Fault tolerance of quantum low-density parity check codes with sublinear distance scaling
Alexey A. Kovalev, Leonid P. Pryadko · Physical Review A · 2013
We study the fault tolerance of quantum low-density parity check (LDPC) codes, such as generalized toric codes with a finite rate suggested by Tillich and Z\'emor [in ISIT 2009: IEEE International Symposium on Information Theory (IEEE, New York, 2009)]. We show that any family of quantum LDPC codes where each syndrome measurement involves a limited number of qubits and each qubit is involved in a limited number of measurements (as well as any similarly limited family of classical LDPC codes), in which distance scales as a positive power $\ensuremath{\alpha}$ of the number of physical qubits ($\ensuremath{\alpha}<1$ for ``bad'' codes), has a finite error probability threshold. We conclude that for sufficiently large quantum computers, quantum LDPC codes can offer an advantage over the toric codes.