A cellular automaton decoder for a noise-bias tailored color code

Jonathan F. San Miguel1, Dominic J. Williamson1,2, and Benjamin J. Brown2

1Department of Physics, Stanford University, Stanford, CA 94305
2Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia

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Abstract

Self-correcting quantum memories demonstrate robust properties that can be exploited to improve active quantum error-correction protocols. Here we propose a cellular automaton decoder for a variation of the color code where the bases of the physical qubits are locally rotated, which we call the XYZ color code. The local transformation means our decoder demonstrates key properties of a two-dimensional fractal code if the noise acting on the system is infinitely biased towards dephasing, namely, no string-like logical operators. As such, in the high-bias limit, our local decoder reproduces the behavior of a partially self-correcting memory. At low error rates, our simulations show that the memory time diverges polynomially with system size without intervention from a global decoder, up to some critical system size that grows as the error rate is lowered. Furthermore, although we find that we cannot reproduce partially self-correcting behavior at finite bias, our numerics demonstrate improved memory times at realistic noise biases. Our results therefore motivate the design of tailored cellular automaton decoders that help to reduce the bandwidth demands of global decoding for realistic noise models.

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[1] 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, Oct 2020. 10.1103/​PhysRevResearch.2.043165. URL https:/​/​doi.org/​10.1103/​PhysRevResearch.2.043165.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.043165

[2] R. Alicki, M. Horodecki, P. Horodecki, and R. Horodecki. On thermal stability of topological qubit in kitaev's 4d model. Open Systems & Information Dynamics, 17 (01): 1–20, 2010. 10.1142/​S1230161210000023. URL https:/​/​doi.org/​10.1142/​S1230161210000023.
https:/​/​doi.org/​10.1142/​S1230161210000023

[3] P Aliferis, F Brito, D P DiVincenzo, J Preskill, M Steffen, and B M Terhal. Fault-tolerant computing with biased-noise superconducting qubits: a case study. New J. Phys, 11 (1): 013061, jan 2009. 10.1088/​1367-2630/​11/​1/​013061. URL https:/​/​doi.org/​10.1088/​1367-2630/​11/​1/​013061.
https:/​/​doi.org/​10.1088/​1367-2630/​11/​1/​013061

[4] Panos Aliferis and John Preskill. Fault-tolerant quantum computation against biased noise. Phys. Rev. A, 78 (5): 052331, November 2008. 10.1103/​PhysRevA.78.052331. URL https:/​/​doi.org/​10.1103/​PhysRevA.78.052331.
https:/​/​doi.org/​10.1103/​PhysRevA.78.052331

[5] Christian Kraglund Andersen, Ants Remm, Stefania Lazar, Sebastian Krinner, Nathan Lacroix, Graham J. Norris, Mihai Gabureac, Christopher Eichler, and Andreas Wallraff. Repeated quantum error detection in a surface code. Nature Physics, 16 (8): 875–880, 2020. 10.1038/​s41567-020-0920-y. URL https:/​/​doi.org/​10.1038/​s41567-020-0920-y.
https:/​/​doi.org/​10.1038/​s41567-020-0920-y

[6] Charles H. Bennett and G. Grinstein. Role of irreversibility in stabilizing complex and nonergodic behavior in locally interacting discrete systems. Phys. Rev. Lett., 55: 657–660, Aug 1985. 10.1103/​PhysRevLett.55.657. URL https:/​/​doi.org/​10.1103/​PhysRevLett.55.657.
https:/​/​doi.org/​10.1103/​PhysRevLett.55.657

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

[8] H. Bombin and M. A. Martin-Delgado. Optimal resources for topological two-dimensional stabilizer codes: Comparative study. Phys. Rev. A, 76: 012305, Jul 2007. 10.1103/​PhysRevA.76.012305. URL https:/​/​doi.org/​10.1103/​PhysRevA.76.012305.
https:/​/​doi.org/​10.1103/​PhysRevA.76.012305

[9] J. Pablo Bonilla Ataides, David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, and Benjamin J. Brown. The xzzx surface code. Nature Communications, 12 (1): 2172, Apr 2021. ISSN 2041-1723. 10.1038/​s41467-021-22274-1. URL https:/​/​doi.org/​10.1038/​s41467-021-22274-1.
https:/​/​doi.org/​10.1038/​s41467-021-22274-1

