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Signatures of associative memory behavior in open quantum systems

Eliana Fiorelli

Published Dec 11, 2020• 0 citations
OpenAlex

Abstract

Nowadays, the research field of Machine Learning (ML) is undergoing a rapid expansion. In addition, in the last decades, within the research area of quantum computation, several quantum protocols and algorithms have shown the ability to outperform their classical counterparts. In the light of these twofold advances, a frontier field investigating the potentiality of quantum implementations of ML architectures arose. Among the latter, many efforts are concentrated on Artificial Neural Networks (ANNs), which represent a highly successful ML architecture. This thesis focuses on characterizing and understanding the role of quantum effects on the behavior of an instance of ANN, the so-called Hopfield Neural Network (HNN). The HNN is an example of an associative memory model and it permits the recognition, or retrieval, of patterns, such as letters of an alphabet. Formally, it can be described in terms of a classical spin network with an all-to-all interaction, ruled by a stochastic dynamics. Its relatively intuitive behavior, and its physical description make the HNN being a good candidate for a theoretical investigation of its possible quantum generalizations. In the first part of our work we deal with the implementation of fully connected Ising models, such as the HNN one, in terms of quantum systems. To this end, we refer to the cases where the theory of Open Quantum System (OQS) is exploited for engineering steady states of out-of-equilibrium dynamics. This paradigm allows us to build a dissipative - yet quantum - dynamics which encodes in its stationary state the thermal state of the original (classical) problem. Such a construction gives us the chance to address the question as to whether the quantum description could lead to a speed-up in reaching the shared stationary state with respect to the classical counterpart. We show that a suitable choice of the parameters allows the system to switch between a classical regime and a quantum one, the latter being characterized by an accelerated approach towards the long-time limit state. In the second part of our work we focus on practically relevant implementations of quantum systems that can show analogies with the HNN model. To this end, we consider spin-boson models as described by the Dicke Hamiltonian. Indeed, at strong spin-boson interaction couplings, and when the latter are additionally characterized by disorder, the equilibrium Dicke model is able to reproduce the retrieval mechanism typical of HNNs. When dissipation is included, the theoretical description becomes more challenging. We analyze the open, disordered Dicke model by employing some perturbative techniques in order to deal with the out-of-equilibrium dynamics at strong coupling. As a result, we are able to highlight similarities and differences between the quantum model and HNNs.

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Signatures of Associative Memory Behavior in a Multimode Dicke Model

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Dicke-like models can describe a variety of physical systems, such as atoms in a cavity or vibrating ion chains. In equilibrium these systems often feature a radical change in their behavior when switching from weak to strong spin-boson interaction. This usually manifests in a transition from a "dark" to a "superradiant" phase. However, understanding the out-of-equilibrium physics of these models is extremely challenging, and even more so for strong spin-boson coupling. Here we show that the nonequilibrium strongly interacting multimode Dicke model can mimic some fundamental properties of an associative memory-a system which permits the recognition of patterns, such as letters of an alphabet. Patterns are encoded in the couplings between spins and bosons, and we discuss the dynamics of the spins from the perspective of pattern retrieval in associative memory models. We identify two phases, a "paramagnetic" and a "ferromagnetic" one, and a crossover behavior between these regimes. The "ferromagnetic" phase is reminiscent of pattern retrieval. We highlight similarities and differences with the thermal dynamics of a Hopfield associative memory and show that indeed elements of "machine learning behavior" emerge in the strongly coupled multimode Dicke model.

