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.

Using an approach inspired from Spin Glasses, we show that the multimode disordered Dicke model is equivalent to a quantum Hopfield network. We propose variational ground states for the system at zero temperature, which we conjecture to be exact in the thermodynamic limit. These ground states contain the information on the disordered qubit-photon couplings. These results lead to two intriguing physical implications. First, once the qubit-photon couplings can be engineered, it should be possible to build scalable pattern-storing systems whose dynamics is governed by quantum laws. Second, we argue with an example how such Dicke quantum simulators might be used as a solver of "hard" combinatorial optimization problems. PACS numbers:The connection of experimentally realizable quantum systems with computation contains promising perspectives from both the fundamental and the technological viewpoint [1,2]. For example, quantum computational capabilities can be implemented by "quantum gates" [3] and by the so-called "adiabatic quantum optimization" technique [4-6]. Today's experimental technology of highly controllable quantum simulators, recently used for testing theoretical predictions in a wide range of areas of physics [7][8][9], offers new opportunities for exploring computing power for quantum systems.In the case of light-matter interaction at the quantum level, the reference benchmark is the Dicke model [10]. Studies of its equilibrium properties have predicted a superradiant transition to occur in the strong coupling and low temperature regime [11][12][13]. The superradiant phase is characterized by a macroscopic number of atoms in the excited state whose collective behaviour produces an enhancement of spontaneous emission (proportional to the number of cooperating atoms in the sample). Crucially, this phenomenology is in direct link with experimentally feasible quantum simulators. Recently, Nagy and coworkers [14] argued that the Dicke model effectively describes the self-organization phase transition of a BoseEinstein condensate (BEC) in an optical cavity [15,16]. Additionally, Dimer and colleagues [17] proposed a Cavity QED realization of the Dicke model based on cavitymediated Raman transitions, closer in spirit to the original Dicke's idea. Evidence of superradiance in this system is reported in [18]. An implementation of generalized Dicke models in hybrid quantum systems has also been put forward [19]. More generally, Dicke-like Hamiltonians describe a variety of physical systems, ranging from Circuit QED [20][21][22][23][24] to Cavity QED with Dirac fermions in graphene [25][26][27]. Additionally, disorder and frustration of the atom-photon couplings have an important role FIG. 1: In the Dicke model, photons (yellow lines) mediate a long range interaction between qubits (green circles). The drawing sketches schematically a six qubits system within its fully-connected graph and its internal level structure. In the standard single-mode Dicke model the exchange coupling is fixed at the same value for every p...

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