Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames

Mischa P. Woods1 and Álvaro M. Alhambra2

1Institute for Theoretical Physics, ETH Zurich, Switzerland
2Perimeter Institute for Theoretical Physics, Waterloo, Canada

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Abstract

Following the introduction of the task of $\textit{reference frame error}$ $\textit{correction}$ [1], we show how, by using reference frame alignment with clocks, one can add a continuous Abelian group of transversal logical gates to $any$ error-correcting code. With this we further explore a way of circumventing the no-go theorem of Eastin and Knill, which states that if local errors are correctable, the group of transversal gates must be of finite order. We are able to do this by introducing a small error on the decoding procedure that decreases with the dimension of the frames used. Furthermore, we show that there is a direct relationship between how small this error can be and how accurate quantum clocks can be: the more accurate the clock, the smaller the error; and the no-go theorem would be violated if time could be measured perfectly in quantum mechanics. The asymptotic scaling of the error is studied under a number of scenarios of reference frames and error models. The scheme is also extended to errors at unknown locations, and we show how to achieve this by simple majority voting related error correction schemes on the reference frames. In the Outlook, we discuss our results in relation to the AdS/CFT correspondence and the Page-Wooters mechanism.

Quantum error correction is the field that aims to devise schemes that protect quantum information as it is being processed in computations. This is a monumental task, which needs to overcome fundamental physical constraints unique to quantum physics such as the no-cloning theorem. The codes must then present different features that makes them both scalable and robust. A crucial one is the existence of "transversal gates", which are logical operations that can be implemented by acting separately on the different subsystems within the code.

Previously, a number of no-go theorems heavily restricted the possibility of such transversal gates, establishing that the set of them must in general be finite. Here, we show that one can in fact enhance these sets to continuous or infinite groups by appending a so-called "quantum clock" to the codes. This is only possible if one allows for a small error in the decoding procedure — an idea that had been overlooked in much of the previous literature. We show that the error here in fact quickly vanishes as the size and precision of the clocks increases, so that in principle it can be made as small as required in the different implementations. Quantum error correcting codes have recently found numerous applications in physics beyond their initial purpose, and we explain what our results could imply for these connections.

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► References

[1] Patrick Hayden, Sepehr Nezami, Sandu Popescu, and Grant Salton. Error correction of quantum reference frame information. ArXiv:1709.04471, 2017. URL https:/​/​arxiv.org/​abs/​1709.04471.
arXiv:1709.04471

[2] Bei Zeng, Andrew Cross, and Isaac L Chuang. Transversality versus universality for additive quantum codes. IEEE Trans. Inf. Theory, 57 (9): 6272–6284, 2011. 10.1109/​TIT.2011.2161917.
https:/​/​doi.org/​10.1109/​TIT.2011.2161917

[3] Sergey Bravyi and Robert König. Classification of topologically protected gates for local stabilizer codes. Phys. Rev. Lett., 110 (17): 170503, 2013. 10.1103/​PhysRevLett.110.170503.
https:/​/​doi.org/​10.1103/​PhysRevLett.110.170503

[4] Fernando Pastawski and Beni Yoshida. Fault-tolerant logical gates in quantum error-correcting codes. Phys. Rev. A, 91 (1): 012305, 2015. 10.1103/​PhysRevA.91.012305.
https:/​/​doi.org/​10.1103/​PhysRevA.91.012305

[5] Tomas Jochym-O'Connor, Aleksander Kubica, and Theodore J Yoder. Disjointness of stabilizer codes and limitations on fault-tolerant logical gates. Phys. Rev. X, 8 (2): 021047, 2018. 10.1103/​PhysRevX.8.021047.
https:/​/​doi.org/​10.1103/​PhysRevX.8.021047

[6] Benjamin J Brown, Daniel Loss, Jiannis K Pachos, Chris N Self, and James R Wootton. Quantum memories at finite temperature. Rev. Mod. Phys., 88 (4): 045005, 2016. 10.1103/​RevModPhys.88.045005.
https:/​/​doi.org/​10.1103/​RevModPhys.88.045005

[7] Bryan Eastin and Emanuel Knill. Restrictions on transversal encoded quantum gate sets. Phys. Rev. Lett., 102 (11): 110502, 2009. 10.1103/​PhysRevLett.102.110502.
https:/​/​doi.org/​10.1103/​PhysRevLett.102.110502

[8] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A, 71 (2): 022316, 2005. 10.1103/​PhysRevA.71.022316.
https:/​/​doi.org/​10.1103/​PhysRevA.71.022316

