Quantum Coding with Low-Depth Random Circuits

• Published in: Physical review. X Vol. 11; no. 3; p. 031066
• Main Authors: Gullans, Michael J., Krastanov, Stefan, Huse, David A., Jiang, Liang, Flammia, Steven T.
• Format: Journal Article
• Language: English
• Published: College Park American Physical Society 01.09.2021
• Subjects: Algorithms; Channel capacity; Circuits; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Codes; Coding; Coding theory; Convergence; Critical phenomena; Data processing; Error analysis; Error correcting codes; Error correction; Error correction & detection; Fault tolerance; Operators; Phase transitions; Physics; quantum channels; quantum computation; Quantum computers; Quantum computing; Quantum entanglement; quantum error correction; Quantum phenomena; Qubits (quantum computing); Random errors; Statistical analysis; Subsystems
• ISSN: 2160-3308; 2160-3308
• DOI: 10.1103/PhysRevX.11.031066

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 inD≥1spatial dimensions to generate quantum error-correcting codes. For random stabilizer codes and the erasure channel, we find strong evidence that a depthO(logN)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 anyD. Previous results on random circuits have only shown thatO(N1/D)depth suffices or thatO(log3N)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 withN. We find that the requisite depth scales likeO(logN)only for dimensionsD≥2and that random circuits requireO(N)depth forD=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 inD≥2spatial 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 inD=2dimensions. 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.