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Fault-tolerant logical gates in quantum error-correcting codes

Pastawski, Fernando and Yoshida, Beni (2015) Fault-tolerant logical gates in quantum error-correcting codes. Physical Review A, 91 (1). Art. No. 012305. ISSN 1050-2947. doi:10.1103/PhysRevA.91.012305.

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Recently, S. Bravyi and R. König [Phys. Rev. Lett. 110, 170503 (2013)] have shown that there is a trade-off between fault-tolerantly implementable logical gates and geometric locality of stabilizer codes. They consider locality-preserving operations which are implemented by a constant-depth geometrically local circuit and are thus fault tolerant by construction. In particular, they show that, for local stabilizer codes in D spatial dimensions, locality-preserving gates are restricted to a set of unitary gates known as the Dth level of the Clifford hierarchy. In this paper, we explore this idea further by providing several extensions and applications of their characterization to qubit stabilizer and subsystem codes. First, we present a no-go theorem for self-correcting quantum memory. Namely, we prove that a three-dimensional stabilizer Hamiltonian with a locality-preserving implementation of a non-Clifford gate cannot have a macroscopic energy barrier. This result implies that non-Clifford gates do not admit such implementations in Haah's cubic code and Michnicki's welded code. Second, we prove that the code distance of a D-dimensional local stabilizer code with a nontrivial locality-preserving mth-level Clifford logical gate is upper bounded by O(L^(D+1−m)). For codes with non-Clifford gates (m>2), this improves the previous best bound by S. Bravyi and B. Terhal [New. J. Phys. 11, 043029 (2009)]. Topological color codes, introduced by H. Bombin and M. A. Martin-Delgado [Phys. Rev. Lett. 97, 180501 (2006); Phys. Rev. Lett. 98, 160502 (2007); Phys. Rev. B 75, 075103 (2007)], saturate the bound for m=D. Third, we prove that the qubit erasure threshold for codes with a nontrivial transversal mth-level Clifford logical gate is upper bounded by 1/m. This implies that no family of fault-tolerant codes with transversal gates in increasing level of the Clifford hierarchy may exist. This result applies to arbitrary stabilizer and subsystem codes and is not restricted to geometrically local codes. Fourth, we extend the result of Bravyi and König to subsystem codes. Unlike stabilizer codes, the so-called union lemma does not apply to subsystem codes. This problem is avoided by assuming the presence of an error threshold in a subsystem code, and a conclusion analogous to that of Bravyi and König is recovered.

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Additional Information:© 2015 American Physical Society. Received 25 August 2014; published 8 January 2015. We would like to thank M. Beverland for pointing out the nongeometric interpretation of BK’s results, and R. König, K. Temme, D. Poulin, and J. Preskill for fruitful discussions. We would also like to thank an anonymous referee for valuable feedback. We acknowledge funding provided by the Institute for Quantum Information and Matter, a NSF Physics Frontiers Center with support from the Gordon and Betty Moore Foundation (Grants No. PHY-0803371 and No. PHY-1125565). B.Y. is supported by the David and Ellen Lee Postdoctoral Fellowship.
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
Institute for Quantum Information and MatterUNSPECIFIED
NSF Physics Frontiers CenterUNSPECIFIED
Gordon and Betty Moore FoundationUNSPECIFIED
David and Ellen Lee Postdoctoral FellowshipUNSPECIFIED
Issue or Number:1
Classification Code:PACS number(s): 03.67.Pp, 03.65.Ud, 03.67.Ac, 03.67.Lx
Record Number:CaltechAUTHORS:20150309-111410905
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Official Citation:Pastawski, F., & Yoshida, B. (2015). Fault-tolerant logical gates in quantum error-correcting codes. Physical Review A, 91(1), 012305.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:55641
Deposited On:09 Mar 2015 20:11
Last Modified:10 Nov 2021 20:48

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