[10] A.B. Bortz, M.H. Kalos, and J.L. Lebowitz. A new algorithm for monte carlo simulation of ising spin systems. Journal of Computational Physics, 17 (1): 10–18, 1975. ISSN 0021-9991. https:/​/​doi.org/​10.1016/​0021-9991(75)90060-1. URL https:/​/​doi.org/​10.1016/​0021-9991(75)90060-1.
https:/​/​doi.org/​10.1016/​0021-9991(75)90060-1

[11] Sergey Bravyi and Jeongwan Haah. Energy landscape of 3d spin hamiltonians with topological order. Phys. Rev. Lett., 107: 150504, Oct 2011. 10.1103/​PhysRevLett.107.150504. URL https:/​/​doi.org/​10.1103/​PhysRevLett.107.150504.
https:/​/​doi.org/​10.1103/​PhysRevLett.107.150504

[12] Sergey Bravyi and Jeongwan Haah. Quantum self-correction in the 3d cubic code model. Phys. Rev. Lett., 111: 200501, Nov 2013a. 10.1103/​PhysRevLett.111.200501. URL https:/​/​doi.org/​10.1103/​PhysRevLett.111.200501.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.200501

[13] Sergey Bravyi and Jeongwan Haah. Quantum self-correction in the 3d cubic code model. Phys. Rev. Lett., 111: 200501, Nov 2013b. 10.1103/​PhysRevLett.111.200501. URL https:/​/​doi.org/​10.1103/​PhysRevLett.111.200501.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.200501

[14] Nikolas P. Breuckmann, Kasper Duivenvoorden, Dominik Michels, and Barbara M. Terhal. Local decoders for the 2D and 4D toric code. Quant. Inf. Comput., 17: 0181, 2016. 10.48550/​ARXIV.1609.00510. URL https:/​/​doi.org/​10.48550/​arXiv.1609.00510.
https:/​/​doi.org/​10.48550/​ARXIV.1609.00510

[15] Peter Brooks and John Preskill. Fault-tolerant quantum computation with asymmetric Bacon-Shor codes. Physical Review A, 87: 032310, Mar 2013. 10.1103/​PhysRevA.87.032310. URL https:/​/​doi.org/​10.1103/​PhysRevA.87.032310.
https:/​/​doi.org/​10.1103/​PhysRevA.87.032310

[16] Benjamin J. Brown and Dominic J. Williamson. Parallelized quantum error correction with fracton topological codes. Phys. Rev. Research, 2: 013303, Mar 2020. 10.1103/​PhysRevResearch.2.013303. URL https:/​/​doi.org/​10.1103/​PhysRevResearch.2.013303.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.013303

[17] Benjamin J. Brown, Daniel Loss, Jiannis K. Pachos, Chris N. Self, and James R. Wootton. Quantum memories at finite temperature. Rev. Mod. Phys., 88: 045005, Nov 2016. 10.1103/​RevModPhys.88.045005. URL https:/​/​doi.org/​10.1103/​RevModPhys.88.045005.
https:/​/​doi.org/​10.1103/​RevModPhys.88.045005

[18] Claudio Castelnovo and Claudio Chamon. Topological quantum glassiness. Philosophical Magazine, 92 (1-3): 304–323, 2012. 10.1080/​14786435.2011.609152. URL https:/​/​doi.org/​10.1080/​14786435.2011.609152.
https:/​/​doi.org/​10.1080/​14786435.2011.609152

[19] Claudio Chamon. Quantum glassiness in strongly correlated clean systems: An example of topological overprotection. Phys. Rev. Lett., 94: 040402, Jan 2005. 10.1103/​PhysRevLett.94.040402. URL https:/​/​doi.org/​10.1103/​PhysRevLett.94.040402.
https:/​/​doi.org/​10.1103/​PhysRevLett.94.040402

