Flag Proxy Networks: Tackling the Architectural, Scheduling, and Decoding Obstacles of Quantum LDPC codes

TL;DR

This paper proposes Flag-Proxy Networks (FPNs), a generalizable architecture for quantum codes that achieves low connectivity through flag and proxy qubits, and proposes a greedy syndrome extraction scheduling algorithm for general quantum codes and uses this algorithm for fault-tolerant syndrome extraction on FPNs.

Abstract

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~($d$), making it space-inefficient for the large-distance codes necessary for promising applications. Thus, {\em 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. {\em First}, we propose {\em Flag-Proxy Networks (FPNs)}, a generalizable architecture for quantum codes that achieves low connectivity through flag and proxy qubits. {\em Second}, we propose a {\em greedy syndrome extraction scheduling} algorithm for general quantum codes and further use this algorithm for fault-tolerant syndrome extraction on FPNs. {\em 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 $2.9\times$ and $5.5\times$ 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 error rates comparable to their planar counterparts.

Figures (21)

• Figure 1: (a) Tradeoff between efficiency and connectivity amongst quantum error correcting codes. (b) Local structures for the planar and hyperbolic surface codes. (c) Syndrome extraction with a flag qubit is measured to detect errors on the data qubits. (d) The proposed Flag-Proxy Network architecture.
• Figure 2: (a) Example of planar surface code. (b) Example of the local structure of a $\{4, 5\}$ hyperbolic surface code. Each edge corresponds to a data qubit, each face corresponds to an $X$ check, and each vertex corresponds to a $Z$ check. (c) Example of a $\{4, 6\}$ hyperbolic color code. Each vertex corresponds to a data qubit, and each face (plaquette) corresponds to both a $Z$ and $X$ check. (c) Example of an undetected propagation error caused by a CNOT error.
• Figure 3: Parity checks for (a) the planar surface code, and (b) QLDPC codes. For illustration purposes only.
• Figure 4: Left: local connectivity required by a flag qubit. Right: corresponding syndrome extraction with a flag qubit $F$, which detects a propagation error.
• Figure 5: An example of a flag $H$ which provides redundant information about a propagation error detected by flags $F$ and $G$.

Theorems & Definitions (2)

• Theorem 1
• proof