Stabilizer Formalism for Operator Algebra Quantum Error Correction

Guillaume Dauphinais1, David W. Kribs1,2, and Michael Vasmer1,3,4

1Xanadu, Toronto, ON M5G 2C8, Canada
2Department of Mathematics & Statistics, University of Guelph, Guelph, ON N1G 2W1, Canada
3Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
4Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada

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

Abstract

We introduce a stabilizer formalism for the general quantum error correction framework called operator algebra quantum error correction (OAQEC), which generalizes Gottesman's formulation for traditional quantum error correcting codes (QEC) and Poulin's for operator quantum error correction and subsystem codes (OQEC). The construction generates hybrid classical-quantum stabilizer codes and we formulate a theorem that fully characterizes the Pauli errors that are correctable for a given code, generalizing the fundamental theorems for the QEC and OQEC stabilizer formalisms. We discover hybrid versions of the Bacon-Shor subsystem codes motivated by the formalism, and we apply the theorem to derive a result that gives the distance of such codes. We show how some recent hybrid subspace code constructions are captured by the formalism, and we also indicate how it extends to qudits.

Quantum error correction is a central topic in the development of new quantum technologies, with origins as an independent field of study going back almost three decades, and now touching on almost every aspect of quantum information science. More recent developments included the introduction of a framework called ‘operator algebra quantum error correction' (OAQEC) that generalized previous approaches, while additionally enabling extensions to full blown infinite-dimensional error correction and providing an error correction framework for hybrid codes used for the simultaneous encoding of classical and quantum information. The last few years have witnessed significant renewed interest in OAQEC from a few different directions, including hybrid classical-quantum coding theory, experimental quantum computing, and, somewhat unexpectedly, from black hole theory.

The ‘stabilizer formalism’ is a bedrock of quantum error correction. With its initial formulation introduced in the early days of the field and a subsequent generalization obtained for important ‘subsystem codes’, it provides a toolbox for the construction and characterization of codes for the central class of Pauli error models. In this paper, we introduce a stabilizer formalism for finite-dimensional OAQEC which generalizes the previous formulations. The resulting codes constructed include hybrid classical-quantum stabilizer codes, and motivated by this, we discover hybrid versions of an important class of subsystem codes. We prove a theorem that fully characterizes the error sets that are correctable for a given stabilizer code, generalizing the fundamental theorems from previous settings. We also present several examples and show how some recent hybrid code constructions are captured by the formalism.

► BibTeX data

► References

[1] Rajeev Acharya, Igor Aleiner, Richard Allen, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Joao Basso, Andreas Bengtsson, Sergio Boixo, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Ben Chiaro, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander L. Crook, Ben Curtin, Dripto M. Debroy, Alexander Del Toro Barba, Sean Demura, Andrew Dunsworth, Daniel Eppens, Catherine Erickson, Lara Faoro, Edward Farhi, Reza Fatemi, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, William Giang, Craig Gidney, Dar Gilboa, Marissa Giustina, Alejandro Grajales Dau, Jonathan A. Gross, Steve Habegger, Michael C. Hamilton, Matthew P. Harrigan, Sean D. Harrington, Oscar Higgott, Jeremy Hilton, Markus Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, William J. Huggins, Lev B. Ioffe, Sergei V. Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Pavol Juhas, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Tanuj Khattar, Mostafa Khezri, Mária Kieferová, Seon Kim, Alexei Kitaev, Paul V. Klimov, Andrey R. Klots, Alexander N. Korotkov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Kim-Ming Lau, Lily Laws, Joonho Lee, Kenny Lee, Brian J. Lester, Alexander Lill, Wayne Liu, Aditya Locharla, Erik Lucero, Fionn D. Malone, Jeffrey Marshall, Orion Martin, Jarrod R. McClean, Trevor Mccourt, Matt McEwen, Anthony Megrant, Bernardo Meurer Costa, Xiao Mi, Kevin C. Miao, Masoud Mohseni, Shirin Montazeri, Alexis Morvan, Emily Mount, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Hartmut Neven, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Murphy Yuezhen Niu, Thomas E. O'Brien, Alex Opremcak, John Platt, Andre Petukhov, Rebecca Potter, Leonid P. Pryadko, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin J. Satzinger, Henry F. Schurkus, Christopher Schuster, Michael J. Shearn, Aaron Shorter, Vladimir Shvarts, Jindra Skruzny, Vadim Smelyanskiy, W. Clarke Smith, George Sterling, Doug Strain, Marco Szalay, Alfredo Torres, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Theodore White, Cheng Xing, Z. Jamie Yao, Ping Yeh, Juhwan Yoo, Grayson Young, Adam Zalcman, Yaxing Zhang, and Ningfeng Zhu. Suppressing quantum errors by scaling a surface code logical qubit. Nature, 614(7949):676–681, 2023. doi:10.1038/​s41586-022-05434-1.
https:/​/​doi.org/​10.1038/​s41586-022-05434-1

