Second, a lower average photon number per mode is required to enjoy a level of protection at least as good as that of the cat-codes. First, monitoring the photon number difference can be done without turning off the currently implementable dissipative stabilizing process. Despite employing more resources, the two-mode scheme enjoys two advantages over its one-mode counterpart with regards to implementation using current circuit QED technology. Additionally, it is possible to protect information from arbitrary photon loss in either (but not simultaneously both) of the modes by continuously monitoring the difference between the expected photon numbers of the logical states. We show how quantum information encoded in a steady-state subspace of this system is exponentially immune to phase drifts (cavity dephasing) in both modes. We introduce a driven-dissipative two-mode bosonic system whose reservoir causes simultaneous loss of two photons in each mode and whose steady states are superpositions of pair-coherent/Barut-Girardello coherent states. I conclude the thesis with discussions on the importance of non-Gaussian resources for continuous-variable quantum information processing. Furthermore, I provide explicit bosonic error correction schemes that nearly achieve the fundamental performance limit set by the quantum capacity. In particular, I present improved bounds on important communication-theoretic quantities such as the quantum capacity of bosonic Gaussian channels. Moreover, I discuss the fundamental aspects of bosonic QEC using the framework of quantum communication theory. I also demonstrate that fault-tolerant bosonic QEC is possible by concatenating a single-mode bosonic code with a multi-qubit error-correcting code. Specifically, I present the benchmark and optimization results of various single-mode bosonic codes against practically relevant excitation loss errors. In this thesis, I provide an overview of bosonic QEC and present my contributions to the field. Recently, bosonic (or continuous-variable) quantum error correction has risen as a promising hardware-efficient alternative to multi-qubit QEC schemes. The required resource overhead associated with the use of conventional multi-qubit QEC schemes, however, is too high for these schemes to be realized at scale with currently available quantum devices. A conventional approach towards large-scale and fault-tolerant quantum information processing is to use multi-qubit quantum error correction (QEC), that is, to encode a logical quantum bit (or a logical qubit) redundantly over many physical qubits such that the redundancy can be used to detect errors. However, noise in realistic quantum devices fundamentally limits the utility of these quantum technologies. Quantum computation and communication are important branches of quantum information science. These schemes provide a robust protection against dominant error channels in the presence of multi-photon driven dissipation. Last, as a follow up of the above results, we present several continuous and/or autonomous QEC schemes using the cat code. In our design, we exploit the strongly nonlinear Hamiltonian of a highimpedance Josephson circuit, coupling ahigh-Q cavity storage cavity mode to a low-Q readout one. We propose a scheme to perform continuous and quantum non-demolition measurements of photon-number parity in a microwave cavity, which corresponds to the error syndrome in the cat code. Through an engineered interaction between these systems, the entropy created by eventual errors is evacuated via the dissipative modes.The second part of this work focus on the recently developed cat codes, through which the logical information is encoded in the large Hilbert space of a harmonic oscillator. We design an autonomous QEC scheme based on quantum reservoir engineering, in which transmon qubits are coupled to lossy modes. In this thesis, we develop several tools in the direction of autonomous Quantum Error Correction (QEC) with superconducting qubits.
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