Quantum computation from dynamic automorphism codes

Margarita Davydova; Nathanan Tantivasadakarn; Shankar Balasubramanian; David Aasen

doi:10.22331/q-2024-08-27-1448

Quantum computation from dynamic automorphism codes

Margarita Davydova^1,2, Nathanan Tantivasadakarn^3,4, Shankar Balasubramanian⁵, and David Aasen⁴

¹Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
²Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
³Walter Burke Institute for Theoretical Physics and Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
⁴Microsoft Quantum, Station Q, Santa Barbara, California, USA
⁵Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Published:	2024-08-27, volume 8, page 1448
Eprint:	arXiv:2307.10353v3
Doi:	https://doi.org/10.22331/q-2024-08-27-1448
Citation:	Quantum 8, 1448 (2024).

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Abstract

We propose a new model of quantum computation comprised of low-weight measurement sequences that simultaneously encode logical information, enable error correction, and apply logical gates. These measurement sequences constitute a new class of quantum error-correcting codes generalizing Floquet codes, which we call dynamic automorphism (DA) codes. We construct an explicit example, the DA color code, which is assembled from short measurement sequences that can realize all 72 automorphisms of the 2D color code. On a stack of $N$ triangular patches, the DA color code encodes $N$ logical qubits and can implement the full logical Clifford group by a sequence of two- and, more rarely, three-qubit Pauli measurements. We also make the first step towards universal quantum computation with DA codes by introducing a 3D DA color code and showing that a non-Clifford logical gate can be realized by adaptive two-qubit measurements.

► BibTeX data

@article{Davydova2024quantumcomputation,
  doi = {10.22331/q-2024-08-27-1448},
  url = {https://doi.org/10.22331/q-2024-08-27-1448},
  title = {Quantum computation from dynamic automorphism codes},
  author = {Davydova, Margarita and Tantivasadakarn, Nathanan and Balasubramanian, Shankar and Aasen, David},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1448},
  month = aug,
  year = {2024}
}

► References

[1] Matthew B. Hastings and Jeongwan Haah. Dynamically Generated Logical Qubits. Quantum, 5: 564, October 2021. ISSN 2521-327X. 10.22331/q-2021-10-19-564.
https://doi.org/10.22331/q-2021-10-19-564

[2] Daniel Gottesman. Opportunities and Challenges in Fault-Tolerant Quantum Computation. arXiv, October 2022. 10.48550/arXiv.2210.15844.
https://doi.org/10.48550/arXiv.2210.15844

[3] Spacetime codes of Clifford circuits. arXiv, April 2023. 10.48550/arXiv.2304.05943.
https://doi.org/10.48550/arXiv.2304.05943

[4] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. Journal of Mathematical Physics, 43 (9): 4452–4505, aug 2002. 10.1063/1.1499754.
https://doi.org/10.1063/1.1499754

[5] David Aasen, Zhenghan Wang, and Matthew B. Hastings. Adiabatic paths of hamiltonians, symmetries of topological order, and automorphism codes. Phys. Rev. B, 106: 085122, Aug 2022. 10.1103/PhysRevB.106.085122.
https://doi.org/10.1103/PhysRevB.106.085122

[6] Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel. Measurement-based quantum computation on cluster states. Phys. Rev. A, 68: 022312, Aug 2003. 10.1103/PhysRevA.68.022312.
https://doi.org/10.1103/PhysRevA.68.022312

[7] R. Raussendorf, J. Harrington, and K. Goyal. A fault-tolerant one-way quantum computer. Annals of Physics, 321 (9): 2242–2270, sep 2006. 10.1016/j.aop.2006.01.012.
https://doi.org/10.1016/j.aop.2006.01.012

[8] H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest. Measurement-based quantum computation. Nature Physics, 5 (1): 19–26, jan 2009. 10.1038/nphys1157.
https://doi.org/10.1038/nphys1157

[9] Benjamin J. Brown and Sam Roberts. Universal fault-tolerant measurement-based quantum computation. Phys. Rev. Res., 2: 033305, Aug 2020. 10.1103/PhysRevResearch.2.033305.
https://doi.org/10.1103/PhysRevResearch.2.033305

