Topological error correcting processes from fixed-point path integrals

Andreas Bauer

Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

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

Abstract

We propose a unifying paradigm for analyzing and constructing topological quantum error correcting codes as dynamical circuits of geometrically local channels and measurements. To this end, we relate such circuits to discrete fixed-point path integrals in Euclidean spacetime, which describe the underlying topological order: If we fix a history of measurement outcomes, we obtain a fixed-point path integral carrying a pattern of topological defects. As an example, we show that the stabilizer toric code, subsystem toric code, and CSS Floquet code can be viewed as one and the same code on different spacetime lattices, and the honeycomb Floquet code is equivalent to the CSS Floquet code under a change of basis. We also use our formalism to derive two new error-correcting codes, namely a Floquet version of the $3+1$-dimensional toric code using only 2-body measurements, as well as a dynamic code based on the double-semion string-net path integral.

Since quantum information is sensitive to noise, scalable quantum computation requires error correction, where the information of a few logical qubits is encoded non-locally in a larger number of physical qubits. A particularly appealing flavor of quantum error correction is topological, where the configurations of physical qubits look like closed-loop pattern. Then, logical quantum information is encoded globally in the homology class, that is, the winding numbers of these loops around non-contractible paths. Traditionally, the codes used for topological error correction are stabilizer codes such as the toric code, consisting of a set of operators that detect errors on the physical qubits. To achieve robustness to noise, these operators are measured over and over again. However, viewing error-correction as a dynamic circuit in spacetime rather than a static stabilizer code offers much richer possibilities for constructing fault-tolerant protocols. This has become apparent especially since the recent discovery so so-called Floquet codes. In this paper, we present a systematic framework to analyze such dynamic fault-tolerant protocols in a unified way and construct new ones. We do this by directly relating error-correcting circuits to discrete path integrals representing the underlying topological phases of matter in spacetime.

► BibTeX data

► References

[1] A. Y. Kitaev. ``Fault-tolerant quantum computation by anyons''. Ann. Phys. 303, 2 – 30 (2003). arXiv:quant-ph/​9707021.
https:/​/​doi.org/​10.1016/​S0003-4916(02)00018-0
arXiv:quant-ph/9707021

[2] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. ``Topological quantum memory''. J. Math. Phys. 43, 4452–4505 (2002). arXiv:quant-ph/​0110143.
https:/​/​doi.org/​10.1063/​1.1499754
arXiv:quant-ph/0110143

[3] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma. ``Non-abelian anyons and topological quantum computation''. Rev. Mod. Phys. 1083, 80 (2008). arXiv:0707.1889.
https:/​/​doi.org/​10.1103/​RevModPhys.80.1083
arXiv:0707.1889

[4] S. Bravyi and M. B. Hastings. ``A short proof of stability of topological order under local perturbations''. Commun. Math. Phys. 307, 609 (2011). arXiv:1001.4363.
https:/​/​doi.org/​10.1007/​s00220-011-1346-2
arXiv:1001.4363

[5] M. Fukuma, S. Hosono, and H. Kawai. ``Lattice topological field theory in two dimensions''. Commun. Math. Phys. 161, 157–176 (1994). arXiv:hep-th/​9212154.
https:/​/​doi.org/​10.1007/​BF02099416
arXiv:hep-th/9212154

[6] R. Dijkgraaf and E. Witten. ``Topological gauge theories and group cohomology''. Commun. Math. Phys. 129, 393–429 (1990).
https:/​/​doi.org/​10.1007/​BF02096988

[7] V. G. Turaev and O. Y. Viro. ``State sum invariants of 3-manifolds and quantum 6j-symbols''. Topology 31, 865–902 (1992).
https:/​/​doi.org/​10.1016/​0040-9383(92)90015-A

[8] John W. Barrett and Bruce W. Westbury. ``Invariants of piecewise-linear 3-manifolds''. Trans. Amer. Math. Soc. 348, 3997–4022 (1996). arXiv:hep-th/​9311155.
https:/​/​doi.org/​10.1090/​S0002-9947-96-01660-1
arXiv:hep-th/9311155

[9] L. Crane and Dd N. Yetter. ``A categorical construction of 4d tqfts''. In Louis Kauffman and Randy Baadhio, editors, Quantum Topology. World Scientific, Singapore (1993). arXiv:hep-th/​9301062.
https:/​/​doi.org/​10.1142/​9789812796387_0005
arXiv:hep-th/9301062