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

[10] Héctor Bombín and Miguel Ángel Martín-Delgado. Topological computation without braiding. Physical review letters, 98 (16): 160502, 2007. 10.1103/​PhysRevLett.98.160502.
https:/​/​doi.org/​10.1103/​PhysRevLett.98.160502

[11] Adam Paetznick and Ben W Reichardt. Universal fault-tolerant quantum computation with only transversal gates and error correction. Phys. Rev. Lett., 111 (9): 090505, 2013. 10.1103/​PhysRevLett.111.090505.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.090505

[12] Tomas Jochym-O'Connor and Raymond Laflamme. Using concatenated quantum codes for universal fault-tolerant quantum gates. Phys. Rev. Lett., 112 (1): 010505, 2014. 10.1103/​PhysRevLett.112.010505.
https:/​/​doi.org/​10.1103/​PhysRevLett.112.010505

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

[14] Theodore J Yoder, Ryuji Takagi, and Isaac L Chuang. Universal fault-tolerant gates on concatenated stabilizer codes. Phys. Rev. X, 6 (3): 031039, 2016. 10.1103/​PhysRevX.6.031039.
https:/​/​doi.org/​10.1103/​PhysRevX.6.031039

[15] Stephen D Bartlett, Terry Rudolph, and Robert W Spekkens. Reference frames, superselection rules, and quantum information. Rev. Mod. Phys., 79 (2): 555, 2007. 10.1103/​RevModPhys.79.555.
https:/​/​doi.org/​10.1103/​RevModPhys.79.555

[16] Wolfgang Pauli. Handbuch der Physik. Springer, Berlin, 24: 83 – 272, 1933. 10.1007/​978-3-642-52619-0_2.
https:/​/​doi.org/​10.1007/​978-3-642-52619-0_2

[17] Wolfgang Pauli. Encyclopedia of Physics. Springer, Berlin, 1: 60, 1958.

[18] John C. Garrison and Jack Wong. Canonically conjugate pairs, uncertainty relations, and phase operators. J. Math. Phys., 11 (8): 2242–2249, Aug 1970. 10.1063/​1.1665388.
https:/​/​doi.org/​10.1063/​1.1665388

[19] Helmut Salecker and EP Wigner. Quantum limitations of the measurement of space-time distances. Phys. Rev., 109 (2): 571, 1958. 10.1103/​PhysRev.109.571.
https:/​/​doi.org/​10.1103/​PhysRev.109.571

[20] Asher Peres. Measurement of time by quantum clocks. American Journal of Physics, 48 (7): 552–557, 1980. 10.1119/​1.12061.
https:/​/​doi.org/​10.1119/​1.12061

[21] Mischa P. Woods, Ralph Silva, and Jonathan Oppenheim. Autonomous Quantum Machines and Finite-Sized Clocks. Annales Henri Poincaré, Oct 2018a. ISSN 1424-0661. 10.1007/​s00023-018-0736-9.
https:/​/​doi.org/​10.1007/​s00023-018-0736-9

[22] Vladimir Bužek, Radoslav Derka, and Serge Massar. Optimal quantum clocks. Phys. Rev. Lett., 82: 2207–2210, Mar 1999. 10.1103/​PhysRevLett.82.2207.
https:/​/​doi.org/​10.1103/​PhysRevLett.82.2207

[23] Paul Erker, Mark T. Mitchison, Ralph Silva, Mischa P. Woods, Nicolas Brunner, and Marcus Huber. Autonomous quantum clocks: Does thermodynamics limit our ability to measure time? Phys. Rev. X, 7: 031022, Aug 2017. 10.1103/​PhysRevX.7.031022. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.7.031022.
https:/​/​doi.org/​10.1103/​PhysRevX.7.031022

[24] Sandra Ranković, Yeong-Cherng Liang, and Renato Renner. Quantum clocks and their synchronisation - the alternate ticks game. ArXiv:1506.01373v1, 2015. URL https:/​/​arxiv.org/​abs/​1506.01373.
arXiv:1506.01373

[25] Mischa P Woods, Ralph Silva, Gilles Pütz, Sandra Stupar, and Renato Renner. Quantum clocks are more accurate than classical ones. ArXiv:1806.00491, 2018b. URL https:/​/​arxiv.org/​abs/​1806.00491.
arXiv:1806.00491

