# The role of coherence theory in attractor quantum neural networks

Carlo Marconi1, Pau Colomer Saus1, María García Díaz1, and Anna Sanpera1,2

1Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
2ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain

### Abstract

We investigate attractor quantum neural networks (aQNNs) within the framework of coherence theory. We show that: i) aQNNs are associated to non-coherence-generating quantum channels; ii) the depth of the network is given by the decohering power of the corresponding quantum map; and iii) the attractor associated to an arbitrary input state is the one minimizing their relative entropy. Further, we examine faulty aQNNs described by noisy quantum channels, derive their physical implementation and analyze under which conditions their performance can be enhanced by using entanglement or coherence as external resources.

Attractor neural networks are a class of neural networks implemented by a collection of $N$ interacting nodes that evolve towards one of the patterns of minimal energy of the system. The importance of these networks is related to their associative memory, that is, the ability to recover, out of a set of stored patterns, the state which is the closest to a noisy input: the larger the number of the patterns, the greater the storage capacity of the network.
When dealing with their quantum analogue, dubbed as attractor quantum neural network (aQNN), the dynamical process leading to the state of minimal energy is described by a quantum channel, $\Lambda$, whose maximum number of stationary states corresponds to the storage capacity of the aQNN. Interestingly, such quantum channels correspond to a set of well known operations in the context of the resource theory of coherence. Such observation motivates our choice to address aQNNs from a coherence-theoretic approach.
Within this framework, we characterize the properties of aQNNs, such as their physical implementation and the depth of the network, i.e, the number of times the map $\Lambda$ has to be applied to retrieve faithfully the state which is closest to the initial input, as schematically depicted in the plot.
In this case we prove that genuinely incoherent operations (GIOs) describe the evolution of aQNNs with maximal storage capacity. Moreover, we show that neither coherence nor entanglement can be exploited as resources to enhance their performance.
Finally, we address the above issues also in the context of faulty aQNNs, that is when some error in the realization of the network is taken into account. In this case, we demonstrate that the corresponding aQNNs are described either by strictly incoherent operations (SIOs) or by maximally incoherent operations (MIOs), thus opening the possibility, in the latter case, to an enhancement of their performance by using coherence as an external resource.

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