Fault tolerance of quantum low-density parity check codes with sublinear distance scaling

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 of the number of physical qubits (<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.