beyond cubic lattices, allowing Z2 QCAs to be defined on arbitrary cellulations. Furthermore, we explicitly construct invertible subalgebras in higher dimensions, obtaining Z2 ISAs in 2l spatial dimensions and Zp ISAs in (4l−2) spatial dimensions. These ISAs give rise to Z2 QCAs in (2l+1) dimensions and Zp QCAs in (4l−1) dimensions. We further prove that the QCAs in 3 spatial dimensions constructed via TQFTs and ISAs are equivalent by identifying their boundary algebras and show that this approach extends to higher dimensions. Together, these results establish a unified and dimension-periodic framework for Clifford QCAs, connecting their explicit lattice realizations to field theories.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Quantum cellular automata (QCAs) are reversible transformations that update quantum systems in discrete time while strictly preserving locality. They capture the broadest form of locality-preserving quantum dynamics and have become central to research at the interface of quantum information, condensed matter physics, and topology. A fundamental question is what QCAs cannot be implemented via a smooth time evolution. Rather, the whole transformation must be enacted all at once in a single discrete step. Rigorous results are only known in one and two spatial dimensions. In higher dimensions, their exact structure and explicit construction have remained elusive, with only partial progress in three dimensions.

In this work, we develop a unified framework for constructing Clifford QCAs using topological quantum field theories and invertible subalgebras. This approach realizes all Clifford QCAs in all admissible dimensions, in exact agreement with predictions from algebraic L-theory. The results reveal a striking dimensional periodicity: qubit (Z2) QCAs appear in every two spatial dimensions, while qupit (Zp for odd prime p) QCAs recur every four spatial dimensions. This is analogous to the Bott periodicity in K-theory.

Beyond classification, our framework connects lattice models, field-theoretic actions, and algebraic invariants, providing both conceptual insights and practical tools with potential applications in quantum error correction, Floquet engineering, and higher-form symmetries.

Article Text

References (97)

