Pauli topological subsystem codes from Abelian anyon theories

Tyler D. Ellison1, Yu-An Chen2, Arpit Dua3, Wilbur Shirley4, Nathanan Tantivasadakarn5,6, and Dominic J. Williamson7

1Department of Physics, Yale University, New Haven, CT 06511, USA
2Department of Physics, Condensed Matter Theory Center, Joint Quantum Institute, and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742, USA
3Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
4School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
5Walter Burke Institute for Theoretical Physics and Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
6Department of Physics, Harvard University, Cambridge, MA 02138, USA
7Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories–this includes anyon theories with degenerate braiding relations and those without a gapped boundary to the vacuum. Our work both extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits and establishes that the classification is at least as rich as that of Abelian anyon theories. We exemplify the construction with topological subsystem codes defined on four-dimensional qudits based on the $\mathbb{Z}_4^{(1)}$ anyon theory with degenerate braiding relations and the chiral semion theory–both of which cannot be captured by topological stabilizer codes. The construction proceeds by "gauging out" certain anyon types of a topological stabilizer code. This amounts to defining a gauge group generated by the stabilizer group of the topological stabilizer code and a set of anyonic string operators for the anyon types that are gauged out. The resulting topological subsystem code is characterized by an anyon theory containing a proper subset of the anyons of the topological stabilizer code. We thereby show that every Abelian anyon theory is a subtheory of a stack of toric codes and a certain family of twisted quantum doubles that generalize the double semion anyon theory. We further prove a number of general statements about the logical operators of translation invariant topological subsystem codes and define their associated anyon theories in terms of higher-form symmetries.

► BibTeX data

► References

[1] S. B. Bravyi and A. Yu. Kitaev. ``Quantum codes on a lattice with boundary'' (1998) arXiv:9811052.

[2] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. ``Topological quantum memory''. Journal of Mathematical Physics 43, 4452–4505 (2002).

[3] A. Yu Kitaev. ``Fault-tolerant quantum computation by anyons''. Annals of Physics 303, 2–30 (2003).

[4] R. Raussendorf, J. Harrington, and K. Goyal. ``A fault-tolerant one-way quantum computer''. Annals of Physics 321, 2242–2270 (2006).

[5] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. ``Surface codes: Towards practical large-scale quantum computation''. Phys. Rev. A 86, 032324 (2012).

[6] David K. Tuckett, Stephen D. Bartlett, and Steven T. Flammia. ``Ultrahigh error threshold for surface codes with biased noise''. Phys. Rev. Lett. 120, 050505 (2018).

[7] H. Bombin. ``Topological order with a twist: Ising anyons from an abelian model''. Phys. Rev. Lett. 105, 030403 (2010).

[8] Benjamin J. Brown, Katharina Laubscher, Markus S. Kesselring, and James R. Wootton. ``Poking holes and cutting corners to achieve clifford gates with the surface code''. Phys. Rev. X 7, 021029 (2017).

[9] Paul Webster and Stephen D. Bartlett. ``Fault-tolerant quantum gates with defects in topological stabilizer codes''. Phys. Rev. A 102, 022403 (2020).

[10] Michael A. Levin and Xiao Gang Wen. ``String-net condensation: A physical mechanism for topological phases''. Physical Review B 71, 045110 (2005). arXiv:0404617.

[11] Daniel Gottesman. ``The heisenberg representation of quantum computers''. Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, eds. S. P. Corney, R. Delbourgo, and P. D. Jarvis, (Cambridge, MA, International Press) Pages 32–43 (1999).

[12] Christopher T. Chubb and Steven T. Flammia. ``Statistical mechanical models for quantum codes with correlated noise''. Annales de L'Institut Henri Poincaré D 8, 269–321 (2021).

[13] David Poulin. ``Stabilizer formalism for operator quantum error correction''. Phys. Rev. Lett. 95, 230504 (2005).

[14] Michael A. Nielsen and David Poulin. ``Algebraic and information-theoretic conditions for operator quantum error correction''. Phys. Rev. A 75, 064304 (2007).

