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Quantum Coding with Low-Depth Random Circuits

Gullans, Michael J. and Krastanov, Stefan and Huse, David A. and Jiang, Liang and Flammia, Steven T. (2021) Quantum Coding with Low-Depth Random Circuits. Physical Review X, 11 (3). Art. No. 031066. ISSN 2160-3308. doi:10.1103/physrevx.11.031066.

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Random quantum circuits have played a central role in establishing the computational advantages of near-term quantum computers over their conventional counterparts. Here, we use ensembles of low-depth random circuits with local connectivity in D ≥ 1 spatial dimensions to generate quantum error-correcting codes. For random stabilizer codes and the erasure channel, we find strong evidence that a depth O(log N) random circuit is necessary and sufficient to converge (with high probability) to zero failure probability for any finite amount below the optimal erasure threshold, set by the channel capacity, for any D. Previous results on random circuits have only shown that O(N^(1/D)) depth suffices or that O(log³ N) depth suffices for all-to-all connectivity (D → ∞). We then study the critical behavior of the erasure threshold in the so-called moderate deviation limit, where both the failure probability and the distance to the optimal threshold converge to zero with N. We find that the requisite depth scales like O(log N) only for dimensions D ≥ 2 and that random circuits require O(√N) depth for D = 1. Finally, we introduce an “expurgation” algorithm that uses quantum measurements to remove logical operators that cause the code to fail by turning them into either additional stabilizers or into gauge operators in a subsystem code. With such targeted measurements, we can achieve sublogarithmic depth in D ≥ 2 spatial dimensions below capacity without increasing the maximum weight of the check operators. We find that for any rate beneath the capacity, high-performing codes with thousands of logical qubits are achievable with depth 4–8 expurgated random circuits in D = 2 dimensions. These results indicate that finite-rate quantum codes are practically relevant for near-term devices and may significantly reduce the resource requirements to achieve fault tolerance for near-term applications.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Gullans, Michael J.0000-0003-3974-2987
Krastanov, Stefan0000-0001-5550-5258
Huse, David A.0000-0003-1008-5178
Jiang, Liang0000-0002-0000-9342
Flammia, Steven T.0000-0002-3975-0226
Additional Information:© 2021 The Author(s). Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. (Received 13 November 2020; revised 11 June 2021; accepted 12 July 2021; published 24 September 2021) We thank Steve Girvin, Pradeep Niroula, and Sarang Gopalakrishnan for helpful discussions. M. J. G. and D. A. H. were supported in part by the Defense Advanced Research Projects Agency (DARPA) Driven and Nonequilibrium Quantum Systems (DRINQS) program. L. J. acknowledges support from the Army Research Office (ARO) (Grants No. W911NF-18-1-0020 and No. W911NF-18-1-0212), ARO MURI (Grant No. W911NF-16-1-0349), Air Force Office of Scientific Research (AFOSR) MURI (Grant No. FA9550-19-1-0399), Department of Energy (DOE) (Grant No. DE-SC0019406), National Science Foundation (NSF) (Grants No. EFMA-1640959, No. OMA-1936118, and No. EEC-1941583), and the Packard Foundation (Grant No. 2013-39273).
Group:AWS Center for Quantum Computing
Funding AgencyGrant Number
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Army Research Office (ARO)W911NF-18-1-0020
Army Research Office (ARO)W911NF-18-1-0212
Army Research Office (ARO)W911NF-16-1-0349
Department of Energy (DOE)DE-SC0019406
Air Force Office of Scientific Research (AFOSR)FA9550-19-1-0399
David and Lucile Packard Foundation2013-39273
Issue or Number:3
Record Number:CaltechAUTHORS:20210927-213255344
Persistent URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111053
Deposited By: George Porter
Deposited On:28 Sep 2021 14:07
Last Modified:28 Sep 2021 14:07

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