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Associative memory refers to the ability to relate a memory with an input and targets the restoration of corrupted patterns. It has been intensively studied in classical physical systems, as in neural networks where an attractor dynamics settles on stable solutions. Several extensions to the quantum domain have been recently reported, displaying different features. In this work, we develop a comprehensive framework for a quantum associative memory based on open quantum system dynamics, which allows us to compare existing models, identify the theoretical prerequisites for performing associative memory tasks, and extend it in different forms. The map that achieves an exponential increase in the number of stored patterns with respect to classical systems is derived. We establish the crucial role of symmetries and dissipation in the operation of quantum associative memory. Our theoretical analysis demonstrates the feasibility of addressing both quantum and classical patterns, orthogonal and non-orthogonal memories, stationary and metastable operating regimes, and measurement-based outputs. Finally, this opens up new avenues for practical applications in quantum computing and machine learning, such as quantum error correction or quantum memories.

We consider a class of open quantum many-body systems that evolves in a Markovian fashion, the dynamical generator being in GKS-Lindblad form. Here, the Hamiltonian contribution is characterized by an all-to-all coupling, and the dissipation features local transitions that depend on collective, operator-valued rates, encoding average properties of the system. These types of generators can be formally obtained by generalizing, to the quantum realm, classical (mean-field) stochastic Markov dynamics, with state-dependent transitions. Focusing on the dynamics emerging in the limit of infinitely large systems, we build on the exactness of the mean-field equations for the dynamics of average operators. In this framework, we derive the dynamics of quantum fluctuation operators, that can be used in turn to understand the fate of quantum correlations in the system. We apply our results to quantum generalized Hopfield associative memories, showing that, asymptotically and at the mesoscopic scale only a very weak amount of quantum correlations, in the form of quantum discord, emerges beyond classical correlations., Comment: 30 pages, 2 Appendices, 4 figures. Comments welcome

We consider a class of open quantum many-body Lindblad dynamics characterized by an all-to-all coupling Hamiltonian and by dissipation featuring collective ``state-dependent" rates. The latter encodes local incoherent transitions that depend on average properties of the system. This type of open quantum dynamics can be seen as a generalization of classical (mean-field) stochastic Markov dynamics, in which transitions depend on the instantaneous configuration of the system, to the quantum domain. We study the time evolution in the limit of infinitely large systems, and we demonstrate the exactness of the mean-field equations for the dynamics of average operators. We further derive the effective dynamical generator governing the time evolution of (quasi-)local operators. Our results allow for a rigorous and systematic investigation of the impact of quantum effects on paradigmatic classical models, such as quantum generalized Hopfield associative memories or (mean-field) kinetically-constrained models., Comment: 43 pages,2 figures

Quantum neural networks form one pillar of the emergent field of quantum machine learning. Here, quantum generalisations 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 non-equilibrium 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., Comment: 6 pages, 2 figures + appendix

In this work, we study the stochastic entropy production in open quantum systems whose time evolution is described by a class of non-unital quantum maps. In particular, as in [Phys. Rev. E 92, 032129 (2015)], we consider Kraus operators that can be related to a nonequilibrium potential. This class accounts for both thermalization and equilibration to a non-thermal state. Unlike unital quantum maps, non-unitality is responsible for an unbalance of the forward and backward dynamics of the open quantum system under scrutiny. Here, concentrating on observables that commute with the invariant state of the evolution, we show how the non-equilibrium potential enters the statistics of the stochastic entropy production. In particular, we prove a fluctuation relation for the latter and we find a convenient way of expressing its average solely in terms of relative entropies. Then, the theoretical results are applied to the thermalization of a qubit with non-Markovian transient, and the phenomenon of irreversibility mitigation, introduced in [Phys. Rev. Research 2, 033250 (2020)], is analyzed in this context., Comment: 17 pages, v2 close to published version

We introduce and analyze an open quantum generalization of the q-state Potts-Hopfield neural network, 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 neural network. 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., Comment: 15 pages, 8 figures

Spin-boson 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 non-equilibrium strongly interacting spin-boson systems 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 strongly coupled spin-boson systems., Comment: 8 pages, 3 figures

Spin-boson Hamiltonians are an effective description for numerous quantum many-body systems such as atoms coupled to cavity modes, quantum electrodynamics in circuits and trapped ion systems. While reaching the limit of strong coupling is possible in current experiments, the understanding of the physics in this parameter regime remains a challenge, especially when disorder and dissipation are taken into account. Here we investigate a regime where the many-body spin dynamics can be related to a Ising energy function defined in terms of the spin-boson couplings. While in the coherent weak coupling regime it is known that an effective description in terms of spin Hamiltonian is possible, we show that a similar viewpoint can be adopted in the presence of dissipation and strong couplings. The resulting many-body dynamics features approximately thermal regimes, separated by out-of-equilibrium ones in which detailed balance is broken. Moreover, we show that under appropriately chosen conditions one can even achieve cooling of the spin degrees of freedom. This points towards the possibility of using strongly coupled dissipative spin-boson systems for engineering complex energy landscapes together with an appropriate cooling dynamics.

We consider a quantum device $D$ interacting with a quantum many-body environment $R$ which features a second-order phase transition at $T=0$. Exploiting the description of the critical slowing down undergone by $R$ according to the Kibble-Zurek mechanism, we explore the possibility to freeze the environment in a configuration such that its impact on the device is significantly reduced. Within this framework, we focus upon the magnetic-domain formation typical of the critical behaviour in spin models, and propose a strategy that allows one to protect the entanglement between different components of $D$ from the detrimental effects of the environment., Comment: 12 pages, 7 figures

We analytically express the loss of entanglement between the components of a quantum device due to the generation of quantum correlations with its environment, and show that such loss diminishes when the latter is macroscopic and displays a semi-classical behaviour. We model the problem as a device made of a couple of qubits with a magnetic environment: this choice allows us to implement the above condition of semi-classical macroscopicity in terms of a large-S condition, according to the well known equivalence between classical and $S \rightarrow \infty$ limit. A possible strategy for protecting internal entanglement exploiting the mechanism of domain-formation typical of critical dynamics is also suggested., Comment: 5 pages, 1 figure

We explore the question as to whether quantum effects can yield a speedup of the non-equilibrium evolution of spin systems towards a classical thermal state. In our approach we exploit the fact that the thermal state of a spin system can be mapped onto a node-free quantum state whose coefficients are given by thermal weights. This perspective permits the construction of a dissipative -- yet quantum -- dynamics which encodes in its stationary state the thermal state of the original problem. We show for the case of an all-to-all connected Ising spin model that an appropriate transformation of this dissipative dynamics allows to interpolate between a regime in which the order parameter obeys the classical equations of motion under Glauber dynamics, to a quantum regime with an accelerated approach to stationarity. We show that this effect enables in principle a speedup of pattern retrieval in a Hopfield neural network., Comment: 4 + 9 pages, 3 figures