[20] Zijun Chen, Kevin J. Satzinger, Juan Atalaya, Alexander N. Korotkov, Andrew Dunsworth, Daniel Sank, Chris Quintana, Matt McEwen, Rami Barends, Paul V. Klimov, Sabrina Hong, Cody Jones, Andre Petukhov, Dvir Kafri, Sean Demura, Brian Burkett, Craig Gidney, Austin G. Fowler, Harald Putterman, Igor Aleiner, Frank Arute, Kunal Arya, Ryan Babbush, Joseph C. Bardin, Andreas Bengtsson, Alexandre Bourassa, Michael Broughton, Bob B. Buckley, David A. Buell, Nicholas Bushnell, Benjamin Chiaro, Roberto Collins, William Courtney, Alan R. Derk, Daniel Eppens, Catherine Erickson, Edward Farhi, Brooks Foxen, Marissa Giustina, Jonathan A. Gross, Matthew P. Harrigan, Sean D. Harrington, Jeremy Hilton, Alan Ho, Trent Huang, William J. Huggins, L. B. Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Kostyantyn Kechedzhi, Seon Kim, Fedor Kostritsa, David Landhuis, Pavel Laptev, Erik Lucero, Orion Martin, Jarrod R. McClean, Trevor McCourt, Xiao Mi, Kevin C. Miao, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Michael Newman, Murphy Yuezhen Niu, Thomas E. O'Brien, Alex Opremcak, Eric Ostby, Bálint Pató, Nicholas Redd, Pedram Roushan, Nicholas C. Rubin, Vladimir Shvarts, Doug Strain, Marco Szalay, Matthew D. Trevithick, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, Sergio Boixo, Vadim Smelyanskiy, Yu Chen, Anthony Megrant, and Julian Kelly. Exponential suppression of bit or phase flip errors with repetitive error correction, 2021. URL https:/​/​doi.org/​10.1038/​s41586-021-03588-y.
https:/​/​doi.org/​10.1038/​s41586-021-03588-y

[21] Andrew S. Darmawan and David Poulin. Tensor-network simulations of the surface code under realistic noise. Phys. Rev. Lett., 119: 040502, Jul 2017. 10.1103/​PhysRevLett.119.040502. URL https:/​/​doi.org/​10.1103/​PhysRevLett.119.040502.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.040502

[22] Andrew S. Darmawan and David Poulin. Linear-time general decoding algorithm for the surface code. Phys. Rev. E, 97: 051302, May 2018. 10.1103/​PhysRevE.97.051302. URL https:/​/​doi.org/​10.1103/​PhysRevE.97.051302.
https:/​/​doi.org/​10.1103/​PhysRevE.97.051302

[23] Andrew S. Darmawan, Benjamin J. Brown, Arne L. Grimsmo, David K. Tuckett, and Shruti Puri. Practical quantum error correction with the xzzx code and kerr-cat qubits. PRX Quantum, 2: 030345, Sep 2021. 10.1103/​PRXQuantum.2.030345. URL https:/​/​doi.org/​10.1103/​PRXQuantum.2.030345.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.030345

[24] Poulami Das, Christopher A. Pattison, Srilatha Manne, Douglas Carmean, Krysta Svore, Moinuddin Qureshi, and Nicolas Delfosse. A scalable decoder micro-architecture for fault-tolerant quantum computing, 2020. URL https:/​/​doi.org/​10.48550/​arXiv.2001.06598.
https:/​/​doi.org/​10.48550/​arXiv.2001.06598

[25] Nicolas Delfosse. Hierarchical decoding to reduce hardware requirements for quantum computing, 2020. URL https:/​/​doi.org/​10.48550/​arXiv.2001.11427.
https:/​/​doi.org/​10.48550/​arXiv.2001.11427

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

[27] Trithep Devakul, Yizhi You, F. J. Burnell, and S. L. Sondhi. Fractal symmetric phases of matter. SciPost Phys., 6: 007, 2019. URL https:/​/​doi.org/​10.21468/​SciPostPhys.6.1.007.
https:/​/​doi.org/​10.21468/​SciPostPhys.6.1.007

[28] Arpit Dua, Aleksander Kubica, Liang Jiang, Steven T. Flammia, and Michael J. Gullans. Clifford-deformed Surface Codes, 1 2022. URL https:/​/​doi.org/​10.48550/​arXiv.2201.07802.
https:/​/​doi.org/​10.48550/​arXiv.2201.07802

