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 simplicity. However, encoding a singular logical qubit with the planar surface code requires physical qubits quadratic in the code distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(d)$</tex>, making it space-inefficient for the large-distance codes necessary for promising applications. Thus, Quantum Low-Density Parity-Check (QLDPC) have emerged as an alternative to the planar surface code but require a higher degree of connectivity. Furthermore, the problems of fault-tolerant syndrome extraction and decoding are understudied for these codes and also 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 challenges mentioned above 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 decode the hyperbolic codes accurately. Our work finds that degree-4 FPNs of the hyperbolic surface and color codes are respectively <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2.9\times$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$5.5\times$</tex> more space-efficient than the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d=5$</tex> planar surface code, and become even more space-efficient when considering higher distances. The hyperbolic codes also have error rates comparable to their planar counterparts.