We acknowledge the Spanish State Research Agency, through the María de Maeztu project CEX2021-001164-M funded by the MCIN/AEI/10.13039/501100011033, through the QUARESC project (PID2019-109094GB-C21/AEI/10.13039/501100011033) and through the COQUSY Projects No. PID2022-140506NB-C21 and No. PID2022-140506NB-C22 funded by MCIN/AEI/10.13039/501100011033, Ministry for Digital Transformation and of Civil Service of the Spanish Government through the QUANTUM ENIA project call - Quantum Spain project, and by the European Union through the Recovery, Transformation and Resilience Plan - NextGenerationEU within the framework of the Digital Spain 2026 Agenda. The CSIC Interdisciplinary Thematic Platform (PTI+) on Quantum Technologies in Spain (QTEP+) is also acknowledged. G.L.G. is funded by the Spanish Ministerio de Educación y Formación Profesional/Ministerio de Universidades and cofunded by the University of the Balearic Islands through the Beatriz Galindo program (BG20/00085). A.L.-M. is funded by the University of the Balearic Islands through the project BGRH-UIB-2021.

Quantum oscillators with nonlinear driving and dissipative terms have gained significant attention due to their ability to stabilize cat-states for universal quantum computation. Recently, superconducting circuits have been employed to realize such long-lived qubits stored in coherent states. We present a generalization of these oscillators, which are not limited to coherent states, in the presence of different nonlinearities in driving and dissipation, exploring different degrees. Specifically, we present an extensive analysis of the asymptotic dynamical features and of the storage of squeezed states. We demonstrate that coherent superpositions of squeezed states are achievable in the presence of a strong symmetry, thereby allowing for the storage of squeezed cat-states. In the weak symmetry regime, accounting for linear dissipation, we investigate the potential application of these nonlinear driven-dissipative resonators for quantum computing and quantum associative memory and analyze the impact of squeezing on their performance.

With funding from the Spanish government through the "Severo Ochoa Centre of Excellence" accreditation (CEX2021-001164-M).

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