Fault Tolerant Decoding of QLDPC-GKP Codes with Circuit Level Soft Information

Concatenated bosonic-stabilizer codes have recently gained prominence as promising candidates for achieving lowoverhead fault-tolerant quantum computing in the long term. In such a system, analog information obtained from the syndrome measurements of an inner bosonic code is used to inform decoding for an outer code layer consisting of a discrete-variable stabilizer code such as a surface code. The use of Quantum Low Density Parity Check (QLDPC) codes as an outer code is of particular interest in view of the significantly higher encoding rates offered by these code families, leading to a further reduction in the overhead requirements for large-scale quantum computing. Recent works have investigated the performance of QLDPC-GKP codes in detail, and indeed, the use of analog information obtained from the inner code significantly boosts the decoder performance. However, the noise models assumed in these works are typically limited to the depolarizing or phenomenological noise models. In this paper, we investigate the performance of QLDPC-GKP concatenated codes under circuit-level noise, based on a noise model introduced by Noh et al. in relation to the surfaceGKP code. To demonstrate the performance boost provided by the use of analog information, we investigate three different scenarios, namely (a) decoding performance in the absence of soft information, (b) decoding performance with precomputed probabilities of error but without real-time soft information, and (c) decoding performance with real-time soft information obtained from round-to-round decoding of the inner GKP code. Results show minimal performance improvement between (a) and (b), but a significant boost in (c), indicating that real-time soft information is critical for concatenated decoding under circuitlevel noise. We also investigate the effect of using measurement schedules of varying depths and show that using a schedule with minimum depth is critical for obtaining reliable soft information from the inner code.