The promise of quantum speedup in information processing is not yet fulfilled in a useful quantum algorithm due to the susceptibility of quantum information to decoherence. This makes quantum error correction (QEC) a vital area of research. The vast majority of QEC protocols, however, come with an overwhelming hardware and software overhead. For superconducting quantum circuits, it is possible to minimize this overhead by hardware-efficient encoding in infinite dimensional Hilbert spaces of high-Q harmonic oscillators, and error mitigation using autonomous feedback. Autonomous quantum error correction (AQEC) is particularly challenging, since it requires specific nonlinear interactions between various modes of a quantum system. In this thesis, we explore the Hamiltonian engineering techniques, geared towards realizing the required interactions for a promising class of hardware efficient QEC codes, namely Schrödinger cat codes.
A four-component Schrödinger cat code, which allows for first-order protection against all error channels, requires a highly nonlinear four-photon driven-dissipative process for autonomous stabilization of the decoherence free manifold. We propose a scheme for engineering such a process through a Raman-assisted cascading of readily available four-wave mixing interactions, and experimentally demonstrate the feasibility of this cascading. Furthermore, an improved four-wave mixing device that cancels unwanted always-on interactions is studied. We also propose an implementation of a new error-correction code, the pair-cat code, which offers autonomous protection against all the error channels using low-order parametric interactions.