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.

We propose new schemes for quantum computation with hybrid qubit-oscillator systems consisting of a certain number of bosonic modes coupled to a constant number of qubits by a Jaynes-Cummings Hamiltonian. We ask how much energy is required to weakly simulate an~$n$-qubit quantum circuit (i.e., produce samples from its output distribution) by a unitary circuit in this model. We find that efficient approximate weak simulation of an~$n$-qubit quantum circuit of polynomial size with inverse polynomial error is possible with (I) a constant number of modes and an exponential amount of energy, or (II) a sublinear (polynomial) number of modes and a subexponential amount of energy, or (III) a linear number of modes and a polynomial amount of energy. Our construction encodes qubits into high-dimensional approximate Gottesman-Kitaev-Preskill (GKP) codes. It provides new insight into the trade-off between system size (i.e., number of modes) and the amount of energy required to perform quantum computation in the continuous-variable setting.