Lukas Brenner

To quantify the accuracy of logical gates in approximate quantum error correction, we introduce the {\em composable logical gate error}. This quantity accounts for both deviation from the target gate and leakage out of the code space. It is subadditive under gate composition, enabling simple circuit analysis, and can be bounded using matrix elements of physical unitaries between (approximate) logical basis states. As a case study, we study the composable logical gate error of linear optics implementations of Paulis and Cliffords in approximate Gottesman-Kitaev-Preskill (GKP) codes. We find that the logical gate error for implementations of Pauli gates depends linearly on the squeezing parameter. This means that their accuracy increases monotonically with the amount of squeezing. In contrast, implementations of some Clifford gates retain a constant logical gate error even in the limit of infinite squeezing. This highlights that results derived for ideal GKP codes do not always translate to physically realistic approximate codes. We propose a way of sidestepping this no-go result in hybrid qubit-oscillator systems with Gaussian, multi-qubit, and qubit-controlled Gaussian unitaries. We propose implementations of logical gates using two oscillators and three qubits, whose logical gate error is bounded by a linear function of the squeezing parameter and scales polynomially with the number of encoded qubits.