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Theoretical investigations of open quantum systems have uncovered a wide variety of unique dynamical phenomena. For instance, current research in quantum information technology aims to control and utilize time-delayed system-reservoir interactions and phonon-induced dynamical dissipative processes for the design of novel chip-integrated quantum optical devices. This thesis develops and advances multiple theoretical approaches for the accurate and efficient description of open quantum systems, where a high degree of information compression is essential to extract crucial information from the open system degrees of freedom which scale exponentially with the system size. The presented methods range from correlation expansions to second-order perturbative master equations, real-time path integral formulations, numerical realizations of high-dimensional tensor networks and, based on the most recent theoretical developments, machine learning implementations in the form of artificial neural networks. We explore all of these techniques with respect to their compression efficiency, performance and representational limits by applying them to a variety of physical setups and scenarios. Firstly, emerging non-Markovian memory effects in open quantum systems are investigated, e.g., resulting from interactions with a structured phonon environment or time-discrete coherent quantum feedback, leading to information exchange and a time-delayed dynamical interplay between system and reservoir. Examined systems range from solid state p-wave topological superconductors coupled to a structured phonon environment, where a memory-induced stabilization and recovery of topological properties is observed, to semiconductor quantum dot nanostructures exhibiting complete population inversion and unidirectional quantum transport as a result of dissipation-induced non-Markovian system-reservoir interactions, and quantum emitters simultaneously subjected to coherent quantum feedback and a decoherence-inducing structured phonon reservoir, leading to the formation of dynamical dissipative structures and population trapping. Secondly, novel machine learning techniques based on artificial neural networks are investigated, enabling simulations of symmetric open quantum spin-1/2 systems with Markovian dynamics. Combining their potential for parallelization and efficient Hilbert space truncation, they facilitate unmatched degrees of information compression and numerical performance for large systems. In this thesis, we expand the representational limits of the restricted Boltzmann machine neural network architecture by developing novel hybrid sampling strategies for a highly customizable compression of configuration space, rendering accurate and efficient simulations of asymmetric systems feasible. Finally, the applicability of the neural network is extended beyond pure spin-1/2 systems by introducing a neural bit encoding scheme for Fock number states, facilitating high performing calculations of large hybrid open quantum systems with bosonic degrees of freedom.

