Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits

We propose an autonomous quantum error correction scheme using squeezed cat (SC) code against the dominant error source, excitation loss, in continuous-variable systems. Through reservoir engineering, we show that a structured dissipation can stabilize a two-component SC while autonomously correcting the errors. The implementation of such dissipation only requires low-order nonlinear couplings among three bosonic modes or between a bosonic mode and a qutrit. While our proposed scheme is device independent, it is readily implementable with current experimental platforms such as superconducting circuits and trapped-ion systems. Compared to the stabilized cat, the stabilized SC has a much lower dominant error rate and a significantly enhanced noise bias. Furthermore, the bias-preserving operations for the SC have much lower error rates. In combination, the stabilized SC leads to substantially better logical performance when concatenating with an outer discrete-variable code. The surface-SC scheme achieves more than one order of magnitude increase in the threshold ratio between the loss rate $\kappa_1$ and the engineered dissipation rate $\kappa_2$. Under a practical noise ratio $\kappa_1/\kappa_2 = 10^{-3}$, the repetition-SC scheme can reach a $10^{-15}$ logical error rate even with a small mean excitation number of 4, which already suffices for practically useful quantum algorithms.