  1. J. Haah, L. Fidkowski, and M. B. Hastings, Nontrivial quantum cellular automata in higher dimensions, Commun. Math. Phys. 398, 469 (2023).
  2. A. Kapustin, Symmetry protected topological phases, anomalies, and cobordisms: Beyond group cohomology, arXiv:1403.1467.
  3. L. Fidkowski, J. Haah, and M. B. Hastings, Exactly solvable model for a 4+1D beyond-cohomology symmetry-protected topological phase, Phys. Rev. B 101, 155124 (2020).
  4. Y.-A. Chen and P.-S. Hsin, Exactly solvable lattice Hamiltonians and gravitational anomalies, SciPost Phys. 14, 089 (2023).
  5. L. Fidkowski, J. Haah, and M. B. Hastings, A quantum cellular automaton for every symmetry protected topological phase, Phys. Rev. B 112, 035123 (2025).
  6. C. Jones, P. Naaijkens, and D. Penneys, Holography for bulk-boundary local topological order, arXiv:2506.19969.
  7. R. Kobayashi, Y. Li, H. Xue, P.-S. Hsin, and Y.-A. Chen, Generalized statistics on lattices, Phys. Rev. X 16, 011010 (2026).
  8. W. Shirley, C. Zhang, W. Ji, and M. Levin, Anomaly-free symmetries with obstructions to gauging and onsite ability, arXiv:2507.21267.
  9. Y.-T. Tu, D. M. Long, and D. V. Else, Anomalies of global symmetries on the lattice, arXiv:2507.21209.
  10. S. Seifnashri and W. Shirley, Disentangling anomaly-free symmetries of quantum spin chains, arXiv:2503.09717.
  11. A. Kapustin, Higher symmetries and anomalies in quantum lattice systems, arXiv:2505.04719.
  12. A. Kapustin and L. Spodyneiko, Higher symmetries, anomalies, and crossed squares in lattice gauge theory, arXiv:2507.16966.
  13. A. Kapustin and N. Sopenko, Anomalous symmetries of quantum spin chains and a generalization of the Lieb-Schultz-Mattis theorem, Commun. Math. Phys. 406, 238 (2025).
  14. R. Ma, Y. Li, and M. Cheng, Quantum cellular automata on symmetric subalgebras, arXiv:2411.19280.
  15. K. Kawagoe and W. Shirley, Anomaly diagnosis via symmetry restriction in two-dimensional lattice systems, arXiv:2507.07430.
  16. T. A. Brun and L. Mlodinow, Quantum cellular automata and quantum field theory in two spatial dimensions, Phys. Rev. A 102, 062222 (2020).
  17. C. Zhang and M. Levin, Bulk-boundary correspondence for interacting Floquet systems in two dimensions, Phys. Rev. X 13, 031038 (2023).
  18. H. C. Po, L. Fidkowski, T. Morimoto, A. C. Potter, and A. Vishwanath, Chiral Floquet phases of many-body localized bosons, Phys. Rev. X 6, 041070 (2016).
  19. H. C. Po, L. Fidkowski, A. Vishwanath, and A. C. Potter, Radical chiral Floquet phases in a periodically driven Kitaev model and beyond, Phys. Rev. B 96, 245116 (2017).
  20. A. C. Potter and T. Morimoto, Dynamically enriched topological orders in driven two-dimensional systems, Phys. Rev. B 95, 155126 (2017).
  21. A. C. Potter, A. Vishwanath, and L. Fidkowski, Infinite family of three-dimensional Floquet topological paramagnets, Phys. Rev. B 97, 245106 (2018).
  22. C. Zhang and M. Levin, Classification of interacting Floquet phases with U(1) symmetry in two dimensions, Phys. Rev. B 103, 064302 (2021).
  23. P. Glorioso, A. Gromov, and S. Ryu, Effective response theory for Floquet topological systems, Phys. Rev. Res. 3, 013117 (2021).
  24. J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, Matrix product unitaries: Structure, symmetries, and topological invariants, J. Stat. Mech.: Theory Exp. 2017, 083105 (2017).
  25. M. B. Şahinoğlu, S. K. Shukla, F. Bi, and X. Chen, Matrix product representation of locality preserving unitaries, Phys. Rev. B 98, 245122 (2018).
  26. Z. Gong, C. Sünderhauf, N. Schuch, and J. I. Cirac, Classification of matrix-product unitaries with symmetries, Phys. Rev. Lett. 124, 100402 (2020).
  27. L. Piroli, A. Turzillo, S. K. Shukla, and J. I. Cirac, Fermionic quantum cellular automata and generalized matrix-product unitaries, J. Stat. Mech.: Theory Exp. 2021, 013107 (2021).
  28. L. Piroli and J. I. Cirac, Quantum cellular automata, tensor networks, and area laws, Phys. Rev. Lett. 125, 190402 (2020).
  29. Z. Gong, L. Piroli, and J. I. Cirac, Topological lower bound on quantum chaos by entanglement growth, Phys. Rev. Lett. 126, 160601 (2021).
  30. D. Ranard, M. Walter, and F. Witteveen, A converse to Lieb-Robinson bounds in one dimension using index theory, Ann. Henri Poincaré 23, 3905 (2022)..
  31. Z. Gong, A. Nahum, and L. Piroli, Coarse-grained entanglement and operator growth in anomalous dynamics, Phys. Rev. Lett. 128 080602 (2022)..
