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The above figure shows performance results for the codes published in our paper. The captions of the relevant figures are given below.

Summary of performances of several quantum codes on the 4-ary symmetric channel (depolarizing channel), treated (by almost all the decoding algorithms shown in this figure) as if the channel were a pair of independent binary symmetric channels. Each point shows the (horizontal axis) marginal noise level at which the block error probability is $10^{-4}$. In the case of dual-containing codes, this is the noise level at which each of the two identical constituent codes has an error probability of $5\times 10^{-5}$. As an aid to the eye, lines have been added between the four unicycle codes (U); between a sequence of bicycle codes (B) all of blocklength $N=3786$ with different rates; and between a sequence of of BCH codes with increasing blocklength. The curve labelled S2 is the Shannon limit if the correlations between X errors and Z errors are neglected. Points `$+$' are codes invented elsewhere. All other point styles denote codes presented for the first time in this paper. Summary of performances of several codes on the 4-ary symmetric channel (depolarizing channel). The additional points at the right and bottom are as follows. 3786(B,4SC): a code of construction B (the same code as its neighbour in the figure) decoded with a decoder that exploits the known correlations between X errors and Z errors. 3786(B,D): the same code as the $N=3786$ code to its left in the figure, simulated with a channel where the qubits have a diversity of known reliabilities; X errors and Z errors occur independently with probabilities determined from a Gaussian distribution; the channel in this case is not the 4-ary symmetric channel, but we plot the performance at the equivalent value of the marginal noise level. [[29,1,11]]: an algebraically constructed quantum code (not a sparse-graph code) from Markus Grassl.

David MacKay Last modified: Sun Sep 12 20:12:39 2004