[29] Laird Egan, Dripto M. Debroy, Crystal Noel, Andrew Risinger, Daiwei Zhu, Debopriyo Biswas, Michael Newman, Muyuan Li, Kenneth R. Brown, Marko Cetina, and Christopher Monroe. Fault-tolerant operation of a quantum error-correction code. Nature, 2021. 10.1038/​s41586-021-03928-y.
https:/​/​doi.org/​10.1038/​s41586-021-03928-y

[30] 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, Sep 2012. 10.1103/​PhysRevA.86.032324. URL https:/​/​doi.org/​10.1103/​PhysRevA.86.032324.
https:/​/​doi.org/​10.1103/​PhysRevA.86.032324

[31] Juan P. Garrahan and M. E. J. Newman. Glassiness and constrained dynamics of a short-range nondisordered spin model. Phys. Rev. E, 62: 7670–7678, Dec 2000. 10.1103/​PhysRevE.62.7670. URL https:/​/​doi.org/​10.1103/​PhysRevE.62.7670.
https:/​/​doi.org/​10.1103/​PhysRevE.62.7670

[32] Michael R. Geller and Zhongyuan Zhou. Efficient error models for fault-tolerant architectures and the pauli twirling approximation. Phys. Rev. A, 88: 012314, Jul 2013. 10.1103/​PhysRevA.88.012314. URL https:/​/​doi.org/​10.1103/​PhysRevA.88.012314.
https:/​/​doi.org/​10.1103/​PhysRevA.88.012314

[33] A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret. Stabilization and operation of a Kerr-cat qubit. Nature, 584 (7820): 205–209, August 2020. 10.1038/​s41586-020-2587-z. URL http:/​/​dx.doi.org/​10.1038/​s41586-020-2587-z.
https:/​/​doi.org/​10.1038/​s41586-020-2587-z

[34] Jérémie Guillaud and Mazyar Mirrahimi. Repetition cat qubits for fault-tolerant quantum computation. Phys. Rev. X, 9: 041053, Dec 2019. 10.1103/​PhysRevX.9.041053. URL http:/​/​dx.doi.org/​10.1103/​PhysRevX.9.041053.
https:/​/​doi.org/​10.1103/​PhysRevX.9.041053

[35] Jérémie Guillaud and Mazyar Mirrahimi. Error rates and resource overheads of repetition cat qubits. Phys. Rev. A, 103: 042413, Apr 2021. 10.1103/​PhysRevA.103.042413. URL https:/​/​doi.org/​10.1103/​PhysRevA.103.042413.
https:/​/​doi.org/​10.1103/​PhysRevA.103.042413

[36] Jeongwan Haah. Local stabilizer codes in three dimensions without string logical operators. Phys. Rev. A, 83: 042330, Apr 2011. 10.1103/​PhysRevA.83.042330. URL https:/​/​doi.org/​10.1103/​PhysRevA.83.042330.
https:/​/​doi.org/​10.1103/​PhysRevA.83.042330

[37] James W. Harrington. Analysis of quantum error-correcting codes: sympletic lattice codes and toric codes. PhD thesis, California Institute of Technology, 2004. URL https:/​/​doi.org/​10.7907/​AHMQ-EG82.
https:/​/​doi.org/​10.7907/​AHMQ-EG82

[38] M. B. Hastings. Decoding in hyperbolic spaces: LDPC codes with linear rate and efficient error correction. Quant. Inf. Comput., 14: 1187, 2014. URL https:/​/​doi.org/​10.48550/​arXiv.1312.2546.
https:/​/​doi.org/​10.48550/​arXiv.1312.2546

[39] Michael Herold, Earl T. Campbell, Jens Eisert, and Michael J. Kastoryano. Cellular-automaton decoders for topological quantum memories. npj Quantum Information, 1 (1): 15010, Oct 2015. ISSN 2056-6387. 10.1038/​npjqi.2015.10. URL https:/​/​doi.org/​10.1038/​npjqi.2015.10.
https:/​/​doi.org/​10.1038/​npjqi.2015.10

