Soft-decision decoder for quantum erasure and probabilistic-gate error models

Abstract

A soft-decision decoder for quantum erasure errors with standard depolarizing errors is proposed for concatenated Calderbank-Shor-Steane codes. This decoding is computationally efficient as long as the block size of each concatenation level is sufficiently small. The numerical simulation with a specific quantum code called the ${C}_{4}/{C}_{6}$ code shows that this decoder can achieve almost optimal performance and the so-called hashing bound for the erasure and depolarizing error model. Since probabilistic-gate errors can be regarded as erasure errors, this decoder is useful for fault-tolerant quantum computation with probabilistic gates. To demonstrate this, we perform numerical simulation with the ${C}_{4}/{C}_{6}$ code. Consequently, it turns out that the error probability of a logical controlled-not gate with probabilistic physical two-qubit gates is improved by three orders of magnitude by the present decoder compared to a previous hard-decision decoder.