Photon-loss tolerance of the surface-GKP code with anti-photon-loss decoding

Abstract

Photon loss somewhat limits fault-tolerant photonic quantum computing. We investigate its effect on finite-energy Gottesman--Kitaev--Preskill (GKP) states by modeling the pure-loss channel directly. We obtain photon-loss threshold estimates for the surface-GKP code using nearest-lattice-point inner GKP decoding and minimum-weight perfect matching for the outer surface code. For GKP states with $ε=0.01$ and $ε=0.05$, corresponding to GKP squeezing of $20$ and $13\,\mathrm{dB}$, the estimated thresholds are $γ^\ast=5.5\%$ ($0.24\,\mathrm{dB}$) and $γ^\ast=5.9\%$ ($0.26\,\mathrm{dB}$), respectively. We then introduce an anti-photon-loss inner decoder that compensates for the contraction of the measured quadrature. It increases the corresponding threshold estimates to $γ^\ast=9.8\%$ ($0.45\,\mathrm{dB}$) and $γ^\ast=6.3\%$ ($0.28\,\mathrm{dB}$). Our results highlight the importance of accounting for loss-induced lattice contraction in inner decoding and demonstrate that quadrature compensation enhances the photon-loss tolerance of surface-GKP codes for both high- and moderate-quality GKP states, providing a route toward more photon-loss-robust optical quantum computing.