[15] H. Bombin, M. Kargarian, and M. A. Martin-Delgado. ``Interacting anyonic fermions in a two-body color code model''. Phys. Rev. B 80, 075111 (2009).

[16] H. Bombin. ``Topological subsystem codes''. Phys. Rev. A 81, 032301 (2010).

[17] H Bombin, Guillaume Duclos-Cianci, and David Poulin. ``Universal topological phase of two-dimensional stabilizer codes''. New Journal of Physics 14, 073048 (2012).

[18] Hector Bombin. ``Structure of 2D Topological Stabilizer Codes''. Communications in Mathematical Physics 327, 387–432 (2014).

[19] Jeongwan Haah. ``Classification of translation invariant topological pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices''. Journal of Mathematical Physics 62, 012201 (2021).

[20] Tyler D. Ellison, Yu-An Chen, Arpit Dua, Wilbur Shirley, Nathanan Tantivasadakarn, and Dominic J. Williamson. ``Pauli stabilizer models of twisted quantum doubles''. PRX Quantum 3, 010353 (2022).

[21] Sergey Bravyi. ``Subsystem codes with spatially local generators''. Phys. Rev. A 83, 012320 (2011).

[22] Martin Suchara, Sergey Bravyi, and Barbara Terhal. ``Constructions and noise threshold of topological subsystem codes''. Journal of Physics A: Mathematical and Theoretical 44, 155301 (2011).

[23] Adam Paetznick and Ben W. Reichardt. ``Universal fault-tolerant quantum computation with only transversal gates and error correction''. Phys. Rev. Lett. 111, 090505 (2013).

[24] Jonas T. Anderson, Guillaume Duclos-Cianci, and David Poulin. ``Fault-tolerant conversion between the steane and reed-muller quantum codes''. Phys. Rev. Lett. 113, 080501 (2014).

[25] Héctor Bombín. ``Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes''. New Journal of Physics 17, 083002 (2015).

[26] Sergey Bravyi, Guillaume Duclos-Cianci, David Poulin, and Martin Suchara. ``Subsystem surface codes with three-qubit check operators''. Quant. Inf. Comp. 13, 0963–0985 (2013).

[27] Christophe Vuillot, Lingling Lao, Ben Criger, Carmen García Almudéver, Koen Bertels, and Barbara M Terhal. ``Code deformation and lattice surgery are gauge fixing''. New Journal of Physics 21, 033028 (2019).

[28] H. Bombin and M. A. Martin-Delgado. ``Exact topological quantum order in $d=3$ and beyond: Branyons and brane-net condensates''. Phys. Rev. B 75, 075103 (2007).

[29] Benjamin J. Brown, Naomi H. Nickerson, and Dan E. Browne. ``Fault-tolerant error correction with the gauge color code''. Nature Communications 7, 12302 (2016).

[30] Benjamin J. Brown. ``A fault-tolerant non-clifford gate for the surface code in two dimensions''. Science Advances 6, eaay4929 (2020).

[31] Paolo Zanardi, Daniel A. Lidar, and Seth Lloyd. ``Quantum tensor product structures are observable induced''. Phys. Rev. Lett. 92, 060402 (2004).

[32] Alexei Kitaev. ``Anyons in an exactly solved model and beyond''. Annals of Physics 321, 2–111 (2006).

[33] Oscar Higgott and Nikolas P. Breuckmann. ``Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead''. Phys. Rev. X 11, 031039 (2021).

[34] Matthew B. Hastings and Jeongwan Haah. ``Dynamically Generated Logical Qubits''. Quantum 5, 564 (2021).

[35] Craig Gidney, Michael Newman, Austin Fowler, and Michael Broughton. ``A Fault-Tolerant Honeycomb Memory''. Quantum 5, 605 (2021).

[36] Jeongwan Haah and Matthew B. Hastings. ``Boundaries for the Honeycomb Code''. Quantum 6, 693 (2022).