[2] Chris Akers, Netta Engelhardt, Daniel Harlow, Geoff Penington, and Shreya Vardhan. The black hole interior from non-isometric codes and complexity. arXiv preprint, 2022. doi:10.48550/​arXiv.2207.06536.
https:/​/​doi.org/​10.48550/​arXiv.2207.06536

[3] Chris Akers and Geoff Penington. Quantum minimal surfaces from quantum error correction. SciPost Phys., 12:157, 2022. doi:10.21468/​SciPostPhys.12.5.157.
https:/​/​doi.org/​10.21468/​SciPostPhys.12.5.157

[4] Chris Akers and Pratik Rath. Holographic Renyi entropy from quantum error correction. JHEP, 2019(5):52, 2019. doi:10.1007/​JHEP05(2019)052.
https:/​/​doi.org/​10.1007/​JHEP05(2019)052

[5] Ahmed Almheiri. Holographic quantum error correction and the projected black hole interior. arXiv preprint, 2018. doi:10.48550/​arXiv.1810.02055.
https:/​/​doi.org/​10.48550/​arXiv.1810.02055

[6] Ahmed Almheiri, Xi Dong, and Daniel Harlow. Bulk locality and quantum error correction in AdS/​CFT. JHEP, 2015(4):163, 2015. doi:10.1007/​JHEP04(2015)163.
https:/​/​doi.org/​10.1007/​JHEP04(2015)163

[7] Salah A. Aly and Andreas Klappenecker. Subsystem code constructions. In 2008 IEEE International Symposium on Information Theory (ISIT), pages 369–373, 2008. doi:10.1109/​ISIT.2008.4595010.
https:/​/​doi.org/​10.1109/​ISIT.2008.4595010

[8] Dave Bacon. Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys. Rev. A, 73:012340, 2006. doi:10.1103/​PhysRevA.73.012340.
https:/​/​doi.org/​10.1103/​PhysRevA.73.012340

[9] Cédric Bény, Achim Kempf, and David W. Kribs. Generalization of quantum error correction via the Heisenberg picture. Phys. Rev. Lett., 98:100502, 2007. doi:10.1103/​PhysRevLett.98.100502.
https:/​/​doi.org/​10.1103/​PhysRevLett.98.100502

[10] Cédric Bény, Achim Kempf, and David W. Kribs. Quantum error correction of observables. Phys. Rev. A, 76:042303, 2007. doi:10.1103/​PhysRevA.76.042303.
https:/​/​doi.org/​10.1103/​PhysRevA.76.042303

[11] Cédric Bény, Achim Kempf, and David W Kribs. Quantum error correction on infinite-dimensional Hilbert spaces. J. Math. Phys., 50(6):062108, 2009. doi:10.1063/​1.3155783.
https:/​/​doi.org/​10.1063/​1.3155783

[12] Marcel Bergmann and Peter van Loock. Quantum error correction against photon loss using NOON states. Phys. Rev. A, 94:012311, 2016. doi:10.1103/​PhysRevA.94.012311.
https:/​/​doi.org/​10.1103/​PhysRevA.94.012311

[13] Héctor Bombín. Single-shot fault-tolerant quantum error correction. Phys. Rev. X, 5:031043, 2015. doi:10.1103/​PhysRevX.5.031043.
https:/​/​doi.org/​10.1103/​PhysRevX.5.031043

