We present sparse-graph codes appropriate for use in quantum error-correction. Quantum error-correcting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparse-graph codes keep the number of quantum interactions associated with the quantum error-correction process small: a constant number per quantum bit, independent of the blocklength. Third, sparse-graph codes often offer great flexibility with respect to blocklength and rate.We believe some of the codes we present are unsurpassed by previously published quantum error-correcting codes.Our aim in this paper is to create useful quantum error-correcting codes. To be useful, we think a quantum code must have a large blocklength (since quantum computation becomes interesting only when the number of entangled qubits is substantial), and it must be able to correct a large number of errors. From a theoretical point of view, we would especially like to find, for any rate R, a family of error-correcting codes with increasing blocklength N, such that, no matter how large N is, the number of errors that can be corrected is proportional to N. (Such codes are called 'good' codes.) From a practical point of view, however, we will settle for a lesser goal: to be able to make codes with blocklengths in the range 500-20, 000 qubits and rates in the range 0.1-0.9 that can correct the largest number of errors possible.While the existence of 'good' quantum error-correcting codes was proved by Calderbank and Shor (1996), their method of proof was non-constructive. Recently, a family of asymptotically good quantum codes based on algebraic geometry has been found by Ashikhmin et al. (2000) (see also Ling et al. (2001) and Matsumoto (2002)); however to the best of our knowledge no practical decoding algorithm (i.e., an algorithm for which the decoding time is polynomial in the blocklength) exists for these codes. Thus, the task of constructing good quantum error-correcting codes for which there exists a practical decoder remains an open challenge.This stands in contrast to the situation for classical error-correction, where practicallydecodable codes exist which, when optimally decoded, achieve information rates close to the Shannon limit. Low-density parity-check codes (Gallager 1962;Gallager 1963) are an example of such codes. A regular low-density parity-check code has a parity-check matrix H in which each column has a small weight j (e.g., j = 3) and the weight per row, k is also uniform (e.g., k = 6). Recently low-density parity-check codes have been shown to have outstanding performance (MacKay and Neal 1996;MacKay 1999) and modifications to their construction have turned them into state-of-the-art codes, both at low rates and large blocklengths (Luby et al. 2001;Richardson et al. 2001;Davey and MacKay 1998) and at high rates and short blocklengths (MacKay and Davey 2000). The sparseness of the paritycheck matrices makes the codes easy to encode and decode, even when co...
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