[37] Adam Paetznick, Christina Knapp, Nicolas Delfosse, Bela Bauer, Jeongwan Haah, Matthew B. Hastings, and Marcus P. da Silva. ``Performance of planar floquet codes with majorana-based qubits''. PRX Quantum 4, 010310 (2023).

[38] Craig Gidney, Michael Newman, and Matt McEwen. ``Benchmarking the Planar Honeycomb Code''. Quantum 6, 813 (2022).

[39] Sergio Doplicher, Rudolf Haag, and John E Roberts. ``Local observables and particle statistics I''. Communications in Mathematical Physics 23, 199–230 (1971).

[40] Sergio Doplicher, Rudolf Haag, and John E Roberts. ``Local observables and particle statistics II''. Communications in Mathematical Physics 35, 49–85 (1974).

[41] Matthew Cha, Pieter Naaijkens, and Bruno Nachtergaele. ``On the stability of charges in infinite quantum spin systems''. Communications in Mathematical Physics 373 (2020).

[42] Kyle Kawagoe and Michael Levin. ``Microscopic definitions of anyon data''. Phys. Rev. B 101, 115113 (2020).

[43] Liang Wang and Zhenghan Wang. ``In and around abelian anyon models''. Journal of Physics A: Mathematical and Theoretical 53, 505203 (2020).

[44] Pieter Naaijkens. ``Quantum spin systems on infinite lattices''. Springer International Publishing. (2017).

[45] Edward Witten. ``Why does quantum field theory in curved spacetime make sense? and what happens to the algebra of observables in the thermodynamic limit?'' (2021) arXiv:2112.11614.

[46] Michael Levin and Xiao-Gang Wen. ``Fermions, strings, and gauge fields in lattice spin models''. Phys. Rev. B 67, 245316 (2003).

[47] Anton Kapustin and Lev Spodyneiko. ``Thermal hall conductance and a relative topological invariant of gapped two-dimensional systems''. Phys. Rev. B 101, 045137 (2020).

[48] Parsa H. Bonderson. ``Non-abelian anyons and interferometry''. PhD thesis. Caltech. (2012).

[49] Maissam Barkeshli, Hong-Chen Jiang, Ronny Thomale, and Xiao-Liang Qi. ``Generalized kitaev models and extrinsic non-abelian twist defects''. Phys. Rev. Lett. 114, 026401 (2015).

[50] Vlad Gheorghiu. ``Standard form of qudit stabilizer groups''. Physics Letters A 378, 505–509 (2014).

[51] Po-Shen Hsin, Ho Tat Lam, and Nathan Seiberg. ``Comments on one-form global symmetries and their gauging in 3d and 4d''. SciPost Phys. 6, 039 (2019).

[52] Yuting Hu, Yidun Wan, and Yong-Shi Wu. ``Twisted quantum double model of topological phases in two dimensions''. Phys. Rev. B 87, 125114 (2013).

[53] Anton Kapustin and Natalia Saulina. ``Topological boundary conditions in abelian chern–simons theory''. Nuclear Physics B 845, 393–435 (2011).

[54] Justin Kaidi, Zohar Komargodski, Kantaro Ohmori, Sahand Seifnashri, and Shu-Heng Shao. ``Higher central charges and topological boundaries in 2+1-dimensional TQFTs''. SciPost Phys. 13, 067 (2022).

[55] Sam Roberts and Dominic J. Williamson. ``3-fermion topological quantum computation'' (2020). arXiv:2011.04693.

[56] Clay Cordova, Po-Shen Hsin, and Nathan Seiberg. ``Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups''. SciPost Phys. 4, 021 (2018).

[57] 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, 155138 (2010).

[58] Alexei Davydov, Michael Müger, Dmitri Nikshych, and Victor Ostrik. ``The Witt group of non-degenerate braided fusion categories''. Journal fur die Reine und Angewandte Mathematik 19, 135–177 (2013). arXiv:1109.5558.

[59] Alexei Davydov, Dmitri Nikshych, and Victor Ostrik. ``On the structure of the Witt group of braided fusion categories''. Selecta Mathematica, New Series 19, 237–269 (2013).