[14] Héctor Bombín. Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes. New J. Phys., 17(8):083002, 2015. doi:10.1088/​1367-2630/​17/​8/​083002.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​8/​083002

[15] 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, 2021. doi:10.22331/​q-2021-02-04-392.
https:/​/​doi.org/​10.22331/​q-2021-02-04-392

[16] Todd Brun, Igor Devetak, and Min-Hsiu Hsieh. Correcting quantum errors with entanglement. Science, 314(5798):436–439, 2006. doi:10.1126/​science.1131563.
https:/​/​doi.org/​10.1126/​science.1131563

[17] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane. Quantum error correction and orthogonal geometry. Phys. Rev. Lett., 78:405–408, 1997. doi:10.1103/​PhysRevLett.78.405.
https:/​/​doi.org/​10.1103/​PhysRevLett.78.405

[18] A. R. Calderbank and Peter W. Shor. Good quantum error-correcting codes exist. Phys. Rev. A, 54:1098–1105, 1996. doi:10.1103/​PhysRevA.54.1098.
https:/​/​doi.org/​10.1103/​PhysRevA.54.1098

[19] Ningping Cao, David W. Kribs, Chi-Kwong Li, Mike I. Nelson, Yiu-Tung Poon, and Bei Zeng. Higher rank matricial ranges and hybrid quantum error correction. Linear Multilinear Alg., 69(5):827–839, 2021. doi:10.1080/​03081087.2020.1748852.
https:/​/​doi.org/​10.1080/​03081087.2020.1748852

[20] Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra Appl., 10(3):285–290, 1975. doi:10.1016/​0024-3795(75)90075-0.
https:/​/​doi.org/​10.1016/​0024-3795(75)90075-0

[21] Kenneth R. Davidson. C*-algebras by example, volume 6 of Fields Institute Monographs. American Mathematical Soc., 1996.

[22] Igor Devetak and Peter W. Shor. The capacity of a quantum channel for simultaneous transmission of classical and quantum information. Commun. Math. Phys., 256(2):287–303, 2005. doi:10.1007/​s00220-005-1317-6.
https:/​/​doi.org/​10.1007/​s00220-005-1317-6

[23] 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. arXiv preprint, 2020. doi:10.48550/​ARXIV.2009.11482.
https:/​/​doi.org/​10.48550/​ARXIV.2009.11482

[24] Vlad Gheorghiu. Standard form of qudit stabilizer groups. Phys. Lett. A, 378(5-6):505–509, 2014. doi:10.1016/​j.physleta.2013.12.009.
https:/​/​doi.org/​10.1016/​j.physleta.2013.12.009

[25] Daniel Gottesman. Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A, 54(3):1862, 1996. doi:10.1103/​PhysRevA.54.1862.
https:/​/​doi.org/​10.1103/​PhysRevA.54.1862

[26] Daniel Gottesman. Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology, 1997. doi:10.48550/​arXiv.quant-ph/​9705052.
https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​9705052
arXiv:quant-ph/9705052

[27] Daniel Gottesman. Fault-tolerant quantum computation with higher-dimensional systems. In Quantum Computing and Quantum Communications: First NASA International Conference, QCQC’98 Palm Springs, California, USA February 17–20, 1998 Selected Papers, pages 302–313. Springer, 1999.

[28] Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator. Phys. Rev. A, 64(1):012310, 2001. doi:/​10.1103/​PhysRevA.64.012310.
https:/​/​doi.org/​10.1103/​PhysRevA.64.012310

[29] Markus Grassl, Sirui Lu, and Bei Zeng. Codes for simultaneous transmission of quantum and classical information. In 2017 IEEE International Symposium on Information Theory (ISIT), pages 1718–1722, 2017. doi:10.1109/​ISIT.2017.8006823.
https:/​/​doi.org/​10.1109/​ISIT.2017.8006823

[30] Daniel Harlow. The Ryu–Takayanagi formula from quantum error correction. Commun. Math. Phys., 354(3):865–912, 2017. doi:10.1007/​s00220-017-2904-z.
https:/​/​doi.org/​10.1007/​s00220-017-2904-z

