Associative memory on qutrits by means of quantum annealing
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Associative memory on qutrits by means of quantum annealing Vladimir Zobov1 · Ivan Pichkovskiy1 Received: 7 November 2019 / Accepted: 29 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract When associative memory is implemented on the well-studied Hopfield network, patterns are recorded in the interaction constants between binary neurons. These constants are chosen so that each pattern should have its own minimum energy of the system described by the Ising model. In the quantum version of the Hopfield network, it was proposed to recall such states by the adiabatic change of the Hamiltonian in time. Qubits, quantum elements with two states, for example, spins with S 1/2 were considered as neurons. In this paper, for the first time, we study the function of associative memory using three-level quantum elements—qutrits, represented by spins with S 1. We record patterns with the help of projection operators. This choice is due to the need to operate with a state with a zero spin projection, whose interaction with the magnetic field vanishes. We recall the state corresponding to one of the patterns recorded in the memory, or superposition of such states by means of quantum annealing. To equalize the probabilities of finding the system in different states of superposition, an auxiliary Hamiltonian is proposed, which is turned off at the end of evolution. Simulations were performed on two and three qutrits and an increase in the memory capacity after replacing qubits with qutrits was shown. Keywords Quantum adiabatic algorithm · Quantum annealing · Qubit · Qutrit · Associative memory · Memory capacity
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11128-02 0-02851-x) contains supplementary material, which is available to authorized users.
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Ivan Pichkovskiy [email protected] Vladimir Zobov [email protected]
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Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Akademgorodok 50, bld. 38, Krasnoyarsk, Russia 0123456789().: V,-vol
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V. Zobov, I. Pichkovskiy
1 Introduction At the present stage of development of information technologies, quantum properties of many-body systems open up new possibilities of artificial neural networks [1, 2], including associative memory [3–7]. When implementing associative memory, several patterns are recorded in a neural network [3]. Then, we give the network a hint, which is a distorted version of the desired pattern. The network should be able to return the true instance. An example of associative memory is the Hopfield network, in which each binary neuron is represented as a spin S 1/2 and interacts with all other spins [3]. The interactions between the spins are described by the classical Ising model. In such a classical Hopfield network, the patterns are recorded in the interaction constants between the spins, which are chosen so that each pattern should have its own minimum system energy. The recognition process is carried out by iteratively changing the
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