[10] Sara Bartolucci, Patrick Birchall, Hector Bombín, 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): 1–9, February 2023. ISSN 2041-1723. 10.1038/s41467-023-36493-1.
https://doi.org/10.1038/s41467-023-36493-1

[11] A. Kitaev. Anyons in an exactly solved model and beyond. Ann. Phys., 321: 2–111, January 2006. 10.1016/j.aop.2005.10.005.
https://doi.org/10.1016/j.aop.2005.10.005

[12] Martin Suchara, Sergey Bravyi, and Barbara Terhal. Constructions and noise threshold of topological subsystem codes. Journal of Physics A: Mathematical and Theoretical, 44 (15): 155301, mar 2011. 10.1088/1751-8113/44/15/155301.
https://doi.org/10.1088/1751-8113/44/15/155301

[13] Craig Gidney, Michael Newman, Austin Fowler, and Michael Broughton. A Fault-Tolerant Honeycomb Memory. Quantum, 5: 605, December 2021. ISSN 2521-327X. 10.22331/q-2021-12-20-605.
https://doi.org/10.22331/q-2021-12-20-605

[14] 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, Jan 2023. 10.1103/PRXQuantum.4.010310.
https://doi.org/10.1103/PRXQuantum.4.010310

[15] Craig Gidney, Michael Newman, and Matt McEwen. Benchmarking the Planar Honeycomb Code. Quantum, 6: 813, September 2022. ISSN 2521-327X. 10.22331/q-2022-09-21-813.
https://doi.org/10.22331/q-2022-09-21-813

[16] Joseph Sullivan, Rui Wen, and Andrew C. Potter. Floquet codes and phases in twist-defect networks. Phys. Rev. B, 108 (19): 195134, November 2023. ISSN 2469-9969. 10.1103/PhysRevB.108.195134.
https://doi.org/10.1103/PhysRevB.108.195134

[17] David Aasen, Jeongwan Haah, Zhi Li, and Roger S. K. Mong. Measurement Quantum Cellular Automata and Anomalies in Floquet Codes. arXiv, April 2023. 10.48550/arXiv.2304.01277.
https://doi.org/10.48550/arXiv.2304.01277

[18] Margarita Davydova, Nathanan Tantivasadakarn, and Shankar Balasubramanian. Floquet codes without parent subsystem codes. PRX Quantum, 4: 020341, Jun 2023. 10.1103/PRXQuantum.4.020341.
https://doi.org/10.1103/PRXQuantum.4.020341

[19] 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. 10.48550/ARXIV.2212.00042.
https://doi.org/10.48550/ARXIV.2212.00042

[20] Hector Bombin, Daniel Litinski, Naomi Nickerson, Fernando Pastawski, and Sam Roberts. Unifying flavors of fault tolerance with the ZX calculus. Quantum, 8: 1379, June 2024. 10.22331/q-2024-06-18-1379.
https://doi.org/10.22331/q-2024-06-18-1379

[21] Alex Townsend-Teague, Julio Magdalena de la Fuente, and Markus Kesselring. Floquetifying the Colour Code. arXiv, July 2023. 10.4204/EPTCS.384.14.
https://doi.org/10.4204/EPTCS.384.14

[22] Arpit Dua, Nathanan Tantivasadakarn, Joseph Sullivan, and Tyler D. Ellison. Engineering 3D Floquet Codes by Rewinding. PRX Quantum, 5 (2): 020305, April 2024. ISSN 2691-3399. 10.1103/PRXQuantum.5.020305.
https://doi.org/10.1103/PRXQuantum.5.020305

[23] Andreas Bauer. Topological error correcting processes from fixed-point path integrals. arXiv, March 2023. 10.22331/q-2024-03-20-1288.
https://doi.org/10.22331/q-2024-03-20-1288

[24] Zhehao Zhang, David Aasen, and Sagar Vijay. $X$-cube Floquet code: A dynamical quantum error correcting code with a subextensive number of logical qubits. Phys. Rev. B, 108 (20): 205116, November 2023. ISSN 2469-9969. 10.1103/PhysRevB.108.205116.
https://doi.org/10.1103/PhysRevB.108.205116

[25] Christophe Vuillot. Planar Floquet Codes. arXiv, October 2021. 10.48550/arXiv.2110.05348.
https://doi.org/10.48550/arXiv.2110.05348

