Composable logical gate error in approximate quantum error correction: reexamining gate implementations in Gottesman-Kitaev-Preskill codes

Quantifying the accuracy of logical gates is paramount in approximate error correction, where perfect implementations are often unachievable with the available set of physical operations. To this end, we introduce a single scalar quantity we call the (composable) logical gate error. It captures both the deviation of the logical action from the desired target gate as well as leakage out of the code space. It is subadditive under successive application of gates, providing a simple means for analyzing circuits. We show how to bound the composable logical gate error in terms of matrix elements of physical unitaries between (approximate) logical computational basis states. In the continuous-variable context, this sidesteps the need for computing energy-bounded norms. As an example, we study the composable logical gate error for linear optics implementations of Paulis and Cliffords in approximate Gottesman-Kitaev-Preskill (GKP) codes. We find that the logical gate error for implementations of Paulis depends linearly on the squeezing parameter. This implies that their accuracy improves monotonically with the amount of squeezing. For some Cliffords, however, linear optics implementations which are exact for ideal GKP codes fail in the approximate case: they have a constant logical gate error even in the limit of infinite squeezing. This is consistent with previous results about the limitations of certain gate implementations for approximate GKP codes. It shows that findings applicable to ideal GKP codes do not always translate to the realm of physically realizable approximate GKP codes.