[10] A. Bauer, J. Eisert, and C. Wille. ``A unified diagrammatic approach to topological fixed point models''. SciPost Phys. Core 5, 38 (2022). arXiv:2011.12064.
https:/​/​doi.org/​10.21468/​SciPostPhysCore.5.3.038
arXiv:2011.12064

[11] Matthew B. Hastings and Jeongwan Haah. ``Dynamically generated logical qubits''. Quantum 5, 564 (2021). arXiv:2107.02194.
https:/​/​doi.org/​10.22331/​q-2021-10-19-564
arXiv:2107.02194

[12] Jeongwan Haah and Matthew B. Hastings. ``Boundaries for the honeycomb code''. Quantum 6, 693 (2022). arXiv:2110.09545.
https:/​/​doi.org/​10.22331/​q-2022-04-21-693
arXiv:2110.09545

[13] 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.
https:/​/​doi.org/​10.1103/​PRXQuantum.5.010342
arXiv:2212.00042

[14] Margarita Davydova, Nathanan Tantivasadakarn, and Shankar Balasubramanian. ``Floquet codes without parent subsystem codes'' (2022). arXiv:2210.02468.
https:/​/​doi.org/​10.1103/​PRXQuantum.4.020341
arXiv:2210.02468

[15] 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). arXiv:2203.11137.
https:/​/​doi.org/​10.1103/​PhysRevB.106.085122
arXiv:2203.11137

[16] David Aasen, Jeongwan Haah, Zhi Li, and Roger S. K. Mong. ``Measurement quantum cellular automata and anomalies in floquet codes'' (2023). arXiv:2304.01277.
arXiv:2304.01277

[17] Joseph Sullivan, Rui Wen, and Andrew C. Potter. ``Floquet codes and phases in twist-defect networks''. Phys. Rev. B 108, 195134 (2023). arXiv:2303.17664.
https:/​/​doi.org/​10.1103/​PhysRevB.108.195134
arXiv:2303.17664

[18] Zhehao Zhang, David Aasen, and Sagar Vijay. ``The x-cube floquet code''. Phys. Rev. B 108, 205116 (2023). arXiv:2211.05784.
https:/​/​doi.org/​10.1103/​PhysRevB.108.205116
arXiv:2211.05784

[19] David Kribs, Raymond Laflamme, and David Poulin. ``A unified and generalized approach to quantum error correction''. Phys. Rev. Lett. 94, 180501 (2005). arXiv:quant-ph/​0412076.
https:/​/​doi.org/​10.1103/​PhysRevLett.94.180501
arXiv:quant-ph/0412076

[20] H. Bombin. ``Topological subsystem codes''. Phys. Rev. A 81, 032301 (2010). arXiv:0908.4246.
https:/​/​doi.org/​10.1103/​PhysRevA.81.032301
arXiv:0908.4246

[21] 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). arXiv:1207.1443.
arXiv:1207.1443

[22] M. A. Levin and X.-G. Wen. ``String-net condensation: A physical mechanism for topological phases''. Phys. Rev. B 71, 045110 (2005).
https:/​/​doi.org/​10.1103/​PhysRevB.71.045110

[23] Yuting Hu, Yidun Wan, and Yong-Shi Wu. ``Twisted quantum double model of topological phases in two dimensions''. Phys. Rev. B 87, 125114 (2013).
https:/​/​doi.org/​10.1103/​PhysRevB.87.125114

[24] U. Pachner. ``P. l. homeomorphic manifolds are equivalent by elementary shellings''. Europ. J. Comb. 12, 129 – 145 (1991).
https:/​/​doi.org/​10.1016/​S0195-6698(13)80080-7

[25] Bob Coecke and Aleks Kissinger. ``Picturing quantum processes: A first course in quantum theory and diagrammatic reasoning''. Cambridge University Press. (2017).
https:/​/​doi.org/​10.1017/​9781316219317

[26] John van de Wetering. ``Zx-calculus for the working quantum computer scientist'' (2020). arXiv:2012.13966.
arXiv:2012.13966

[27] Andreas Bauer. ``Quantum mechanics is *-algebras and tensor networks'' (2020). arXiv:2003.07976.
arXiv:2003.07976

