# Quantum Error Correcting Code Empirical Performance

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.