  32. D. Gross, V. Nesme, H. Vogts, and R. F. Werner, Index theory of one dimensional quantum walks and cellular automata, Commun. Math. Phys. 310, 419 (2012).
  33. M. Freedman and M. B. Hastings, Classification of quantum cellular automata, Commun. Math. Phys. 376, 1171 (2020).
  34. J. Haah, Clifford quantum cellular automata: Trivial group in 2D and Witt group in 3D, J. Math. Phys. 62, 092202 (2021).
  35. W. Shirley, Y.-A. Chen, A. Dua, T. D. Ellison, N. Tantivasadakarn, and D. J. Williamson, Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond, PRX Quantum 3, 030326 (2022).
  36. J. Haah, Topological phases of unitary dynamics: Classification in Clifford category, Commun. Math. Phys. 406, 76 (2025).
  37. B. Yang, Categorifying Clifford QCA, arXiv:2504.14811.
  38. S. Roberts and D. J. Williamson, 3-fermion topological quantum computation, PRX Quantum 5, 010315 (2024).
  39. H. Bombin, D. Litinski, N. Nickerson, F. Pastawski, and S. Roberts, Unifying flavors of fault tolerance with the ZX calculus, Quantum 8, 1379 (2024).
  40. H. Bombin, C. Dawson, T. Farrelly, Y. Liu, N. Nickerson, M. Pant, F. Pastawski, and S. Roberts, Fault-tolerant complexes, arXiv:2308.07844.
  41. D. T. Stephen, H. P. Nautrup, J. Bermejo-Vega, J. Eisert, and R. Raussendorf, Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter, Quantum 3, 142 (2019).
  42. Y.-A. Chen and S. Tata, Higher cup products on hypercubic lattices: Application to lattice models of topological phases, J. Math. Phys. 64, 091902 (2023).
  43. J. Haah, Invertible subalgebras, Commun. Math. Phys. 403, 661 (2023).
  44. K. Walker and Z. Wang, (3+1)-TQFTs and topological insulators, Front. Phys. 7, 150 (2012).
  45. Y.-A. Chen, T. D. Ellison, and N. Tantivasadakarn, Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions, Phys. Rev. Res. 3, 013056 (2021).
  46. X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry-protected topological orders in interacting bosonic systems, Science 338, 1604 (2012).
  47. A. Kapustin and R. Thorngren, in Algebra, Geometry, and Physics in the 21st Century, edited by D. Auroux, L. Katzarkov, T. Pantev, Y. Soibelman, and Y. Tschinkel, Vol. 324 (Birkhäuser, Cham, 2017), pp. 177–202.
  48. D.-M. Schlingemann, H. Vogts, and R. F. Werner, On the structure of Clifford quantum cellular automata, J. Math. Phys. 49, 112104 (2008).
  49. J. Gütschow, Entanglement generation of Clifford quantum cellular automata, Appl. Phys. B 98, 623 (2010).
  50. J. Haah, Commuting Pauli Hamiltonians as maps between free modules, Commun. Math. Phys. 324, 351 (2013).
  51. J. Haah, Algebraic methods for quantum codes on lattices, Rev. Colomb. Mat. 50, 299 (2017).
  52. N. E. Steenrod, Products of cocycles and extensions of mappings, Ann. Math. 48, 290 (1947).
  53. D. Gaiotto and A. Kapustin, Spin TQFTs and fermionic phases of matter, Int. J. Mod. Phys. A 31, 1645044 (2016).
  54. Y.-A. Chen, Exact bosonization in arbitrary dimensions, Phys. Rev. Res. 2, 033527 (2020).
  55. L. Tsui and X.-G. Wen, Lattice models that realize Zn−1 symmetry-protected topological states for even n, Phys. Rev. B 101, 035101 (2020).
  56. P. Panteleev and G. Kalachev, Quantum LDPC codes with almost linear minimum distance, IEEE Trans. Inf. Theory 68, 213 (2022).
  57. Z. Liang, Y. Xu, J. T. Iosue, and Y.-A. Chen, Extracting topological orders of generalized Pauli stabilizer codes in two dimensions, PRX Quantum 5, 030328 (2024).
  58. C. W. von Keyserlingk, F. J. Burnell, and S. H. Simon, Three-dimensional topological lattice models with surface anyons, Phys. Rev. B 87, 045107 (2013).
  59. L. Wang and Z. Wang, In and around abelian anyon models, J. Phys. A: Math. Theor. 53, 505203 (2020).
  60. S. Eilenberg and S. MacLane, On the groups H(Π,n), I, Ann. Math. 58, 55 (1953).
  61. S. Eilenberg and S. MacLane, On the groups H(Π,n), II: Methods of computation, Ann. Math. 60, 49 (1954).
  62. A. Kapustin and R. Thorngren, Topological field theory on a lattice, discrete theta-angles and confinement, Adv. Theor. Math. Phys. 18, 1233 (2014).
  63. C. Delcamp and A. Tiwari, On 2-form gauge models of topological phases, J. High Energy Phys. 2019, 64 (2019).
  64. R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129, 393 (1990).
  65. A. Davydov, M. Müger, D. Nikshych, and V. Ostrik, The Witt group of non-degenerate braided fusion categories, J. Reine Angew. Math. 677, 135 (2013).