[28] Aleksander Kubica and John Preskill. ``Cellular-automaton decoders with provable thresholds for topological codes''. Phys. Rev. Lett. 123, 020501 (2019). arXiv:1809.10145.
https:/​/​doi.org/​10.1103/​PhysRevLett.123.020501
arXiv:1809.10145

[29] Jack Edmonds. ``Paths, trees, and flowers''. Canadian Journal of Mathematics 17, 449–467 (1965).
https:/​/​doi.org/​10.4153/​CJM-1965-045-4

[30] Craig Gidney. ``A pair measurement surface code on pentagons''. Quantum 7, 1156 (2023). arXiv:2206.12780.
https:/​/​doi.org/​10.22331/​q-2023-10-25-1156
arXiv:2206.12780

[31] Aleks Kissinger. ``Phase-free zx diagrams are css codes (...or how to graphically grok the surface code)'' (2022). arXiv:2204.14038.
arXiv:2204.14038

[32] Hector Bombin, Daniel Litinski, Naomi Nickerson, Fernando Pastawski, and Sam Roberts. ``Unifying flavors of fault tolerance with the zx calculus'' (2023). arXiv:2303.08829.
arXiv:2303.08829

[33] Alexei Kitaev. ``Anyons in an exactly solved model and beyond''. Ann. Phys. 321, 2–111 (2006). arXiv:cond-mat/​0506438.
https:/​/​doi.org/​10.1016/​j.aop.2005.10.005
arXiv:cond-mat/0506438

[34] 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). arXiv:2202.11829.
https:/​/​doi.org/​10.1103/​PRXQuantum.4.010310
arXiv:2202.11829

[35] 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). arXiv:cond-mat/​0607736.
https:/​/​doi.org/​10.1103/​PhysRevB.75.075103
arXiv:cond-mat/0607736

[36] Wikipedia. ``Bitruncated cubic honeycomb''.

[37] Guillaume Dauphinais, Laura Ortiz, Santiago Varona, and Miguel Angel Martin-Delgado. ``Quantum error correction with the semion code''. New J. Phys. 21, 053035 (2019). arXiv:1810.08204.
https:/​/​doi.org/​10.1088/​1367-2630/​ab1ed8
arXiv:1810.08204

[38] Julio Carlos Magdalena de la Fuente, Nicolas Tarantino, and Jens Eisert. ``Non-Pauli topological stabilizer codes from twisted quantum doubles''. Quantum 5, 398 (2021). arXiv:2001.11516.
https:/​/​doi.org/​10.22331/​q-2021-02-17-398
arXiv:2001.11516

[39] 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). arXiv:2112.11394.
https:/​/​doi.org/​10.1103/​PRXQuantum.3.010353
arXiv:2112.11394

[40] Alexis Schotte, Guanyu Zhu, Lander Burgelman, and Frank Verstraete. ``Quantum error correction thresholds for the universal fibonacci turaev-viro code''. Phys. Rev. X 12, 021012 (2022). arXiv:2012.04610.
https:/​/​doi.org/​10.1103/​PhysRevX.12.021012
arXiv:2012.04610

[41] Alex Bullivant and Clement Delcamp. ``Tube algebras, excitations statistics and compactification in gauge models of topological phases''. JHEP 2019, 1–77 (2019). arXiv:1905.08673.
https:/​/​doi.org/​10.1007/​JHEP10(2019)216
arXiv:1905.08673

[42] Tian Lan and Xiao-Gang Wen. ``Topological quasiparticles and the holographic bulk-edge relation in 2+1d string-net models''. Phys. Rev. B 90, 115119 (2014). arXiv:1311.1784.
https:/​/​doi.org/​10.1103/​PhysRevB.90.115119
arXiv:1311.1784

[43] Julio C. Magdalena de la Fuente, Jens Eisert, and Andreas Bauer. ``Bulk-to-boundary anyon fusion from microscopic models''. J. Math. Phys. 64, 111904 (2023). arXiv:2302.01835.
https:/​/​doi.org/​10.1063/​5.0147335
arXiv:2302.01835

[44] Yuting Hu, Nathan Geer, and Yong-Shi Wu. ``Full dyon excitation spectrum in generalized levin-wen models''. Phys. Rev. B 97, 195154 (2018). arXiv:1502.03433.
https:/​/​doi.org/​10.1103/​PhysRevB.97.195154
arXiv:1502.03433