[26] Jeongwan Haah and Matthew B. Hastings. Boundaries for the Honeycomb Code. Quantum, 6: 693, April 2022. ISSN 2521-327X. 10.22331/q-2022-04-21-693.
https://doi.org/10.22331/q-2022-04-21-693

[27] Tyler D. Ellison, Joseph Sullivan, and Arpit Dua. Floquet codes with a twist. arXiv, June 2023a. 10.48550/arXiv.2306.08027.
https://doi.org/10.48550/arXiv.2306.08027

[28] Stefano Paesani and Benjamin J. Brown. High-Threshold Quantum Computing by Fusing One-Dimensional Cluster States. Phys. Rev. Lett., 131 (12): 120603, September 2023. ISSN 1079-7114. 10.1103/PhysRevLett.131.120603.
https://doi.org/10.1103/PhysRevLett.131.120603

[29] Beni Yoshida. Topological color code and symmetry-protected topological phases. Phys. Rev. B, 91: 245131, Jun 2015. 10.1103/PhysRevB.91.245131.
https://doi.org/10.1103/PhysRevB.91.245131

[30] Markus S. Kesselring, Fernando Pastawski, Jens Eisert, and Benjamin J. Brown. The boundaries and twist defects of the color code and their applications to topological quantum computation. Quantum, 2: 101, oct 2018. 10.22331/q-2018-10-19-101.
https://doi.org/10.22331/q-2018-10-19-101

[31] Daniel E. Browne and Terry Rudolph. Resource-efficient linear optical quantum computation. Phys. Rev. Lett., 95: 010501, Jun 2005. 10.1103/PhysRevLett.95.010501.
https://doi.org/10.1103/PhysRevLett.95.010501

[32] J. Eli Bourassa, Rafael N. Alexander, Michael Vasmer, Ashlesha Patil, Ilan Tzitrin, Takaya Matsuura, Daiqin Su, Ben Q. Baragiola, Saikat Guha, Guillaume Dauphinais, Krishna K. Sabapathy, Nicolas C. Menicucci, and Ish Dhand. Blueprint for a scalable photonic fault-tolerant quantum computer. Quantum, 5: 392, feb 2021. 10.22331/q-2021-02-04-392.
https://doi.org/10.22331/q-2021-02-04-392

[33] Torsten Karzig, Christina Knapp, Roman M. Lutchyn, Parsa Bonderson, Matthew B. Hastings, Chetan Nayak, Jason Alicea, Karsten Flensberg, Stephan Plugge, Yuval Oreg, Charles M. Marcus, and Michael H. Freedman. Scalable designs for quasiparticle-poisoning-protected topological quantum computation with majorana zero modes. Phys. Rev. B, 95: 235305, Jun 2017. 10.1103/PhysRevB.95.235305.
https://doi.org/10.1103/PhysRevB.95.235305

[34] Héctor Bombín. Dimensional jump in quantum error correction. New Journal of Physics, 18 (4): 043038, apr 2016. 10.1088/1367-2630/18/4/043038.
https://doi.org/10.1088/1367-2630/18/4/043038

[35] Hector Bombin. 2D quantum computation with 3D topological codes. arXiv, October 2018. 10.48550/arXiv.1810.09571.
https://doi.org/10.48550/arXiv.1810.09571

[36] Benjamin J. Brown. A fault-tolerant non-Clifford gate for the surface code in two dimensions. Sci. Adv., 6 (21), May 2020. ISSN 2375-2548. 10.1126/sciadv.aay4929.
https://doi.org/10.1126/sciadv.aay4929

[37] T. R. Scruby, D. E. Browne, P. Webster, and M. Vasmer. Numerical implementation of just-in-time decoding in novel lattice slices through the three-dimensional surface code. Quantum, 6: 721, may 2022. 10.22331/q-2022-05-24-721.
https://doi.org/10.22331/q-2022-05-24-721

[38] Sergey Bravyi and Andrew Cross. Doubled Color Codes. arXiv, September 2015. 10.48550/arXiv.1509.03239.
https://doi.org/10.48550/arXiv.1509.03239

