Optimal and efficient decoding of concatenated quantum block codes

We consider the problem of optimally decoding a quantum error correction code—that is, to find the optimal recovery procedure given the outcomes of partial “check” measurements on the system. In general, this problem is NP hard. However, we demonstrate that for concatenated block codes, the optimal decoding can be efficiently computed using a message-passing algorithm. We compare the performance of the message-passing algorithm to that of the widespread blockwise hard decoding technique. Our Monte Carlo results using the five-qubit and Steane’s code on a depolarizing channel demonstrate significant advantages of the message-passing algorithms in two respects: (i) Optimal decoding increases by as much as 94$\%$ the error threshold below which the error correction procedure can be used to reliably send information over a noisy channel; and (ii) for noise levels below these thresholds, the probability of error after optimal decoding is suppressed at a significantly higher rate, leading to a substantial reduction of the error correction overhead.

Figure 1

Monte Carlo results for the five-qubit code showing the probability of erroneous decoding $p_e$ as a function of the level of concatenation $\ell$ for different depolarization rate $p=0.13$, 0.15, 0.17, 0.18, 0.1885, and $0.19$. The diamonds are from the message-passing algorithm, and the circles are from the blockwise decoding. All data are from samples of $2\times {10}^4$ encoded qubits.

Figure 2

As in Fig. 1 for $p=0.1$ and 0.05. Diamonds obtained from samples of ${10}^8$ encoded qubits. Circles were produced using an exact numerical technique similar to that of Ref. 14.