[40] Michael Herold, Michael J. Kastoryano, Earl T. Campbell, and Jens Eisert. Cellular automaton decoder for topological quantum memories in the fault tolerant setting. New J. Phys., 19: 063012, 2017. URL https:/​/​doi.org/​10.1088/​1367-2630/​aa7099.
https:/​/​doi.org/​10.1088/​1367-2630/​aa7099

[41] Oscar Higgott and Nikolas P. Breuckmann. Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead. Phys. Rev. X, 11: 031039, Aug 2021. 10.1103/​PhysRevX.11.031039. URL https:/​/​doi.org/​10.1103/​PhysRevX.11.031039.
https:/​/​doi.org/​10.1103/​PhysRevX.11.031039

[42] Oscar Higgott, Thomas C. Bohdanowicz, Aleksander Kubica, Steven T. Flammia, and Earl T. Campbell. Fragile boundaries of tailored surface codes, 3 2022. URL https:/​/​doi.org/​10.48550/​arXiv.2203.04948.
https:/​/​doi.org/​10.48550/​arXiv.2203.04948

[43] Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58 (301): 13–30, 1963. 10.1080/​01621459.1963.10500830. URL https:/​/​doi.org/​10.1080/​01621459.1963.10500830.
https:/​/​doi.org/​10.1080/​01621459.1963.10500830

[44] A.Yu. Kitaev. Fault-tolerant quantum computation by anyons. Ann. Phys., 303 (1): 2–30, Jan 2003. ISSN 0003-4916. 10.1016/​S0003-4916(02)00018-0. URL http:/​/​dx.doi.org/​10.1016/​S0003-4916(02)00018-0.
https:/​/​doi.org/​10.1016/​S0003-4916(02)00018-0

[45] Morten Kjaergaard, Mollie E. Schwartz, Jochen Braumüller, Philip Krantz, Joel I.-J. Wang, Simon Gustavsson, and William D. Oliver. Superconducting qubits: Current state of play. Annual Review of Condensed Matter Physics, 11 (1): 369–395, 2020. 10.1146/​annurev-conmatphys-031119-050605. URL https:/​/​doi.org/​10.1146/​annurev-conmatphys-031119-050605.
https:/​/​doi.org/​10.1146/​annurev-conmatphys-031119-050605

[46] Andrzej Kossakowski, Alberto Frigerio, Vittorio Gorini, and Maurizio Verri. Quantum detailed balance and KMS condition. Communications in Mathematical Physics, 57: 97–110, 1977. URL https:/​/​doi.org/​10.1007/​BF01625769.
https:/​/​doi.org/​10.1007/​BF01625769

[47] Aleksander Kubica and Michael E. Beverland. Universal transversal gates with color codes: A simplified approach. Phys. Rev. A, 91: 032330, Mar 2015. 10.1103/​PhysRevA.91.032330. URL https:/​/​doi.org/​10.1103/​PhysRevA.91.032330.
https:/​/​doi.org/​10.1103/​PhysRevA.91.032330

[48] Muyuan Li, Daniel Miller, Michael Newman, Yukai Wu, and Kenneth R. Brown. 2D compass codes. Phys. Rev. X, 9 (2): 021041, May 2019. 10.1103/​PhysRevX.9.021041. URL http:/​/​dx.doi.org/​10.1103/​physrevx.9.021041.
https:/​/​doi.org/​10.1103/​PhysRevX.9.021041

[49] Norbert M. Linke, Mauricio Gutierrez, Kevin A. Landsman, Caroline Figgatt, Shantanu Debnath, Kenneth R. Brown, and Christopher Monroe. Fault-tolerant quantum error detection. Science Advances, 3 (10), 2017. 10.1126/​sciadv.1701074. URL https:/​/​doi.org/​10.1126/​sciadv.1701074.
https:/​/​doi.org/​10.1126/​sciadv.1701074

[50] Olivier Martin, Andrew M. Odlyzko, and Stephen Wolfram. Algebraic properties of cellular automata. Communications in Mathematical Physics, 93 (2): 219 – 258, 1984. . URL.
https:/​/​doi.org/​10.1007/​BF01223745