[31] Matthew B. Hastings and Jeongwan Haah. Dynamically generated logical qubits. Quantum, 5:564, 2021. doi:10.22331/​q-2021-10-19-564.
https:/​/​doi.org/​10.22331/​q-2021-10-19-564

[32] Patrick Hayden, Sepehr Nezami, Xiao-Liang Qi, Nathaniel Thomas, Michael Walter, and Zhao Yang. Holographic duality from random tensor networks. JHEP, 2016(11):9, 2016. doi:10.1007/​JHEP11(2016)009.
https:/​/​doi.org/​10.1007/​JHEP11(2016)009

[33] Patrick Hayden and Geoffrey Penington. Learning the alpha-bits of black holes. JHEP, 2019(12):7, 2019. doi:10.1007/​JHEP12(2019)007.
https:/​/​doi.org/​10.1007/​JHEP12(2019)007

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

[35] Alexander S. Holevo. Quantum Systems, Channels, Information. De Gruyter, Berlin, Boston, 2013. doi:doi:10.1515/​9783110273403.
https:/​/​doi.org/​10.1515/​9783110273403

[36] Jeffrey Holzgrafe, Jan Beitner, Dhiren Kara, Helena S. Knowles, and Mete Atatüre. Error corrected spin-state readout in a nanodiamond. npj Quantum Inf., 5(1):13, 2019. doi:10.1038/​s41534-019-0126-2.
https:/​/​doi.org/​10.1038/​s41534-019-0126-2

[37] Min-Hsiu Hsieh, Igor Devetak, and Todd Brun. General entanglement-assisted quantum error-correcting codes. Phys. Rev. A, 76(6):062313, 2007. doi:10.1103/​PhysRevA.76.062313.
https:/​/​doi.org/​10.1103/​PhysRevA.76.062313

[38] Min-Hsiu Hsieh and Mark M. Wilde. Entanglement-assisted communication of classical and quantum information. IEEE Trans. Inf. Theory, 56(9):4682–4704, 2010. doi:10.1109/​TIT.2010.2053903.
https:/​/​doi.org/​10.1109/​TIT.2010.2053903

[39] Min-Hsiu Hsieh and Mark M. Wilde. Trading classical communication, quantum communication, and entanglement in quantum Shannon theory. IEEE Trans. Inf. Theory, 56(9):4705–4730, 2010. doi:10.1109/​TIT.2010.2054532.
https:/​/​doi.org/​10.1109/​TIT.2010.2054532

[40] Helia Kamal and Geoffrey Penington. The Ryu-Takayanagi formula from quantum error correction: an algebraic treatment of the boundary CFT. arXiv preprint, 2019. doi:10.48550/​arXiv.1912.02240.
https:/​/​doi.org/​10.48550/​arXiv.1912.02240

[41] Alexei Kitaev. Fault-tolerant quantum computation by anyons. Ann. Phys., 303(1):2–30, 2003. doi:10.1016/​S0003-4916(02)00018-0.
https:/​/​doi.org/​10.1016/​S0003-4916(02)00018-0

[42] Emanuel Knill and Raymond Laflamme. Theory of quantum error-correcting codes. Phys. Rev. A, 55(2):900, 1997. doi:10.1103/​PhysRevA.55.900.
https:/​/​doi.org/​10.1103/​PhysRevA.55.900

[43] Isaac Kremsky, Min-Hsiu Hsieh, and Todd A. Brun. Classical enhancement of quantum-error-correcting codes. Phys. Rev. A, 78(1):012341, 2008. doi:10.1103/​PhysRevA.78.012341.
https:/​/​doi.org/​10.1103/​PhysRevA.78.012341

[44] David Kribs, Raymond Laflamme, and David Poulin. Unified and generalized approach to quantum error correction. Phys. Rev. Lett., 94:180501, 2005. doi:10.1103/​PhysRevLett.94.180501.
https:/​/​doi.org/​10.1103/​PhysRevLett.94.180501