[39] H Bombin, Guillaume Duclos-Cianci, and David Poulin. Universal topological phase of two-dimensional stabilizer codes. New Journal of Physics, 14 (7): 073048, jul 2012. 10.1088/1367-2630/14/7/073048.
https://doi.org/10.1088/1367-2630/14/7/073048

[40] Aleksander Kubica, Beni Yoshida, and Fernando Pastawski. Unfolding the color code. New Journal of Physics, 17 (8): 083026, aug 2015. 10.1088/1367-2630/17/8/083026.
https://doi.org/10.1088/1367-2630/17/8/083026

[41] N. David Mermin. Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett., 65: 3373–3376, Dec 1990. 10.1103/PhysRevLett.65.3373.
https://doi.org/10.1103/PhysRevLett.65.3373

[42] A. Peres. Two simple proofs of the Kochen-Specker theorem. J. Phys. A: Math. Gen., 24 (4): L175, February 1991. ISSN 0305-4470. 10.1088/0305-4470/24/4/003.
https://doi.org/10.1088/0305-4470/24/4/003

[43] Liang Kong. Anyon condensation and tensor categories. Nucl. Phys. B, 886: 436–482, September 2014. ISSN 0550-3213. 10.1016/j.nuclphysb.2014.07.003.
https://doi.org/10.1016/j.nuclphysb.2014.07.003

[44] Fiona J Burnell. Anyon condensation and its applications. Annual Review of Condensed Matter Physics, 9: 307–327, 2018. 10.1146/annurev-conmatphys-033117-054154.
https://doi.org/10.1146/annurev-conmatphys-033117-054154

[45] H. Bombin and M. A. Martin-Delgado. Topological quantum distillation. Phys. Rev. Lett., 97: 180501, Oct 2006. 10.1103/PhysRevLett.97.180501.
https://doi.org/10.1103/PhysRevLett.97.180501

[46] H. Bombin, M. Kargarian, and M. A. Martin-Delgado. Quantum 2-body Hamiltonian for topological color codes. Fortschr. Phys., 57 (11-12): 1103–1110, November 2009. ISSN 0015-8208. 10.1002/prop.200900084.
https://doi.org/10.1002/prop.200900084

[47] A.Yu. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303 (1): 2–30, 2003. ISSN 0003-4916. https://doi.org/10.1016/S0003-4916(02)00018-0.
https://doi.org/10.1016/S0003-4916(02)00018-0

[48] Héctor Bombín, Chris Dawson, Ryan V. Mishmash, Naomi Nickerson, Fernando Pastawski, and Sam Roberts. Logical Blocks for Fault-Tolerant Topological Quantum Computation. PRX Quantum, 4 (2): 020303, April 2023. ISSN 2691-3399. 10.1103/PRXQuantum.4.020303.
https://doi.org/10.1103/PRXQuantum.4.020303

[49] Alexey A. Kovalev, Sanjay Prabhakar, Ilya Dumer, and Leonid P. Pryadko. Numerical and analytical bounds on threshold error rates for hypergraph-product codes. Phys. Rev. A, 97: 062320, Jun 2018. 10.1103/PhysRevA.97.062320.
https://doi.org/10.1103/PhysRevA.97.062320

[50] Aleksander Kubica and Nicolas Delfosse. Efficient color code decoders in $d\geq 2$ dimensions from toric code decoders. Quantum, 7: 929, February 2023. ISSN 2521-327X. 10.22331/q-2023-02-21-929.
https://doi.org/10.22331/q-2023-02-21-929

[51] Nicolas Delfosse. Decoding color codes by projection onto surface codes. Phys. Rev. A, 89: 012317, Jan 2014. 10.1103/PhysRevA.89.012317.
https://doi.org/10.1103/PhysRevA.89.012317

[52] Pradeep Sarvepalli and Robert Raussendorf. Efficient decoding of topological color codes. Phys. Rev. A, 85: 022317, Feb 2012. 10.1103/PhysRevA.85.022317.
https://doi.org/10.1103/PhysRevA.85.022317

[53] Christopher Chamberland, Aleksander Kubica, Theodore J Yoder, and Guanyu Zhu. Triangular color codes on trivalent graphs with flag qubits. New Journal of Physics, 22 (2): 023019, feb 2020. 10.1088/1367-2630/ab68fd.
https://doi.org/10.1088/1367-2630/ab68fd

