Fault tolerance of ``bad'' quantum low-density parity check codes
Alexey A. Kovalev and Leonid P. Pryadko
Accepted
We study fault-tolerance of quantum low-density parity check (LDPC) codes such as generalized toric codes with finite rate suggested by Tillich and Zémor. 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), where distance scales as a positive power a of the number of physical qubits (a less than one 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.