[45] David W. Kribs, Raymond Laflamme, and David Poulin. Operator quantum error correction. Quantum Inf. Comput., 6:383–399, 2006. doi:10.26421/​QIC6.4-5-6.
https:/​/​doi.org/​10.26421/​QIC6.4-5-6

[46] Sebastian Krinner, Nathan Lacroix, Ants Remm, Agustin Di Paolo, Elie Genois, Catherine Leroux, Christoph Hellings, Stefania Lazar, Francois Swiadek, Johannes Herrmann, Graham J. Norris, Christian Kraglund Andersen, Markus Müller, Alexandre Blais, Christopher Eichler, and Andreas Wallraff. Realizing repeated quantum error correction in a distance-three surface code. Nature, 605(7911):669–674, 2022. doi:10.1038/​s41586-022-04566-8.
https:/​/​doi.org/​10.1038/​s41586-022-04566-8

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

[48] Greg Kuperberg. The capacity of hybrid quantum memory. IEEE Trans. Inf. Theory, 49(6):1465–1473, 2003. doi:10.1109/​TIT.2003.811917.
https:/​/​doi.org/​10.1109/​TIT.2003.811917

[49] Chi-Kwong Li, Seth Lyles, and Yiu-Tung Poon. Error correction schemes for fully correlated quantum channels protecting both quantum and classical information. Quantum Inf. Process., 19(5):1–17, 2020. doi:10.1007/​s11128-020-02639-z.
https:/​/​doi.org/​10.1007/​s11128-020-02639-z

[50] Muyuan Li, Daniel Miller, Michael Newman, Yukai Wu, and Kenneth R. Brown. 2D Compass Codes. Phys. Rev. X, 9:021041, 2019. doi:10.1103/​PhysRevX.9.021041.
https:/​/​doi.org/​10.1103/​PhysRevX.9.021041

[51] Shayan Majidy. A unification of the coding theory and OAQEC perspective on hybrid codes. Int. J. Theor. Phys., 62:177, 2023. doi:10.1007/​s10773-023-05439-0.
https:/​/​doi.org/​10.1007/​s10773-023-05439-0

[52] Daniel Miller. Small quantum networks in the qudit stabilizer formalism. arXiv preprint, 2019. doi:10.48550/​arXiv.1910.09551.
https:/​/​doi.org/​10.48550/​arXiv.1910.09551

[53] Andrew Nemec and Andreas Klappenecker. Infinite families of quantum-classical hybrid codes. IEEE Trans. Inf. Theory, 67(5):2847–2856, 2021. doi:10.1109/​TIT.2021.3051037.
https:/​/​doi.org/​10.1109/​TIT.2021.3051037

[54] Andrew Nemec and Andreas Klappenecker. Encoding classical information in gauge subsystems of quantum codes. Int. J. Quantum Inf., 20(02):2150041, 2022. doi:10.1142/​S0219749921500416.
https:/​/​doi.org/​10.1142/​S0219749921500416

[55] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010. doi:10.1017/​CBO9780511976667.
https:/​/​doi.org/​10.1017/​CBO9780511976667

[56] Pavel Panteleev and Gleb Kalachev. Asymptotically good quantum and locally testable classical LDPC codes. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC), page 375–388, 2022. doi:10.1145/​3519935.3520017.
https:/​/​doi.org/​10.1145/​3519935.3520017

[57] Fernando Pastawski, Beni Yoshida, Daniel Harlow, and John Preskill. Holographic quantum error-correcting codes: toy models for the bulk/​boundary correspondence. JHEP, 2015(6):149, 2015. doi:10.1007/​JHEP06(2015)149.
https:/​/​doi.org/​10.1007/​JHEP06(2015)149

[58] Vern Paulsen. Completely Bounded Maps and Operator Algebras. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2003. doi:10.1017/​CBO9780511546631.
https:/​/​doi.org/​10.1017/​CBO9780511546631

[59] Geoffrey Penington. Entanglement wedge reconstruction and the information paradox. JHEP, 2020(9):2, 2020. doi:10.1007/​JHEP09(2020)002.
https:/​/​doi.org/​10.1007/​JHEP09(2020)002

