Flag-Proxy Networks: Overcoming the Architectural, Scheduling and Decoding Obstacles of Quantum LDPC Codes

Quantum error correction is necessary for achieving exponential speedups on important applications. The planar surface code has remained the most studied error-correcting code for the last two decades because of its relative simplicitly. However, encoding a sin- gular logical qubit with the planar surface code requires physical qubits quadratic in the code distance (𝑑), making it space-inefficient for the large-distance codes necessary for promising applications. Thus, Quantum Low-Density Parity-Check (QLDPC) codes have emerged as an alternative to the planar surface code but require a higher degree of connectivity to implement. Furthermore, the problems of fault-tolerant syndrome extraction and decoding are understudied for these codes and remain obstacles to their usage. In this paper, we consider two under-studied families of QLDPC codes: hyperbolic surface codes and hyperbolic color codes. We tackle the three aforementioned challenges as follows. First, we propose Flag-Proxy Networks (FPNs), a generalizable architecture for quantum codes that achieves low connectivity through flag and proxy qubits. Second, we propose a greedy syndrome extraction scheduling algorithm for general quantum codes and further use this algorithm for fault-tolerant syndrome extraction on FPNs. Third, we present two decoders that leverage flag measurements to accurately decode the hyperbolic codes. Our work finds that degree-4 FPNs of the hyperbolic surface and color codes are respectively 2.9× and 5.5× more space-efficient than the 𝑑 = 5 planar surface code, and become even more space-efficient when considering higher distances. The hyperbolic codes also have comparable error rates to their planar counterparts.