[54] Kaavya Sahay and Benjamin J. Brown. Decoder for the triangular color code by matching on a möbius strip. PRX Quantum, 3: 010310, Jan 2022. 10.1103/PRXQuantum.3.010310.
https://doi.org/10.1103/PRXQuantum.3.010310

[55] Héctor Bombín. Structure of 2D Topological Stabilizer Codes. Communications in Mathematical Physics, 327 (2): 387–432, 2014. ISSN 14320916. 10.1007/s00220-014-1893-4.
https://doi.org/10.1007/s00220-014-1893-4

[56] Jeongwan Haah. Classification of translation invariant topological pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices. Journal of Mathematical Physics, 62 (1): 012201, 2021. 10.1063/5.0021068.
https://doi.org/10.1063/5.0021068

[57] David Aasen, Shankar Balasubramanian, Margarita Davydova, and Nathanan Tantivasadakarn. to appear. 2024.

[58] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. Tensor categories, volume 205. American Mathematical Soc., 2016. 10.1090/surv/205.
https://doi.org/10.1090/surv/205

[59] Beni Yoshida. Gapped boundaries, group cohomology and fault-tolerant logical gates. Annals of Physics, 377: 387–413, 2017. ISSN 0003-4916. https://doi.org/10.1016/j.aop.2016.12.014.
https://doi.org/10.1016/j.aop.2016.12.014

[60] Michael Müger. On the structure of modular categories. Proceedings of the London Mathematical Society, 87 (2): 291–308, 2003. 10.1112/S0024611503014187.
https://doi.org/10.1112/S0024611503014187

[61] 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, Feb 2007a. 10.1103/PhysRevB.75.075103.
https://doi.org/10.1103/PhysRevB.75.075103

[62] Héctor Bombín. Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes. New Journal of Physics, 17 (8): 083002, aug 2015. 10.1088/1367-2630/17/8/083002.
https://doi.org/10.1088/1367-2630/17/8/083002

[63] Michael Vasmer and Dan E. Browne. Three-dimensional surface codes: Transversal gates and fault-tolerant architectures. Physical Review A, 100 (1), jul 2019. 10.1103/physreva.100.012312.
https://doi.org/10.1103/physreva.100.012312

[64] Joseph Kramer Iverson. Aspects of Fault-Tolerant Quantum Computation. California Institute of Technology, 2020. 10.1088/1367-2630/ab8e5c.
https://doi.org/10.1088/1367-2630/ab8e5c

[65] Andrew J. Landahl and Ciaran Ryan-Anderson. Quantum computing by color-code lattice surgery. arXiv, July 2014. 10.48550/arXiv.1407.5103.
https://doi.org/10.48550/arXiv.1407.5103

[66] Theodore J. Yoder, Ryuji Takagi, and Isaac L. Chuang. Universal fault-tolerant gates on concatenated stabilizer codes. Phys. Rev. X, 6: 031039, Sep 2016. 10.1103/PhysRevX.6.031039.
https://doi.org/10.1103/PhysRevX.6.031039

[67] Konstantinos Roumpedakis, Sahand Seifnashri, and Shu-Heng Shao. Higher Gauging and Non-invertible Condensation Defects. arXiv, April 2022. 10.1007/s00220-023-04706-9.
https://doi.org/10.1007/s00220-023-04706-9

[68] Maissam Barkeshli, Yu-An Chen, Sheng-Jie Huang, Ryohei Kobayashi, Nathanan Tantivasadakarn, and Guanyu Zhu. Codimension-2 defects and higher symmetries in (3+1)D topological phases. SciPost Phys., 14 (4): 065, April 2023. ISSN 2542-4653. 10.21468/SciPostPhys.14.4.065.
https://doi.org/10.21468/SciPostPhys.14.4.065

[69] Maissam Barkeshli, Yu-An Chen, Po-Shen Hsin, and Ryohei Kobayashi. Higher-group symmetry in finite gauge theory and stabilizer codes. SciPost Phys., 16 (4): 089, April 2024. ISSN 2542-4653. 10.21468/SciPostPhys.16.4.089.
https://doi.org/10.21468/SciPostPhys.16.4.089

[70] H. Bombin and M. A. Martin-Delgado. Topological computation without braiding. Phys. Rev. Lett., 98: 160502, Apr 2007b. 10.1103/PhysRevLett.98.160502.
https://doi.org/10.1103/PhysRevLett.98.160502

[71] Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Phys. Rev. A, 86: 052329, Nov 2012. 10.1103/PhysRevA.86.052329.
https://doi.org/10.1103/PhysRevA.86.052329

[72] 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, Mar 2022. 10.1103/PRXQuantum.3.010353.
https://doi.org/10.1103/PRXQuantum.3.010353

[73] 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, October 2023b. 10.22331/q-2023-10-12-1137.
https://doi.org/10.22331/q-2023-10-12-1137

[74] Aleksander Kubica and Michael Vasmer. Single-shot quantum error correction with the three-dimensional subsystem toric code. Nature Communications, 13 (1): 6272, 2022. 10.1038/s41467-022-33923-4.
https://doi.org/10.1038/s41467-022-33923-4

[75] Jacob C. Bridgeman, Aleksander Kubica, and Michael Vasmer. Lifting Topological Codes: Three-Dimensional Subsystem Codes from Two-Dimensional Anyon Models. PRX Quantum, 5 (2): 020310, April 2024. ISSN 2691-3399. 10.1103/PRXQuantum.5.020310.
https://doi.org/10.1103/PRXQuantum.5.020310

[76] Nikolas P. Breuckmann and Jens Niklas Eberhardt. Quantum low-density parity-check codes. PRX Quantum, 2: 040101, Oct 2021. 10.1103/PRXQuantum.2.040101.
https://doi.org/10.1103/PRXQuantum.2.040101

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[15] Andreas Bauer, "x+y Floquet code: A simple example for topological quantum computation in the path-integral approach", Physical Review A 111 3, 032413 (2025).

[16] Julio C. Magdalena de la Fuente, Josias Old, Alex Townsend-Teague, Manuel Rispler, Jens Eisert, and Markus Müller, "XYZ Ruby Code: Making a Case for a Three-Colored Graphical Calculus for Quantum Error Correction in Spacetime", PRX Quantum 6 1, 010360 (2025).

[17] Ali Fahimniya, Hossein Dehghani, Kishor Bharti, Sheryl Mathew, Alicia J. Kollár, Alexey V. Gorshkov, and Michael J. Gullans, "Fault-tolerant hyperbolic Floquet quantum error correcting codes", Quantum 9, 1849 (2025).

[18] M. Sohaib Alam and Eleanor Rieffel, "Dynamical logical qubits in the Bacon-Shor code", Physical Review A 112 2, 022436 (2025).

[19] Oscar Higgott and Nikolas P. Breuckmann, "Constructions and Performance of Hyperbolic and Semi-Hyperbolic Floquet Codes", PRX Quantum 5 4, 040327 (2024).

[20] Rahul Sahay and Ruben Verresen, "Classifying One-Dimensional Quantum States Prepared by a Single Round of Measurements", PRX Quantum 6 1, 010329 (2025).

[21] Benedikt Placke and S. A. Parameswaran, "Slow measurement-only dynamics of entanglement in Pauli subsystem codes", Physical Review B 111 14, 144308 (2025).

[22] Gabrielle Tournaire, Marvin Schwiering, Robert Raussendorf, and Sven Bachmann, "A 3D lattice defect and efficient computations in topological MBQC", Quantum 10, 1997 (2026).

[23] David F. Locher, Josias Old, Katharina Brechtelsbauer, Jakob Holschbach, Hans Peter Büchler, Sebastian Weber, and Markus Müller, "Multiqubit Rydberg Gates for Quantum Error Correction", PRX Quantum 7 2, 020354 (2026).

[24] Yixu Wang, Yijia Xu, and Zi-Wen Liu, "Tessellation Codes: Encoded Quantum Gates by Geometric Rotation", Physical Review Letters 135 14, 140602 (2025).

[25] Nianrui Fu, Yu Zhao, and Yidun Wan, "Symmetry-enriched topological phases and their gauging: a string-net model realization", Journal of High Energy Physics 2026 5, 16 (2026).