[26] Philippe Faist, Sepehr Nezami, Victor V. Albert, Grant Salton, Fernando Pastawski, Patrick Hayden, and John Preskill. Continuous symmetries and approximate quantum error correction. ArXiv:1902.07714, 2019. URL https:/​/​arxiv.org/​abs/​1902.07714.
arXiv:1902.07714

[27] José T. Lunardi, Luiz A. Manzoni, and Andrew T. Nystrom. Salecker Wigner Peres clock and average tunneling times. Phys. Lett. A, 375 (3): 415 – 421, 2011. ISSN 0375-9601. 10.1016/​j.physleta.2010.11.055.
https:/​/​doi.org/​10.1016/​j.physleta.2010.11.055

[28] Dmitri Sokolovski. Salecker-wigner-peres clock, feynman paths, and a tunneling time that should not exist. Phys. Rev. A, 96: 022120, Aug 2017. 10.1103/​PhysRevA.96.022120.
https:/​/​doi.org/​10.1103/​PhysRevA.96.022120

[29] Marcos Calçada, José T. Lunardi, and Luiz A. Manzoni. Salecker-Wigner-Peres clock and double-barrier tunneling. Phys. Rev. A, 79: 012110, Jan 2009. 10.1103/​PhysRevA.79.012110.
https:/​/​doi.org/​10.1103/​PhysRevA.79.012110

[30] Nicolas Teeny, Christoph H. Keitel, and Heiko Bauke. Salecker-Wigner-Peres quantum clock applied to strong-field tunnel ionization. ArXiv:1608.02854, Aug 2016. URL http:/​/​arxiv.org/​abs/​1608.02854.
arXiv:1608.02854

[31] Stephen D Bartlett, Terry Rudolph, Robert W Spekkens, and Peter S Turner. Quantum communication using a bounded-size quantum reference frame. New J. Phys., 11 (6): 063013, 2009. 10.1088/​1367-2630/​11/​6/​063013.
https:/​/​doi.org/​10.1088/​1367-2630/​11/​6/​063013

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

[33] Michael A Nielsen. The entanglement fidelity and quantum error correction. ArXiv: quant-ph/​9606012, 1996. URL https:/​/​arxiv.org/​abs/​quant-ph/​9606012.
arXiv:quant-ph/9606012

[34] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Advances in quantum metrology. Nat. Photonics, 5 (222), Mar 2011. 10.1038/​nphoton.2011.35.
https:/​/​doi.org/​10.1038/​nphoton.2011.35

[35] Iman Marvian Mashhad. Symmetry, asymmetry and quantum information. PhD thesis, University of Waterloo, 2012. URL http:/​/​hdl.handle.net/​10012/​7088.
http:/​/​hdl.handle.net/​10012/​7088

[36] Iman Marvian and Robert W Spekkens. Extending noether's theorem by quantifying the asymmetry of quantum states. Nat. Commun., 5, 2014. 10.1038/​ncomms4821.
https:/​/​doi.org/​10.1038/​ncomms4821

[37] Rafał Demkowicz-Dobrzański, Jan Czajkowski, and Pavel Sekatski. Adaptive quantum metrology under general markovian noise. Phys. Rev. X, 7 (4): 041009, 2017. 10.1103/​PhysRevX.7.041009.
https:/​/​doi.org/​10.1103/​PhysRevX.7.041009

[38] Sisi Zhou, Mengzhen Zhang, John Preskill, and Liang Jiang. Achieving the heisenberg limit in quantum metrology using quantum error correction. Nat. Commun., 9 (1): 78, 2018. 10.1038/​s41467-017-02510-3.
https:/​/​doi.org/​10.1038/​s41467-017-02510-3

[39] David Layden, Sisi Zhou, Paola Cappellaro, and Liang Jiang. Ancilla-free quantum error correction codes for quantum metrology. Physical review letters, 122 (4): 040502, 2019. 10.1103/​PhysRevLett.122.040502.
https:/​/​doi.org/​10.1103/​PhysRevLett.122.040502

[40] Ahmed Almheiri, Xi Dong, and Daniel Harlow. Bulk locality and quantum error correction in AdS/​CFT. J. High Energy Phys., 2015 (4): 163, Apr 2015. ISSN 1029-8479. 10.1007/​JHEP04(2015)163.
https:/​/​doi.org/​10.1007/​JHEP04(2015)163

[41] Daniel Harlow and Hirosi Ooguri. Constraints on symmetries from holography. Physical review letters, 122 (19): 191601, 2019. 10.1103/​PhysRevLett.122.191601.
https:/​/​doi.org/​10.1103/​PhysRevLett.122.191601

