Fault-tolerant quantum computing in the Pauli or Clifford frame with slow error diagnostics

Christopher Chamberland1, Pavithran Iyer2, and David Poulin2

1Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
2Département de Physique and Institut Quantique, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1 Canada

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We consider the problem of fault-tolerant quantum computation in the presence of slow error diagnostics, either caused by measurement latencies or slow decoding algorithms. Our scheme offers a few improvements over previously existing solutions, for instance it does not require active error correction and results in a reduced error-correction overhead when error diagnostics is much slower than the gate time. In addition, we adapt our protocol to cases where the underlying error correction strategy chooses the optimal correction amongst all Clifford gates instead of the usual Pauli gates. The resulting Clifford frame protocol is of independent interest as it can increase error thresholds and could find applications in other areas of quantum computation.

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[1] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, et al. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature, 508 (7497): 500–503, 2014. 10.1038/​nature13171.

[2] Evan Jeffrey, Daniel Sank, J. Y. Mutus, T. C. White, J. Kelly, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Megrant, P. J. J. O'Malley, C. Neill, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and John M. Martinis. Fast accurate state measurement with superconducting qubits. Phys. Rev. Lett., 112: 190504, May 2014. 10.1103/​PhysRevLett.112.190504.

[3] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, et al. Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science, 309 (5744): 2180–2184, 2005. 10.1126/​science.1116955.

[4] M. Veldhorst, J. C. C. Hwang, C H. Yang, A. W. Leenstra, B. de Ronde, J. P. Deholla, J. T. Muhonen, F. E. Hudson, K. M. Itoh, A. Morella, and A. S. Dzurak. An addressable quantum dot qubit with fault-tolerant control-fidelity. Nature nanotechnology, 9 (12): 981–985, 2014. 10.1038/​nnano.2014.216.

[5] J. Stehlik, Y.-Y. Liu, C. M. Quintana, C. Eichler, T. R. Hartke, and J. R. Petta. Fast charge sensing of a cavity-coupled double quantum dot using a josephson parametric amplifier. Phys. Rev. Applied, 4: 014018, Jul 2015. 10.1103/​PhysRevApplied.4.014018.

[6] S. Olmschenk, D. Hayes, D. N. Matsukevich, P. Maunz, D. L. Moehring, K. C. Younge, and C. Monroe. Measurement of the lifetime of the $6p\text{ }{^{2}P}_{1/​2}^{o}$ level of ${\text{yb}}^{+}$. Phys. Rev. A, 80: 022502, Aug 2009. 10.1103/​PhysRevA.80.022502.

[7] H. Häffner, C.F. Roos, and R. Blatt. Quantum computing with trapped ions. Physics Reports, 469 (4): 155 – 203, 2008. ISSN 0370-1573. http:/​/​doi.org/​10.1016/​j.physrep.2008.09.003.

[8] Emanuel Knill. Quantum computing with realistically noisy devices. Nature, 434 (7029): 39–44, 2005. 10.1038/​nature03350.

[9] David P. DiVincenzo and Panos Aliferis. Effective fault-tolerant quantum computation with slow measurements. Phys. Rev. Lett., 98: 020501, Jan 2007. 10.1103/​PhysRevLett.98.020501.

[10] 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.

[11] Emanuel Knill, Raymond Laflamme, and Wojciech H. Zurek. Threshold accuracy for quantum computation. arXiv: quant-ph/​9610011, 1996. URL https:/​/​arxiv.org/​abs/​quant-ph/​9610011.

[12] Tomas Jochym-O'Connor and Raymond Laflamme. Using concatenated quantum codes for universal fault-tolerant quantum gates. Phys. Rev. Lett., page 010505. 10.1103/​PhysRevLett.112.010505.

[13] Adam Paetznick and Ben W. Reichardt. Universal fault-tolerant quantum computation with only transversal gates and error correction. Phys. Rev. Lett., 111: 090505, Aug 2013. 10.1103/​PhysRevLett.111.090505.

[14] Jonas T. Anderson, Guillaume Duclos-Cianci, and David Poulin. Fault-tolerant conversion between the steane and reed-muller quantum codes. Phys. Rev. Lett., page 080501. 10.1103/​PhysRevLett.113.080501.

[15] Héctor Bombín. Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes. New J. Phys., 18: 043038, 2016. URL http:/​/​stacks.iop.org/​1367-2630/​17/​i=8/​a=083002.