[51] Jonathan San Miguel. Decoders for XYZ color codes, March 2023. URL https:/​/​doi.org/​10.5281/​zenodo.7702731.
https:/​/​doi.org/​10.5281/​zenodo.7702731

[52] Mazyar Mirrahimi, Zaki Leghtas, Victor V Albert, Steven Touzard, Robert J Schoelkopf, Liang Jiang, and Michel H Devoret. Dynamically protected cat-qubits: a new paradigm for universal quantum computation. New J. Phys., 16 (4): 045014, 2014. 10.1088/​1367-2630/​16/​4/​045014. URL https:/​/​doi.org/​10.1088/​1367-2630/​16/​4/​045014.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​4/​045014

[53] Rahul M. Nandkishore and Michael Hermele. Fractons, mar 2019. ISSN 19475462. URL https:/​/​doi.org/​10.1146/​annurev-conmatphys-031218-013604.
https:/​/​doi.org/​10.1146/​annurev-conmatphys-031218-013604

[54] M. E. J. Newman and Cristopher Moore. Glassy dynamics and aging in an exactly solvable spin model. Phys. Rev. E, 60: 5068–5072, Nov 1999. 10.1103/​PhysRevE.60.5068. URL https:/​/​doi.org/​10.1103/​PhysRevE.60.5068.
https:/​/​doi.org/​10.1103/​PhysRevE.60.5068

[55] Naomi H. Nickerson and Benjamin J. Brown. Analysing correlated noise in the surface code using adaptive decoding algorithms. Quantum, 3: 131, 2019. 10.22331/​q-2019-04-08-131. URL https:/​/​doi.org/​10.22331/​q-2019-04-08-131.
https:/​/​doi.org/​10.22331/​q-2019-04-08-131

[56] Georgia M. Nixon and Benjamin J. Brown. Correcting spanning errors with a fractal code. IEEE Transactions on Information Theory, 67 (7): 4504–4516, 2021. 10.1109/​TIT.2021.3068359. URL https:/​/​doi.org/​10.1109/​TIT.2021.3068359.
https:/​/​doi.org/​10.1109/​TIT.2021.3068359

[57] J. R. Norris. Continuous-time Markov chains 1, page 60–107. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 1997. 10.1017/​CBO9780511810633.004.
https:/​/​doi.org/​10.1017/​CBO9780511810633.004

[58] Fernando Pastawski, Lucas Clemente, and Juan Ignacio Cirac. Quantum memories based on engineered dissipation. Phys. Rev. A, 83: 012304, 2011. URL https:/​/​doi.org/​10.1103/​PhysRevA.83.012304.
https:/​/​doi.org/​10.1103/​PhysRevA.83.012304

[59] Abhinav Prem, Jeongwan Haah, and Rahul Nandkishore. Glassy quantum dynamics in translation invariant fracton models. Physical Review B, 95 (15): 155133, apr 2017. ISSN 24699969. 10.1103/​PhysRevB.95.155133. URL https:/​/​doi.org/​10.1103.
https:/​/​doi.org/​10.1103/​PhysRevB.95.155133

[60] Michael Pretko, Xie Chen, and Yizhi You. Fracton phases of matter. International Journal of Modern Physics A, 35 (6), jan 2020. ISSN 0217751X. 10.1142/​S0217751X20300033. URL http:/​/​dx.doi.org/​10.1142/​S0217751X20300033.
https:/​/​doi.org/​10.1142/​S0217751X20300033

[61] Shruti Puri, Samuel Boutin, and Alexandre Blais. Engineering the quantum states of light in a kerr-nonlinear resonator by two-photon driving. npj Quantum Inf., 3 (1): 18–25, 2017. 10.1038/​s41534-017-0019-1. URL https:/​/​doi.org/​10.1038/​s41534-017-0019-1.
https:/​/​doi.org/​10.1038/​s41534-017-0019-1

[62] Shruti Puri, Lucas St-Jean, Jonathan A. Gross, Alexander Grimm, Nicholas E. Frattini, Pavithran S. Iyer, Anirudh Krishna, Steven Touzard, Liang Jiang, Alexandre Blais, et al. Bias-preserving gates with stabilized cat qubits. Sci. Adv., 6 (34): eaay5901, August 2020. 10.1126/​sciadv.aay5901. URL http:/​/​dx.doi.org/​10.1126/​sciadv.aay5901.
https:/​/​doi.org/​10.1126/​sciadv.aay5901

