Logical channels in approximate Gottesman-Kitaev-Preskill error correction

The Gottesman-Kitaev-Preskill (GKP) encoding is a top contender among bosonic codes for fault-tolerant quantum computation. However, analysis of the code is complicated by the fact that finite-energy code states leak out of the ideal GKP code space and are not orthogonal. We analyze a variant of the GKP stabilizer measurement circuit that virtually projects onto the ideal GKP code space between rounds of error correction, even when damped, finite-energy GKP states are used. This allows us to identify logical maps between projectors; however, due to finite-energy effects, these maps fail to resolve completely positive, trace-preserving (CPTP) channels on the logical GKP code space. We present two solutions to this problem based on channel twirling the damping operator. The first uses symmetries of standard binning (SB) decoding to passively twirl over the full stabilizer group. Doing so converts small amounts of damping into stochastic Gaussian random noise (GRN). The second uses active twirling over a minimal set of representative Pauli shifts that keeps the energy in the code finite and allows for arbitrary decoding. This approach is not limited to small damping and allows the study of decoding more general than SB, which can be optimized to the noise in the circuit. Focusing on damping, we compare decoding strategies tailored to different levels of effective squeezing. While our results indicate that SB decoding is suboptimal for finite-energy GKP states, we observe that the advantage of optimized decoding shrinks as the energy in the code increases. Moreover, the performance of both decoding strategies converges to that of the stabilizer-twirled GRN logical channel. These studies provide stronger arguments for two commonplace procedures in the analysis of GKP error correction in the fault-tolerance regime: $(i)$ using stochastically shifted GKP states in place of coherently damped ones, and $(ii)$ the use of SB decoding.