  66. B. Ruba and B. Yang, Witt groups and bulk-boundary correspondence for stabilizer states, arXiv:2509.10418.
  67. A. Davydov, D. Nikshych, and V. Ostrik, On the structure of the Witt group of braided fusion categories, Sel. Math. 19, 237 (2013).
  68. P. H. Bonderson, Non-abelian anyons and interferometry, Ph.D. thesis, California Institute of Technology, 2007.
  69. J. Haah, L. Fidkowski, and M. B. Hastings, Nontrivial quantum cellular automata in higher dimensions, Commun. Math. Phys. 398, 469 (2023).
  70. X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013).
  71. Y.-A. Chen and A. Kapustin, Bosonization in three spatial dimensions and a 2-form gauge theory, Phys. Rev. B 100, 245127 (2019).
  72. A. Bauer, Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases, Quantum 9, 1673 (2025).
  73. Z. Liang, B. Yang, J. T. Iosue, and Y.-A. Chen, Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes, arXiv:2410.11942.
  74. T. Schuster, N. Tantivasadakarn, A. Vishwanath, and N. Y. Yao, A holographic view of topological stabilizer codes, arXiv:2312.04617.
  75. L. Kong, X.-G. Wen, and H. Zheng, Boundary-bulk relation in topological orders, Nucl. Phys. B 922, 62 (2017).
  76. A. Kitaev and L. Kong, Models for gapped boundaries and domain walls, Commun. Math. Phys. 313, 351 (2012).
  77. T. Lan and X.-G. Wen, Topological quasiparticles and the holographic bulk-edge relation in (2+1)-dimensional string-net models, Phys. Rev. B 90, 115119 (2014).
  78. Z. Wang, X. Ma, D. T. Stephen, M. Hermele, and X. Chen, Renormalization of Ising cage-net model and generalized foliation, Phys. Rev. B 108, 035148 (2023).
  79. X. Chen, A. Dua, M. Hermele, D. T. Stephen, N. Tantivasadakarn, R. Vanhove, and J.-Y. Zhao, Sequential quantum circuits as maps between gapped phases, Phys. Rev. B 109, 075116 (2024).
  80. J. Haah, Commuting Pauli Hamiltonians as maps between free modules, Commun. Math. Phys. 324, 351 (2013).
  81. A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum error correction and orthogonal geometry, Phys. Rev. Lett. 78, 405 (1997).
  82. M. H. Freedman and Z. Wang, in Prospects in Topology: Proceedings of a Conference in Honor of William Browder (Princeton University, Princeton, New Jersey, 1995), p. 138.
  83. A permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0.
  84. E. K. Pedersen and C. A. Weibel, in Algebraic Topology: Proceedings of an International Conference held in Arcata, California, July 27–August 2, 1986 (Springer, Berlin, Heidelberg, 1989), pp. 346–361.
  85. E. K. Pedersen and C. A. Weibel, in Algebraic and Geometric Topology: Proceedings of a Conference held at Rutgers University, New Brunswick, USA, July 6–13, 1983 (Springer, Berlin, Heidelberg, 2006), pp. 166–181.
  86. E. K. Pedersen, J. Algebra 90, 461 (1984).
  87. B. Ruba and B. Yang, Homological invariants of Pauli stabilizer codes, Commun. Math. Phys. 405, 126 (2024).
  88. This algebraic invariant was first defined this way in Ref. [87] but appeared already in Ref. [50].
  89. L. Fidkowski and M. B. Hastings, Pumping chirality in three dimensions, Phys. Rev. B 109, 235142 (2024).
  90. C. Jones, K. Schatz, and D. J. Williamson, Quantum cellular automata and categorical dualities of spin chains, arXiv:2410.08884.
  91. D. Eliezer and G. Semenoff, Anyonization of lattice Chern-Simons theory, Ann. Phys. 217, 66 (1992).
  92. D. Eliezer and G. W. Semenoff, Intersection forms and the geometry of lattice Chern-Simons theory, Phys. Lett. B 286, 118 (1992).
  93. K. Sun, K. Kumar, and E. Fradkin, Discretized abelian Chern-Simons gauge theory on arbitrary graphs, Phys. Rev. B 92, 115148 (2015).
  94. J. C. M. de la Fuente, T. D. Ellison, M. Cheng, and D. J. Williamson, Topological stabilizer models on continuous variables, arXiv:2411.04993.
  95. Z. Liang, K. Liu, H. Song, and Y.-A. Chen, Generalized toric codes on twisted tori for quantum error correction, PRX Quantum 6, 020357 (2025).
  96. Z. Liang, J. N. Eberhardt, and Y.-A. Chen, Planar quantum low-density parity-check codes with open boundaries, arXiv:2504.08887.
  97. Note that our sign convention differs from Ref. [42], which has a typo in the overall factor.

Outline

Information

PRX Quantum 7, 010362– Published 31 March, 2026
  • Received 26 September 2025
  • Revised 4 February 2026
  • Accepted 27 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation