Quantum Coding with Low-Depth Random Circuits

Journal Article · Fri Sep 24 00:00:00 EDT 2021 · Physical Review. X

DOI:https://doi.org/10.1103/PhysRevX.11.031066· OSTI ID:1822498

Gullans, Michael J. ; Krastanov, Stefan; Huse, David A. ; Jiang, Liang ; Flammia, Steven T.

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 $D\ge 1$ spatial dimensions to generate quantum error-correcting codes. For random stabilizer codes and the erasure channel, we find strong evidence that a depth $O\left(\log N\right)$ 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 $O\left(N^{1/D}\right)$ depth suffices or that $O\left({\log }^{3}N\right)$ depth suffices for all-to-all connectivity ( $D\to \infty$). 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 $O\left(\log N\right)$ only for dimensions $D\ge 2$ and that random circuits require $O\left(\sqrt{N}\right)$ 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 $D\ge 2$ 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.

• Research Organization: Yale Univ., New Haven, CT (United States)
• Sponsoring Organization: USDOE Office of Science (SC); US Army Research Office (ARO); US Air Force Office of Scientific Research (AFOSR); Packard Foundation; National Science Foundation (NSF)
• Grant/Contract Number: SC0019406; W911NF-18-1-0020; W911NF-18-1-0212; W911NF-16-1-0349; FA9550-19-1-0399; EFMA-1640959; 2013-39273; OMA-1936118; EEC-1941583
• OSTI ID: 1822498
• Alternate ID(s): OSTI ID: 1853215
• Journal Information: Physical Review. X, Journal Name: Physical Review. X Vol. 11 Journal Issue: 3; ISSN 2160-3308
• Publisher: American Physical Society
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
Physics
quantum channels
quantum computation
quantum error correction