[16] 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.

[17] S.J. Devitt, A.G. Fowler, T. Tilma, W.J. Munro, and K. Nemoto. Classical processing requirements for a topological quantum computing system. Int. J. Quant. Inf., 8: 1–27, 2010. 10.1142/​S021974991000637X.

[18] Barbara M. Terhal. Quantum error correction for quantum memories. Rev. Mod. Phys., 87: 307–346, Apr 2015. 10.1103/​RevModPhys.87.307.

[19] Christopher Chamberland, Joel Wallman, Stefanie Beale, and Raymond Laflamme. Hard decoding algorithm for optimizing thresholds under general markovian noise. Phys. Rev. A, 95: 042332, Apr 2017. 10.1103/​PhysRevA.95.042332.

[20] 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.

[21] Joel J. Wallman and Joseph Emerson. Noise tailoring for scalable quantum computation via randomized compiling. Phys. Rev. A, 94: 052325, Nov 2016. 10.1103/​PhysRevA.94.052325.

[22] Daniel Gottesman. The heisenberg representation of quantum computers, talk at. In International Conference on Group Theoretic Methods in Physics. Citeseer, 1998. URL https:/​/​arxiv.org/​abs/​quant-ph/​9807006.

[23] Panos Aliferis, Daniel Gottesman, and John Preskill. Quantum accuracy threshold for concatenated distance-3 codes. Quantum Info. Comput., 6 (2): 97–165, March 2006. ISSN 1533-7146. URL http:/​/​dl.acm.org/​citation.cfm?id=2011665.2011666.

[24] P. Oscar Boykin, Tal Mor, Matthew Pulver, Vwani Roychowdhury, and Farrokh Vatan. On universal and fault-tolerant quantum computing: A novel basis and a new constructive proof of universality for shor's basis. In Foundations of Computer Science, 1999. 40th Annual Symposium on, pages 486–494. IEEE, 1999. 10.1109/​SFFCS.1999.814621.

[25] Raymond Laflamme, Cesar Miquel, Juan Pablo Paz, and Wojciech Hubert Zurek. Perfect quantum error correcting code. Phys. Rev. Lett., 77: 198–201, Jul 1996. 10.1103/​PhysRevLett.77.198.

[26] David Poulin. Optimal and efficient decoding of concatenated quantum block codes. Phys. Rev. A, 74: 052333, Nov 2006. 10.1103/​PhysRevA.74.052333.

[27] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction. Phys. Rev. A, 54: 3824–3851, Nov 1996. 10.1103/​PhysRevA.54.3824.

[28] Thomas Koshy. Catalan Numbers with Applications. Oxford University Press, Oxford, England, 2008. ISBN 0-19-533454-X. URL https:/​/​global.oup.com/​academic/​product/​catalan-numbers-with-applications-9780195334548?cc=nl&lang=en&.

[29] George E Andrews, Richard Askey, and Ranjan Roy. Special functions (encyclopedia of mathematics and its applications vol 71), 1999.

[30] Easwar Magesan, J. M. Gambetta, and Joseph Emerson. Scalable and robust randomized benchmarking of quantum processes. Phys. Rev. Lett., 106: 180504, May 2011. 10.1103/​PhysRevLett.106.180504.

[31] Easwar Magesan, Jay M. Gambetta, and Joseph Emerson. Characterizing quantum gates via randomized benchmarking. Phys. Rev. A, 85: 042311, Apr 2012. 10.1103/​PhysRevA.85.042311.

[32] Benjamin Rahn, Andrew C. Doherty, and Hideo Mabuchi. Exact performance of concatenated quantum codes. Phys. Rev. A, 66: 032304, Sep 2002. 10.1103/​PhysRevA.66.032304.

[33] A. Jamiołkowski. Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics, (4): 275 – 278. ISSN 0034-4877. http:/​/​dx.doi.org/​10.1016/​0034-4877(72)90011-0.

[34] Christopher J. Wood, Jacob D. Biamonte, and David G. Cory. Tensor networks and graphical calculus for open quantum systems. Quantum Info. Comput., 15 (9-10): 759–811, July 2015. ISSN 1533-7146. URL http:/​/​dl.acm.org/​citation.cfm?id=2871422.2871425.

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