[26] Esther Xiaozhen Fu and Daniel Gottesman, "Error Correction in Dynamical Codes", Quantum 9, 1886 (2025).

[27] Ming Lai Chan, Aliki Anna Capatos, Peter Lodahl, Anders Søndberg Sørensen, and Stefano Paesani, "Practical blueprint for low-depth photonic quantum computing with quantum dots", arXiv:2507.16152, (2025).

[28] Vedant Motamarri, Campbell McLauchlan, and Benjamin Béri, "SymTFT out of equilibrium: from time crystals to braided drives and Floquet codes", arXiv:2312.17176, (2023).

[29] Michael Liaofan Liu, Nathanan Tantivasadakarn, and Victor V. Albert, "Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma", Quantum 8, 1403 (2024).

[30] Arpit Dua, Nathanan Tantivasadakarn, Joseph Sullivan, and Tyler D. Ellison, "Engineering 3D Floquet Codes by Rewinding", PRX Quantum 5 2, 020305 (2024).

[31] Zijian Liang, Yijia Xu, Joseph T. Iosue, and Yu-An Chen, "Extracting Topological Orders of Generalized Pauli Stabilizer Codes in Two Dimensions", PRX Quantum 5 3, 030328 (2024).

[32] Ryohei Kobayashi and Guanyu Zhu, "Cross-Cap Defects and Fault-Tolerant Logical Gates in the Surface Code and the Honeycomb Floquet Code", PRX Quantum 5 2, 020360 (2024).

[33] Matthew Buican and Rajath Radhakrishnan, "Qudit stabilizer codes, CFTs, and topological surfaces", Physical Review D 110 8, 085021 (2024).

[34] Evan Sutcliffe, Bhargavi Jonnadula, Claire Le Gall, Alexandra E. Moylett, and Coral M. Westoby, "Distributed quantum error correction based on hyperbolic Floquet codes", arXiv:2501.14029, (2025).

[35] Esther Xiaozhen Fu and Daniel Gottesman, "Error Correction in Dynamical Codes", arXiv:2403.04163, (2024).

[36] Andrew Tanggara, Mile Gu, and Kishor Bharti, "Simple Construction of Qudit Floquet Codes on a Family of Lattices", arXiv:2410.02022, (2024).

[37] Yuanjie Ren and Peter Shor, "Topological quantum computation assisted by phase transitions", arXiv:2311.00103, (2023).

[38] Clément Poirson, Joschka Roffe, and Robert I. Booth, "Engineering CSS surgery: compiling any CNOT in any code", arXiv:2505.01370, (2025).

[39] Andrew Tanggara, Mile Gu, and Kishor Bharti, "Strategic Code: A Unified Spatio-Temporal Framework for Quantum Error-Correction", arXiv:2405.17567, (2024).

[40] Alison Warman and Sakura Schafer-Nameki, "Constant-Depth Clifford-Hierarchy Gates via Non-Abelian Surface Codes", arXiv:2512.13777, (2025).

[41] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).

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

[43] Nicolas Delfosse, Adam Paetznick, Jeongwan Haah, and Matthew B. Hastings, "Splitting decoders for correcting hypergraph faults", arXiv:2309.15354, (2023).

[44] Nicolas Delfosse and Adam Paetznick, "Simulation of noisy Clifford circuits without fault propagation", arXiv:2309.15345, (2023).

[45] Hideyuki Ozawa, Isamu Kudo, Yuki Takeuchi, and Tsuyoshi Yoshida, "Hyperbolic Floquet code with graph-edge syndromes", arXiv:2509.24110, (2025).

[46] Rajath Radhakrishnan, Adar Sharon, and Nathanan Tantivasadakarn, "Unveiling dynamical quantum error correcting codes via non-invertible symmetries", arXiv:2510.09565, (2025).

[47] Dominic J. Williamson and Bence Hetényi, "Dynamical quantum codes and logic gates on a lattice with sparse connectivity", arXiv:2510.05225, (2025).

[48] Jiachen Shen and Hui Zhong, "A conditional no-go for resource-free magic-axis measurement on a static surface code", arXiv:2607.16968, (2026).

[49] Shiyu Cao, Zhian Jia, and Sheng Tan, "Topological quantum color code model on infinite lattice", arXiv:2601.12409, (2026).

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