[45] Sara Bartolucci, Patrick Birchall, Hector Bombin, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, Fernando Pastawski, Terry Rudolph, and Chris Sparrow. ``Fusion-based quantum computation''. Nat Commun 14, 912 (2023). arXiv:2101.09310.
https:/​/​doi.org/​10.1038/​s41467-023-36493-1
arXiv:2101.09310

[46] Robert Raussendorf, Jim Harrington, and Kovid Goyal. ``Topological fault-tolerance in cluster state quantum computation''. New Journal of Physics 9, 199 (2007). arXiv:quant-ph/​0703143.
https:/​/​doi.org/​10.1088/​1367-2630/​9/​6/​199
arXiv:quant-ph/0703143

[47] Stefano Paesani and Benjamin J. Brown. ``High-threshold quantum computing by fusing one-dimensional cluster states''. Phys. Rev. Lett. 131, 120603 (2023). arXiv:2212.06775.
https:/​/​doi.org/​10.1103/​PhysRevLett.131.120603
arXiv:2212.06775

[48] David Aasen, Daniel Bulmash, Abhinav Prem, Kevin Slagle, and Dominic J. Williamson. ``Topological defect networks for fractons of all types''. Phys. Rev. Research 2, 043165 (2020). arXiv:2002.05166.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.043165
arXiv:2002.05166

[49] Dominic Williamson. ``Spacetime topological defect networks and floquet codes'' (2022). KITP Conference: Noisy Intermediate-Scale Quantum Systems: Advances and Applications.

[50] Guillaume Dauphinais and David Poulin. ``Fault-tolerant quantum error correction for non-abelian anyons''. Commun. Math. Phys. 355, 519–560 (2017). arXiv:1607.02159.
https:/​/​doi.org/​10.1007/​s00220-017-2923-9
arXiv:1607.02159

[51] Alexis Schotte, Lander Burgelman, and Guanyu Zhu. ``Fault-tolerant error correction for a universal non-abelian topological quantum computer at finite temperature'' (2022). arXiv:2301.00054.
arXiv:2301.00054

[52] Anton Kapustin and Lev Spodyneiko. ``Thermal hall conductance and a relative topological invariant of gapped two-dimensional systems''. Phys. Rev. B 101, 045137 (2020). arXiv:1905.06488.
https:/​/​doi.org/​10.1103/​PhysRevB.101.045137
arXiv:1905.06488

[53] Andreas Bauer, Jens Eisert, and Carolin Wille. ``Towards topological fixed-point models beyond gappable boundaries''. Phys. Rev. B 106, 125143 (2022). arXiv:2111.14868.
https:/​/​doi.org/​10.1103/​PhysRevB.106.125143
arXiv:2111.14868

[54] Tyler D. Ellison, Yu-An Chen, Arpit Dua, Wilbur Shirley, Nathanan Tantivasadakarn, and Dominic J. Williamson. ``Pauli topological subsystem codes from abelian anyon theories''. Quantum 7, 1137 (2023). arXiv:2211.03798.
https:/​/​doi.org/​10.22331/​q-2023-10-12-1137
arXiv:2211.03798

Cited by

[1] Oscar Higgott and Nikolas P. Breuckmann, "Constructions and performance of hyperbolic and semi-hyperbolic Floquet codes", arXiv:2308.03750, (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 3D Floquet codes by rewinding", arXiv:2307.13668, (2023).

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

[5] Michael Liaofan Liu, Nathanan Tantivasadakarn, and Victor V. Albert, "Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma", arXiv:2311.18003, (2023).

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

[7] Brenden Roberts, Sagar Vijay, and Arpit Dua, "Geometric phases in generalized radical Floquet dynamics", arXiv:2312.04500, (2023).

[8] Alex Townsend-Teague, Julio Magdalena de la Fuente, and Markus Kesselring, "Floquetifying the Colour Code", arXiv:2307.11136, (2023).

[9] Andreas Bauer, "Low-overhead non-Clifford topological fault-tolerant circuits for all non-chiral abelian topological phases", arXiv:2403.12119, (2024).

The above citations are from SAO/NASA ADS (last updated successfully 2024-04-12 02:39:58). 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 2024-04-12 02:39:57).