[63] Robert Raussendorf and Jim Harrington. Fault-tolerant quantum computation with high threshold in two dimensions. Phys. Rev. Lett., 98: 190504, May 2007. 10.1103/​PhysRevLett.98.190504. URL https:/​/​doi.org/​10.1103/​PhysRevLett.98.190504.
https:/​/​doi.org/​10.1103/​PhysRevLett.98.190504

[64] D. J. Reilly. Challenges in scaling-up the control interface of a quantum computer, 2019. URL https:/​/​doi.org/​10.48550/​arXiv.1912.05114.
https:/​/​doi.org/​10.48550/​arXiv.1912.05114

[65] Joschka Roffe, Lawrence Z. Cohen, Armanda O. Quintivalle, Daryus Chandra, and Earl T. Campbell. Bias-tailored quantum LDPC codes, 2 2022. URL https:/​/​doi.org/​10.48550/​arXiv.2202.01702.
https:/​/​doi.org/​10.48550/​arXiv.2202.01702

[66] Hao Song, Janik Schönmeier-Kromer, Ke Liu, Oscar Viyuela, Lode Pollet, and M. A. Martin-Delgado. Optimal thresholds for fracton codes and random spin models with subsystem symmetry. Phys. Rev. Lett., 129: 230502, Nov 2022. 10.1103/​PhysRevLett.129.230502. URL https:/​/​doi.org/​10.1103/​PhysRevLett.129.230502.
https:/​/​doi.org/​10.1103/​PhysRevLett.129.230502

[67] Basudha Srivastava, Anton Frisk Kockum, and Mats Granath. The XYZ$^2$ hexagonal stabilizer code. Quantum, 6: 698, April 2022. ISSN 2521-327X. 10.22331/​q-2022-04-27-698. URL https:/​/​doi.org/​10.22331/​q-2022-04-27-698.
https:/​/​doi.org/​10.22331/​q-2022-04-27-698

[68] David T. Stephen, Arpit Dua, José Garre-Rubio, Dominic J. Williamson, and Michael Hermele. Fractionalization of subsystem symmetries in two dimensions, mar 2022. URL https:/​/​doi.org/​10.1103/​PhysRevB.106.085104.
https:/​/​doi.org/​10.1103/​PhysRevB.106.085104

[69] Ashley M. Stephens, William J. Munro, and Kae Nemoto. High-threshold topological quantum error correction against biased noise. Phys. Rev. A, 88: 060301(R), 2013. 10.1103/​physreva.88.060301. URL https:/​/​doi.org/​10.1103/​PhysRevA.88.060301.
https:/​/​doi.org/​10.1103/​physreva.88.060301

[70] Maika Takita, Andrew W. Cross, A. D. Córcoles, Jerry M. Chow, and Jay M. Gambetta. Experimental demonstration of fault-tolerant state preparation with superconducting qubits. Phys. Rev. Lett., 119: 180501, Oct 2017. 10.1103/​PhysRevLett.119.180501. URL https:/​/​doi.org/​10.1103/​PhysRevLett.119.180501.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.180501

[71] Jeffrey C. Y. Teo, Abhishek Roy, and Xiao Chen. Unconventional fusion and braiding of topological defects in a lattice model. Phys. Rev. B, 90: 115118, Sep 2014. 10.1103/​PhysRevB.90.115118. URL https:/​/​doi.org/​10.1103/​PhysRevB.90.115118.
https:/​/​doi.org/​10.1103/​PhysRevB.90.115118

[72] David K. Tuckett, Stephen D. Bartlett, and Steven T. Flammia. Ultrahigh error threshold for surface codes with biased noise. Phys. Rev. Lett., 120: 050505, Jan 2018. 10.1103/​PhysRevLett.120.050505. URL https:/​/​doi.org/​10.1103/​PhysRevLett.120.050505.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.050505