[60] Lukas Postler, Sascha Heußen, Ivan Pogorelov, Manuel Rispler, Thomas Feldker, Michael Meth, Christian D. Marciniak, Roman Stricker, Martin Ringbauer, Rainer Blatt, Philipp Schindler, Markus Müller, and Thomas Monz. Demonstration of fault-tolerant universal quantum gate operations. Nature, 605(7911):675–680, 2022. doi:10.1038/​s41586-022-04721-1.
https:/​/​doi.org/​10.1038/​s41586-022-04721-1

[61] David Poulin. Stabilizer formalism for operator quantum error correction. Phys. Rev. Lett., 95:230504, 2005. doi:10.1103/​PhysRevLett.95.230504.
https:/​/​doi.org/​10.1103/​PhysRevLett.95.230504

[62] John Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2:79, 2018. doi:10.22331/​q-2018-08-06-79.
https:/​/​doi.org/​10.22331/​q-2018-08-06-79

[63] Mark de Wild Propitius and Alexander F. Bais. Discrete gauge theories. arXiv preprint, 1995. doi:10.48550/​axiv.hep-th/​9511201.
https:/​/​doi.org/​10.48550/​axiv.hep-th/​9511201

[64] Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A, 52:R2493–R2496, 1995. doi:10.1103/​PhysRevA.52.R2493.
https:/​/​doi.org/​10.1103/​PhysRevA.52.R2493

[65] Peter W. Shor. Fault-tolerant quantum computation. In Proceedings of 37th Conference on Foundations of Computer Science, pages 56–65, 1996. doi:10.1109/​SFCS.1996.548464.
https:/​/​doi.org/​10.1109/​SFCS.1996.548464

[66] Andrew Steane. Multiple-particle interference and quantum error correction. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 452(1954):2551–2577, 1996. doi:10.1098/​rspa.1996.0136.
https:/​/​doi.org/​10.1098/​rspa.1996.0136

[67] Thomas Unden, Priya Balasubramanian, Daniel Louzon, Yuval Vinkler, Martin B. Plenio, Matthew Markham, Daniel Twitchen, Alastair Stacey, Igor Lovchinsky, Alexander O. Sushkov, Mikhail D. Lukin, Alex Retzker, Boris Naydenov, Liam P. McGuinness, and Fedor Jelezko. Quantum metrology enhanced by repetitive quantum error correction. Phys. Rev. Lett., 116:230502, 2016. doi:10.1103/​PhysRevLett.116.230502.
https:/​/​doi.org/​10.1103/​PhysRevLett.116.230502

[68] Erik Verlinde and Herman Verlinde. Black hole entanglement and quantum error correction. JHEP, 2013(10):107, 2013. doi:10.1007/​JHEP10(2013)107.
https:/​/​doi.org/​10.1007/​JHEP10(2013)107

[69] W. Wang, Z. J. Chen, X. Liu, W. Cai, Y. Ma, X. Mu, X. Pan, Z. Hua, L. Hu, Y. Xu, H. Wang, Y. P. Song, X. B. Zou, C. L. Zou, and L. Sun. Quantum-enhanced radiometry via approximate quantum error correction. Nat. Commun., 13(1):3214, 2022. doi:10.1038/​s41467-022-30410-8.
https:/​/​doi.org/​10.1038/​s41467-022-30410-8

[70] Quntao Zhuang, John Preskill, and Liang Jiang. Distributed quantum sensing enhanced by continuous-variable error correction. New J. Phys., 22(2):022001, 2020. doi:10.1088/​1367-2630/​ab7257.
https:/​/​doi.org/​10.1088/​1367-2630/​ab7257

Cited by

[1] 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).

[2] ChunJun Cao, "Stabilizer Codes Have Trivial Area Operators", arXiv:2306.14996, (2023).

[3] Abhijeet Alase, Kevin D. Stubbs, Barry C. Sanders, and David L. Feder, "Exponential suppression of Pauli errors in Majorana qubits via quasiparticle detection", arXiv:2307.08896, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-04-12 08:15:18). 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 08:15:17).