•Jan 1, 2021•0 citations
Nowadays, the research field of Machine Learning (ML) is undergoing a rapid expansion. In addition, in the last decades, within the research area of quantum computation, several quantum protocols and algorithms have shown the ability to outperform their classical counterparts. In the light of these twofold advances, a frontier field investigating the potentiality of quantum implementations of ML architectures arose. Among the latter, many efforts are concentrated on Artificial Neural Networks (ANNs), which represent a highly successful ML architecture. This thesis focuses on characterizing and understanding the role of quantum effects on the behavior of an instance of ANN, the so-called Hopfield Neural Network (HNN). The HNN is an example of an associative memory model and it permits the recognition, or retrieval, of patterns, such as letters of an alphabet. Formally, it can be described in terms of a classical spin network with an all-to-all interaction, ruled by a stochastic dynamics. Its relatively intuitive behavior, and its physical description make the HNN being a good candidate for a theoretical investigation of its possible quantum generalizations. In the first part of our work we deal with the implementation of fully connected Ising models, such as the HNN one, in terms of quantum systems. To this end, we refer to the cases where the theory of Open Quantum System (OQS) is exploited for engineering steady states of out-of-equilibrium dynamics. This paradigm allows us to build a dissipative - yet quantum - dynamics which encodes in its stationary state the thermal state of the original (classical) problem. Such a construction gives us the chance to address the question as to whether the quantum description could lead to a speed-up in reaching the shared stationary state with respect to the classical counterpart. We show that a suitable choice of the parameters allows the system to switch between a classical regime and a quantum one, the latter being characterized by an accelerated approach towards the long-time limit state. In the second part of our work we focus on practically relevant implementations of quantum systems that can show analogies with the HNN model. To this end, we consider spin-boson models as described by the Dicke Hamiltonian. Indeed, at strong spin-boson interaction couplings, and when the latter are additionally characterized by disorder, the equilibrium Dicke model is able to reproduce the retrieval mechanism typical of HNNs. When dissipation is included, the theoretical description becomes more challenging. We analyze the open, disordered Dicke model by employing some perturbative techniques in order to deal with the out-of-equilibrium dynamics at strong coupling. As a result, we are able to highlight similarities and differences between the quantum model and HNNs.
Abstract We introduce and analyze an open quantum generalization of the q-state Potts-Hopfield neural network (NN), which is an associative memory model based on multi-level classical spins. The dynamics of this many-body system is formulated in terms of a Markovian master equation of Lindblad type, which allows to incorporate both probabilistic classical and coherent quantum processes on an equal footing. By employing a mean field description we investigate how classical fluctuations due to temperature and quantum fluctuations effectuated by coherent spin rotations affect the ability of the network to retrieve stored memory patterns. We construct the corresponding phase diagram, which in the low temperature regime displays pattern retrieval in analogy to the classical Potts-Hopfield NN. When increasing quantum fluctuations, however, a limit cycle phase emerges, which has no classical counterpart. This shows that quantum effects can qualitatively alter the structure of the stationary state manifold with respect to the classical model, and potentially allow one to encode and retrieve novel types of patterns.
Dicke-like models can describe a variety of physical systems, such as atoms in a cavity or vibrating ion chains. In equilibrium these systems often feature a radical change in their behavior when switching from weak to strong spin-boson interaction. This usually manifests in a transition from a "dark" to a "superradiant" phase. However, understanding the out-of-equilibrium physics of these models is extremely challenging, and even more so for strong spin-boson coupling. Here we show that the nonequilibrium strongly interacting multimode Dicke model can mimic some fundamental properties of an associative memory-a system which permits the recognition of patterns, such as letters of an alphabet. Patterns are encoded in the couplings between spins and bosons, and we discuss the dynamics of the spins from the perspective of pattern retrieval in associative memory models. We identify two phases, a "paramagnetic" and a "ferromagnetic" one, and a crossover behavior between these regimes. The "ferromagnetic" phase is reminiscent of pattern retrieval. We highlight similarities and differences with the thermal dynamics of a Hopfield associative memory and show that indeed elements of "machine learning behavior" emerge in the strongly coupled multimode Dicke model.
Quantum neural networks form one pillar of the emergent field of quantum machine learning. Here quantum generalizations of classical networks realizing associative memories---capable of retrieving patterns, or memories, from corrupted initial states---have been proposed. It is a challenging open problem to analyze quantum associative memories with an extensive number of patterns and to determine the maximal number of patterns the quantum networks can reliably store, i.e., their storage capacity. In this work, we propose and explore a general method for evaluating the maximal storage capacity of quantum neural network models. By generalizing what is known as Gardner's approach in the classical realm, we exploit the theory of classical spin glasses for deriving the optimal storage capacity of quantum networks with quenched pattern variables. As an example, we apply our method to an open-system quantum associative memory formed of interacting spin-1/2 particles realizing coupled artificial neurons. The system undergoes a Markovian time evolution resulting from a dissipative retrieval dynamics that competes with a coherent quantum dynamics. We map out the nonequilibrium phase diagram and study the effect of temperature and Hamiltonian dynamics on the storage capacity. Our method opens an avenue for a systematic characterization of the storage capacity of quantum associative memories.
[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.
Abstract We propose a new framework to understand how quantum effects may impact on the dynamics of neural networks. We implement the dynamics of neural networks in terms of Markovian open quantum systems, which allows us to treat thermal and quantum coherent effects on the same footing. In particular, we propose an open quantum generalisation of the Hopfield neural network, the simplest toy model of associative memory. We determine its phase diagram and show that quantum fluctuations give rise to a qualitatively new non-equilibrium phase. This novel phase is characterised by limit cycles corresponding to high-dimensional stationary manifolds that may be regarded as a generalisation of storage patterns to the quantum domain.
Theoretical investigations of open quantum systems have uncovered a wide variety of unique dynamical phenomena. For instance, current research in quantum information technology aims to control and utilize time-delayed system-reservoir interactions and phonon-induced dynamical dissipative processes for the design of novel chip-integrated quantum optical devices. This thesis develops and advances multiple theoretical approaches for the accurate and efficient description of open quantum systems, where a high degree of information compression is essential to extract crucial information from the open system degrees of freedom which scale exponentially with the system size. The presented methods range from correlation expansions to second-order perturbative master equations, real-time path integral formulations, numerical realizations of high-dimensional tensor networks and, based on the most recent theoretical developments, machine learning implementations in the form of artificial neural networks. We explore all of these techniques with respect to their compression efficiency, performance and representational limits by applying them to a variety of physical setups and scenarios. Firstly, emerging non-Markovian memory effects in open quantum systems are investigated, e.g., resulting from interactions with a structured phonon environment or time-discrete coherent quantum feedback, leading to information exchange and a time-delayed dynamical interplay between system and reservoir. Examined systems range from solid state p-wave topological superconductors coupled to a structured phonon environment, where a memory-induced stabilization and recovery of topological properties is observed, to semiconductor quantum dot nanostructures exhibiting complete population inversion and unidirectional quantum transport as a result of dissipation-induced non-Markovian system-reservoir interactions, and quantum emitters simultaneously subjected to coherent quantum feedback and a decoherence-inducing structured phonon reservoir, leading to the formation of dynamical dissipative structures and population trapping. Secondly, novel machine learning techniques based on artificial neural networks are investigated, enabling simulations of symmetric open quantum spin-1/2 systems with Markovian dynamics. Combining their potential for parallelization and efficient Hilbert space truncation, they facilitate unmatched degrees of information compression and numerical performance for large systems. In this thesis, we expand the representational limits of the restricted Boltzmann machine neural network architecture by developing novel hybrid sampling strategies for a highly customizable compression of configuration space, rendering accurate and efficient simulations of asymmetric systems feasible. Finally, the applicability of the neural network is extended beyond pure spin-1/2 systems by introducing a neural bit encoding scheme for Fock number states, facilitating high performing calculations of large hybrid open quantum systems with bosonic degrees of freedom.