[42] Daniel Harlow and Hirosi Ooguri. Symmetries in quantum field theory and quantum gravity. ArXiv:1810.05338, Oct 2018. URL http:/​/​arxiv.org/​abs/​1810.05338.
arXiv:1810.05338

[43] Tamara Kohler and Toby Cubitt. Toy models of holographic duality between local hamiltonians. Journal of High Energy Physics, 2019 (8): 17, Aug 2019. ISSN 1029-8479. 10.1007/​JHEP08(2019)017.
https:/​/​doi.org/​10.1007/​JHEP08(2019)017

[44] Toby S. Cubitt, Ashley Montanaro, and Stephen Piddock. Universal quantum Hamiltonians. Proc. Natl. Acad. Sci. U.S.A, 115 (38): 9497–9502, 2018. ISSN 0027-8424. 10.1073/​pnas.1804949115.
https:/​/​doi.org/​10.1073/​pnas.1804949115

[45] Bryce S. DeWitt. Quantum theory of gravity. i. the canonical theory. Phys. Rev., 160: 1113–1148, Aug 1967. 10.1103/​PhysRev.160.1113.
https:/​/​doi.org/​10.1103/​PhysRev.160.1113

[46] Don N. Page and William K. Wootters. Evolution without evolution: Dynamics described by stationary observables. Phys. Rev. D, 27: 2885–2892, Jun 1983. 10.1103/​PhysRevD.27.2885.
https:/​/​doi.org/​10.1103/​PhysRevD.27.2885

[47] Michael M Wolf. Quantum channels & operations: A Guided tour. Lecture notes, July 2012. URL https:/​/​www-m5.ma.tum.de/​foswiki/​pub/​M5/​Allgemeines/​MichaelWolf/​QChannelLecture.pdf.
https:/​/​www-m5.ma.tum.de/​foswiki/​pub/​M5/​Allgemeines/​MichaelWolf/​QChannelLecture.pdf

[48] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum time. Phys. Rev. D, 92: 045033, Aug 2015. 10.1103/​PhysRevD.92.045033.
https:/​/​doi.org/​10.1103/​PhysRevD.92.045033

[49] Rodolfo Gambini, Rafael A. Porto, Jorge Pullin, and Sebastián Torterolo. Conditional probabilities with dirac observables and the problem of time in quantum gravity. Phys. Rev. D, 79: 041501, Feb 2009. 10.1103/​PhysRevD.79.041501.
https:/​/​doi.org/​10.1103/​PhysRevD.79.041501

[50] Chiara Marletto and Vlatko Vedral. Evolution without evolution and without ambiguities. Phys. Rev. D, 95: 043510, Feb 2017. 10.1103/​PhysRevD.95.043510.
https:/​/​doi.org/​10.1103/​PhysRevD.95.043510

[51] Ekaterina Moreva, Giorgio Brida, Marco Gramegna, Vittorio Giovannetti, Lorenzo Maccone, and Marco Genovese. Time from quantum entanglement: An experimental illustration. Phys. Rev. A, 89: 052122, May 2014. 10.1103/​PhysRevA.89.052122.
https:/​/​doi.org/​10.1103/​PhysRevA.89.052122

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[2] Hyukjoon Kwon and M. S. Kim, "Fluctuation Theorems for a Quantum Channel", Physical Review X 9 3, 031029 (2019).

[3] Tamara Kohler and Toby Cubitt, "Toy models of holographic duality between local Hamiltonians", Journal of High Energy Physics 2019 8, 17 (2019).

[4] Shishir Khandelwal, Maximilian P. E. Lock, and Mischa P. Woods, "Universal quantum modifications to general relativistic time dilation in delocalised clocks", arXiv:1904.02178.

[5] Ning Bao and Newton Cheng, "Eigenstate thermalization hypothesis and approximate quantum error correction", Journal of High Energy Physics 2019 8, 152 (2019).

[6] Dong-Sheng Wang, Guanyu Zhu, Cihan Okay, and Raymond Laflamme, "Quasi-exact quantum computation", arXiv:1910.00038.

[7] Sami Boulebnane, Mischa P. Woods, and Joseph M. Renes, "Approximate quantum non-demolition measurements", arXiv:1909.05265.

[8] Mischa P. Woods and Michał Horodecki, "The Resource Theoretic Paradigm of Quantum Thermodynamics with Control", arXiv:1912.05562.

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