This thesis lies at the intersection of three burgeoning fields: quantum computation, neural networks and bosonic systems. These disciplines have recently attracted significant attention due to their potential to revolutionize information processing. Quantum computation promises to solve problems that are intractable for classical computers, while neural networks have achieved remarkable success in a variety of applications, ranging from image recognition to natural language processing. Finally, bosonic systems offer unique advantages for quantum information processing thanks to their versatility and resilience against certain errors.
Motivated by the challenges of preserving quantum information in the Noisy Intermediate-Scale Quantum (NISQ) era and by the potential of continuous-variable platforms, this thesis explores noise-biased bosonic encodings that exploit high-order nonlinearities. Two distinct memory architectures are studied in detail, distinguished by the nature of the nonlinearities: a \emph{squeezed cat} code stabilized by competing driving and dissipation processes of different orders, and a \emph{chiral cat} code that uses intrinsic Hamiltonian nonlinearities of superconducting circuits. The former, owing to the squeezed nature of the code states, provides additional protection against bit-flip errors without compromising the decoherence time. The latter, through competing high-order Hamiltonian nonlinearities, generates two quasi-orthogonal manifolds endowed in a chiral flow in phase space, which allow to construct an error-correcting mechanism for bit-flip errors. Both schemes demonstrate how engineered nonlinearities can be harnessed as practical tools for realizing fault-tolerant quantum memories in continuous-variable platforms.
In complement to the study of robust quantum memories, this thesis presents a general theoretical framework for quantum associative memories (QAMs) within open quantum systems. This attractor-based neural network algorithm can store and retrieve patterns from partial or noisy inputs. The study addresses the broadest possible scenarios, encompassing both quantum and classical patterns, orthogonal and non-orthogonal memories, stationary and metastable operational regimes, and measurement-based outputs. Several physical mechanisms for implementing QAMs are proposed and analysed, with a focus on the importance of symmetries and dissipation in QAM functioning. We identify strong symmetries that allow for multiple steady states, as well as metastable regimes that enable the rapid retrieval of patterns. This general framework provides the context for an original proposal of a QAM in a bosonic system featuring high-order nonlinearities, which allow patterns to be encoded in presence of strong symmetries or metastable regimes. This specific model exposes all the ingredients developed in the general framework, demonstrate their potential to increase storage capacity beyond that of classical associative memories and introduce the possibility to store genuine quantum states as patterns.