[73] David K. Tuckett, Andrew S. Darmawan, Christopher T. Chubb, Sergey Bravyi, Stephen D. Bartlett, and Steven T. Flammia. Tailoring surface codes for highly biased noise. Phys. Rev. X, 9 (4): 041031, Nov 2019. ISSN 2160-3308. 10.1103/​PhysRevX.9.041031. URL http:/​/​dx.doi.org/​10.1103/​PhysRevX.9.041031.
https:/​/​doi.org/​10.1103/​PhysRevX.9.041031

[74] David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, and Benjamin J. Brown. Fault-tolerant thresholds for the surface code in excess of $5\%$ under biased noise. Phys. Rev. Lett., 124: 130501, Mar 2020. 10.1103/​PhysRevLett.124.130501. URL https:/​/​doi.org/​10.1103/​PhysRevLett.124.130501.
https:/​/​doi.org/​10.1103/​PhysRevLett.124.130501

[75] Sagar Vijay, Jeongwan Haah, and Liang Fu. Fracton topological order, generalized lattice gauge theory, and duality. Phys. Rev. B, 94: 235157, Dec 2016. 10.1103/​PhysRevB.94.235157. URL https:/​/​doi.org/​10.1103/​PhysRevB.94.235157.
https:/​/​doi.org/​10.1103/​PhysRevB.94.235157

[76] Xiao-Gang Wen. Quantum orders in an exact soluble model. Physical Review Letters, 90 (1): 016803, Jan 2003. ISSN 1079-7114. 10.1103/​physrevlett.90.016803. URL http:/​/​dx.doi.org/​10.1103/​PhysRevLett.90.016803.
https:/​/​doi.org/​10.1103/​physrevlett.90.016803

[77] Stephen Wolfram. Statistical mechanics of cellular automata. Rev. Mod. Phys., 55: 601–644, Jul 1983. 10.1103/​RevModPhys.55.601. URL https:/​/​doi.org/​10.1103/​RevModPhys.55.601.
https:/​/​doi.org/​10.1103/​RevModPhys.55.601

[78] Xiaosi Xu, Qi Ahao, Xiao Yuan, and Simon C. Benjamin. A high threshold code for modular hardware with asymmetric noise. arXiv:1812.01505, 2018. 10.1103/​physrevapplied.12.064006. URL https:/​/​doi.org/​10.1103/​PhysRevApplied.12.064006.
https:/​/​doi.org/​10.1103/​physrevapplied.12.064006
arXiv:1812.01505

[79] Beni Yoshida. Information storage capacity of discrete spin systems. Ann. Phys., 338: 134, 2013a. URL https:/​/​doi.org/​10.1016/​j.aop.2013.07.009.
https:/​/​doi.org/​10.1016/​j.aop.2013.07.009

[80] Beni Yoshida. Exotic topological order in fractal spin liquids. Phys. Rev. B, 88: 125122, Sep 2013b. 10.1103/​PhysRevB.88.125122. URL https:/​/​doi.org/​10.1103/​PhysRevB.88.125122.
https:/​/​doi.org/​10.1103/​PhysRevB.88.125122

Cited by

[1] Konstantin Tiurev, Peter-Jan H. S. Derks, Joschka Roffe, Jens Eisert, and Jan-Michael Reiner, "Correcting non-independent and non-identically distributed errors with surface codes", Quantum 7, 1123 (2023).

[2] 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", arXiv:2212.00042, (2022).

[3] Margarita Davydova, Nathanan Tantivasadakarn, and Shankar Balasubramanian, "Floquet Codes without Parent Subsystem Codes", PRX Quantum 4 2, 020341 (2023).

[4] Samuel C. Smith, Benjamin J. Brown, and Stephen D. Bartlett, "Local Predecoder to Reduce the Bandwidth and Latency of Quantum Error Correction", Physical Review Applied 19 3, 034050 (2023).

[5] Konstantin Tiurev, Arthur Pesah, Peter-Jan H. S. Derks, Joschka Roffe, Jens Eisert, Markus S. Kesselring, and Jan-Michael Reiner, "The domain wall color code", arXiv:2307.00054, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-02-26 02:35:29) and SAO/NASA ADS (last updated successfully 2024-02-26 02:35:30). The list may be incomplete as not all publishers provide suitable and complete citation data.