[60] Wilbur Shirley, Yu-An Chen, Arpit Dua, Tyler D. Ellison, Nathanan Tantivasadakarn, and Dominic J. Williamson. ``Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond''. PRX Quantum 3, 030326 (2022).

[61] Andreas Bauer. ``Disentangling modular walker-wang models via fermionic invertible boundaries''. Phys. Rev. B 107, 085134 (2023).

[62] Jeongwan Haah, Lukasz Fidkowski, and Matthew B. Hastings. ``Nontrivial quantum cellular automata in higher dimensions''. Communications in Mathematical Physics 398, 469–540 (2023).

[63] Jeongwan Haah. ``Clifford quantum cellular automata: Trivial group in 2d and witt group in 3d''. Journal of Mathematical Physics 62, 092202 (2021).

[64] Jeongwan Haah. ``Topological phases of unitary dynamics: Classification in clifford category'' (2022) arXiv:2205.09141.

[65] Theo Johnson-Freyd and David Reutter. ``Minimal nondegenerate extensions''. J. Amer. Math. Soc. (2023).

[66] Alexei Kitaev and Liang Kong. ``Models for Gapped Boundaries and Domain Walls''. Communications in Mathematical Physics 313, 351–373 (2012). arXiv:1104.5047.

[67] Daniel Gottesman and Isaac L. Chuang. ``Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations''. Nature 402, 390–393 (1999).

[68] Fernando Pastawski and Beni Yoshida. ``Fault-tolerant logical gates in quantum error-correcting codes''. Phys. Rev. A 91, 012305 (2015).

[69] Konstantinos Roumpedakis, Sahand Seifnashri, and Shu-Heng Shao. ``Higher gauging and non-invertible condensation defects''. Communications in Mathematical Physics 401, 3043–3107 (2023).

[70] Rahul M. Nandkishore and Michael Hermele. ``Fractons''. Annual Review of Condensed Matter Physics 10, 295–313 (2019).

[71] Aleksander Kubica and Michael Vasmer. ``Single-shot quantum error correction with the three-dimensional subsystem toric code''. Nature Communications 13, 6272 (2022).

[72] Theo Johnson-Freyd. ``(3+1)d topological orders with only a $\mathbb{Z}_2$-charged particle'' (2020) arXiv:2011.11165.

[73] Lukasz Fidkowski, Jeongwan Haah, and Matthew B. Hastings. ``Gravitational anomaly of $(3+1)$-dimensional ${\mathbb{z}}_{2}$ toric code with fermionic charges and fermionic loop self-statistics''. Phys. Rev. B 106, 165135 (2022).

[74] Yu-An Chen and Po-Shen Hsin. ``Exactly solvable lattice Hamiltonians and gravitational anomalies''. SciPost Phys. 14, 089 (2023).

[75] David Aasen, Zhenghan Wang, and Matthew B. Hastings. ``Adiabatic paths of hamiltonians, symmetries of topological order, and automorphism codes''. Phys. Rev. B 106, 085122 (2022).

[76] Margarita Davydova, Nathanan Tantivasadakarn, and Shankar Balasubramanian. ``Floquet codes without parent subsystem codes''. PRX Quantum 4, 020341 (2023).

[77] Markus S. Kesselring, Julio C. Magdalena de la Fuente, Felix Thomsen, Jens Eisert, Stephen D. Bartlett, and Benjamin J. Brown. ``Anyon condensation and the color code'' (2022). arXiv:2212.00042.

[78] Adithya Sriram, Tibor Rakovszky, Vedika Khemani, and Matteo Ippoliti. ``Topology, criticality, and dynamically generated qubits in a stochastic measurement-only kitaev model''. Phys. Rev. B 108, 094304 (2023).

[79] Ali Lavasani, Zhu-Xi Luo, and Sagar Vijay. ``Monitored quantum dynamics and the kitaev spin liquid'' (2022) arXiv:2207.02877.

[80] Sanjay Moudgalya and Olexei I. Motrunich. ``Hilbert space fragmentation and commutant algebras''. Phys. Rev. X 12, 011050 (2022).

