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Fault tolerance of "bad" quantum low-density parity check codes

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In my talk, I will discuss various families of quantum
low-density parity check
(LDPC) codes and their fault tolerance. Such codes yield finite code rates and
at the same time
simplify error correction and encoding due to low-weight stabilizer
generators. As an example, a large family of

hypergraph-product codes is considered. Of particular interest are families of
quantum LDPC codes with finite rate and distance scaling as square root of
blocklength since this represents the best known exponent in
distance scaling, even for codes of dimensionality 1. In relation to such
codes, we show that any family of LDPC codes, quantum or classical,
where distance scales as a positive power of the block length, $d
\propto n^\alpha$, $\alpha>0$ ($\alpha

codes), can correct all errors with certainty if the
error rate per qubit is sufficiently small. We specifically
analyze the case of LDPC version of the quantum
hypergraph-product codes recently suggested by Tillich and Z\'emor. These codes are a
finite-rate generalization of the toric codes, and, for sufficiently
large quantum computers, offer an advantage over the toric codes.