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arXiv:2309.06300 · Physical Review A

Quantum memories for squeezed and coherent superpositions in a driven-dissipative nonlinear oscillator

Adrià Labay-Mora, Roberta Zambrini, Gian Luca Giorgi

Published Mar 8, 2024• 9 citations
arXivHTMLPDF

Abstract

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. The key ingredient lies in the presence of different nonlinearities in driving and dissipation, beyond the quadratic one. Through an extensive analysis of the asymptotic dynamical features for different nonlinearities, we identify the conditions for the storage and retrieval of quantum states, such as squeezed states, in both coherent and incoherent superpositions. We explore their applications in quantum computing, where squeezing prolongs the lifetime of memory storage for qubits encoded in the superposition of two symmetric squeezed states, and in quantum associative memory, which has so far been limited to the storage of classical patterns.

DOI
10.1103/physreva.109.032407

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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. The key ingredient lies in the presence of different nonlinearities in driving and dissipation, beyond the quadratic one. Through an extensive analysis of the asymptotic dynamical features for different nonlinearities, we identify the conditions for the storage and retrieval of quantum states, such as squeezed states, in both coherent and incoherent superpositions. We explore their applications in quantum computing, where squeezing prolongs the lifetime of memory storage for qubits encoded in the superposition of two symmetric squeezed states, and in quantum associative memory, which has so far been limited to the storage of classical patterns.
We propose a scheme to dissipatively stabilize superpositions of squeezed coherent states for single- and two-resonator modes in superconducting circuits, where a superconducting qubit is coupled to a single or two parametrically driven transmission-line resonators with different circuit designs. A modulated magnetic flux applied to the qubit can periodically break the inversion symmetry of the qubit and induce both the parametrically enhanced transverse and longitudinal couplings to the resonators, resulting in strong nonlinear two-photon interactions between the qubit and the resonators and enabling the scheme to be implemented in a relatively weak coupling regime. With an additional microwave drive applied to the qubit, the dissipation of the qubit used as a resource can help drive the resonators into the desired superpositions of squeezed coherent states at a steady state with high speed. Numerical simulations show that the target states with high fidelity and highly nonlinear properties can be created for a long time even in the presence of photon leakage out of the resonators. The scheme can be generalized to other quantum platforms and may have potential applications in the field of quantum information processing and the improvement of estimation precision in quantum metrology.
Parametrically driven Kerr nonlinear quantum oscillator plays a central role in quantum optics, superconducting circuits, and in engineered light-matter interactions. For the development of future quantum technologies, it is crucial to understand the behavior of this quantum system under realistic dissipative conditions. The objective of this thesis is to investigate the phase-space dynamics of a dissipative Kerr nonlinear oscillator under different driving regimes and corresponding dissipation channels, with a particular focus on the emergence of nonclassical features in phase space. The aim is also to understand how nonlinearity and different driving mechanisms influence quantum properties such as coherence, nonclassicality, and phase-space structure. The analysis was carried out using the Wigner phase-space formalism and numerical simulations of the Lindblad master equation. The results show that in the case of finite nonlinearity the coherence time is prolonged compared to the harmonic case. In the single photon regime, the interplay between the photon drive and nonlinearity can lead to coherence revivals after which the system relaxes into a squeezed steady state. In the two photon and dissipation regime, the system exhibits parity-protected dynamics, leading to very long coherence times.
Algorithms for associative memory typically rely on a network of many connected units. The prototypical example is the Hopfield model, whose generalizations to the quantum realm are mainly based on open quantum Ising models. We propose a realization of associative memory with a single driven-dissipative quantum oscillator exploiting its infinite degrees of freedom in phase space. The model can improve the storage capacity of discrete neuron-based systems in a large regime and we prove successful state discrimination between n coherent states, which represent the stored patterns of the system. These can be tuned continuously by modifying the driving strength, constituting a modified learning rule. We show that the associative-memory capability is inherently related to the existence of a spectral separation in the Liouvillian superoperator, which results in a long timescale separation in the dynamics corresponding to a metastable phase.
Out-of-equilibrium driven nonlinear oscillators have attracted interest as a platform for tailoring Floquet Hamiltonian spectra and stabilizing quantum states for quantum information. However, it is usually ignored that the necessary parametric drive can amplify environmental fluctuations limiting applications. In this work, we capture this effect by computing the effective Lindbladian of a Kerr parametric oscillator beyond the rotating wave approximation. We find that the relevant dissipators depend strongly on the nonlinearity and the mean-photon number in the oscillator. In particular, we address recent measurements of the logical lifetime on a Schr\"odinger's-cat qubit stabilized by a parametric drive, thereby providing an explanation for the observed three-order-of-magnitude discrepancy between experimental data and an ordinary Lindblad treatment. Our systematic approach paves the path to treat dissipative effects in Floquet-engineered nonlinear quantum systems.
[eng] 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 squeezed cat code stabilized by competing driving and dissipation processes of different orders, and a 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.
Dissipation engineering offers a powerful tool for quantum technologies. Recently, new superconducting devices have achieved an engineered two-photon dissipation rate exceeding all other relevant timescales. In particular, they have proven most useful in preventing transitions between the logical states $|\ifmmode\pm\else\textpm\fi{}\ensuremath{\alpha}⟩$ of a cat qubit. Here, we present three key applications of strong two-photon dissipation for quantum measurement and control, beyond cat qubit stabilization. Firstly, we demonstrate its efficacy in overcoming limitations encountered in Wigner tomography at high photon numbers. Secondly, we showcase its potential for realizing universal gates on cat qubits, exploiting the coherent mapping between cat qubit states and superpositions of 0 and 1 photons. Finally, we harness the transient dynamics of a cat state under two-photon dissipation to prepare squeezed cat states with a squeezing factor exceeding $3.96\ifmmode\pm\else\textpm\fi{}0.07$ dB.