[81] Sanjay Moudgalya and Olexei I. Motrunich. ``Exhaustive characterization of quantum many-body scars using commutant algebras'' (2022) arXiv:2209.03377.

[82] Sanjay Moudgalya and Olexei I. Motrunich. ``From symmetries to commutant algebras in standard hamiltonians'' (2022) arXiv:2209.03370.

[83] Julia Wildeboer, Thomas Iadecola, and Dominic J. Williamson. ``Symmetry-protected infinite-temperature quantum memory from subsystem codes''. PRX Quantum 3, 020330 (2022).

[84] Sergey Bravyi and Barbara Terhal. ``A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes''. New Journal of Physics 11, 43029 (2009). arXiv:0810.1983.

[85] Jeongwan Haah and John Preskill. ``Logical-operator tradeoff for local quantum codes''. Phys. Rev. A 86, 032308 (2012).

[86] Marvin Qi, Leo Radzihovsky, and Michael Hermele. ``Fracton phases via exotic higher-form symmetry-breaking''. Annals of Physics 424, 168360 (2021).

[87] Allen Hatcher. ``Algebraic topology''. Algebraic Topology. Cambridge University Press. (2002).

[88] Chenjie Wang and Michael Levin. ``Topological invariants for gauge theories and symmetry-protected topological phases''. Phys. Rev. B 91, 165119 (2015).

[89] Kevin Walker and Zhenghan Wang. ``(3+1)-TQFTs and topological insulators''. Frontiers of Physics 7, 150–159 (2012). arXiv:1104.2632.

[90] Clement Delcamp and Apoorv Tiwari. ``From gauge to higher gauge models of topological phases''. Journal of High Energy Physics 2018 (2018). arXiv:1802.10104.

Cited by

[1] Hector Bombin, Chris Dawson, Terry Farrelly, Yehua Liu, Naomi Nickerson, Mihir Pant, Fernando Pastawski, and Sam Roberts, "Fault-tolerant complexes", arXiv:2308.07844, (2023).

[2] Tyler D. Ellison, Joseph Sullivan, and Arpit Dua, "Floquet codes with a twist", arXiv:2306.08027, (2023).

[3] Arpit Dua, Nathanan Tantivasadakarn, Joseph Sullivan, and Tyler D. Ellison, "Engineering Floquet codes by rewinding", arXiv:2307.13668, (2023).

[4] Jacob C. Bridgeman, Aleksander Kubica, and Michael Vasmer, "Lifting topological codes: Three-dimensional subsystem codes from two-dimensional anyon models", arXiv:2305.06365, (2023).

[5] Li-Mei Chen, Tyler D. Ellison, Meng Cheng, Peng Ye, and Ji-Yao Chen, "Chiral Fibonacci spin liquid in a $\mathbb{Z}_3$ Kitaev model", arXiv:2302.05060, (2023).

[6] Daniel Bulmash, Oliver Hart, and Rahul Nandkishore, "Multipole groups and fracton phenomena on arbitrary crystalline lattices", arXiv:2301.10782, (2023).

[7] Margarita Davydova, Nathanan Tantivasadakarn, Shankar Balasubramanian, and David Aasen, "Quantum computation from dynamic automorphism codes", arXiv:2307.10353, (2023).

[8] Po-Shen Hsin and Zhenghan Wang, "On topology of the moduli space of gapped Hamiltonians for topological phases", Journal of Mathematical Physics 64 4, 041901 (2023).

[9] Andreas Bauer, "Topological error correcting processes from fixed-point path integrals", arXiv:2303.16405, (2023).

[10] Dominic J. Williamson and Nouédyn Baspin, "Layer Codes", arXiv:2309.16503, (2023).

[11] Rahul Sarkar and Theodore J. Yoder, "The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems", arXiv:2302.07966, (2023).

[12] Matthew Buican and Rajath Radhakrishnan, "Qudit Stabilizer Codes, CFTs, and Topological Surfaces", arXiv:2311.13680, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2023-12-07 00:58:00). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2023-12-07 00:57:58).