Logical channels in approximate Gottesman-Kitaev-Preskill error correction

The GKP encoding is a top contender among bosonic codes for fault-tolerant quantum computation. Analysis of the GKP 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 using damped, approximate GKP states that virtually project onto the ideal GKP code space between rounds of error correction even when 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 channels on the logical code space. We present two solutions to this problem based on channel twirling the damping operator. The first twirls over the full stabilizer group motivated by standard binning (SB) decoding that converts small amounts of damping into Gaussian random noise. The second twirls over a set of representative Pauli shifts that keeps the energy in the code finite and allow for arbitrary decoding. This approach is not limited to small damping, can be applied when logical GKP unitaries or other sources of CV noise are present, and allows us to study general decoding, 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, the advantage of optimized decoding over SB decoding shrinks as the energy in the code increases, and moreover the performance of both strategies converges to that of the stabilizer-twirled logical channel. These studies provide stronger arguments for commonplace procedures in the analysis of GKP error correction:(i) using stochastically shifted GKP states in place of coherently damped ones, and(ii) the use of SB decoding.