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arXiv:2405.13240 · Physical Review A

Quantum Hopfield model with dilute memories

Rongfeng Xie, Alex Kamenev

Published Sep 13, 2024• 2 citations
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Abstract

We discuss adiabatic spectra and dynamics of the quantum, i.e., transverse field, Hopfield model with dilute memories (the number of stored patterns, $p<{log}_{2}N$, where $N$ is the number of qubits). At some critical transverse field, the model undergoes the quantum phase transition from the ordered to the paramagnetic state. The corresponding critical exponents are calculated and used to determine the efficiency of quantum annealing protocols. We also discuss implications of these results for the quantum annealing of generic spin glass models.

DOI
10.1103/physreva.110.032418

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We discuss adiabatic spectra and dynamics of the quantum, i.e., transverse field, Hopfield model with dilute memories (the number of stored patterns, p<log2Np<{log}_{2}Np<log2​N, where NNN is the number of qubits). At some critical transverse field, the model undergoes the quantum phase transition from the ordered to the paramagnetic state. The corresponding critical exponents are calculated and used to determine the efficiency of quantum annealing protocols. We also discuss implications of these results for the quantum annealing of generic spin glass models.
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A quantum Hopfield spin-glass model with transverse fields is presented. A set of equations for the order parameters of a large class of spin-glass models with transverse fields is obtained, and the phase transitions are analyzed with a finite number of patterns p. We find that a reentrant phenomenon is possible for a quantum Mattis model with transverse fields, contrary to the quantum transverse Sherrington-Kirkpatrick-like model. The effects of quantum fluctuations (induced by the transverse field) on the ordered phases of systems with competing interactions are also discussed.
A promising approach to solving hard binary optimisation problems is quantum adiabatic annealing (QA) in a transverse magnetic field. An instantaneous ground state --- initially a symmetric superposition of all possible assignments of NNN qubits --- is closely tracked as it becomes more and more localised near the global minimum of the classical energy. Regions where the energy gap to excited states is small (e.g. at the phase transition) are the algorithm's bottlenecks. Here I show how for large problems the complexity becomes dominated by O(log⁡N)O(\log N)O(logN) bottlenecks inside the spin glass phase, where the gap scales as a stretched exponential. For smaller NNN, only the gap at the critical point is relevant, where it scales polynomially, as long as the phase transition is second order. This phenomenon is demonstrated rigorously for the two-pattern Gaussian Hopfield Model. Qualitative comparison with the Sherrington-Kirkpatrick Model leads to similar conclusions.
We introduce the transverse Ising model as a prototype for discussing quantum phase transition. Next we introduce Suzuki-Trotter formalism to show the correspondence between ddd-dimensional quantum system with a (d+1)(d+1)(d+1)-dimensional classical system. We then discuss transverse Ising spin glass models, namely S-K model, E-A model, and the ±J\pm J±J model with Ising spin in transverse field. We briefly discuss the mean field, exact diagonalization, and quantum Monte Carlo results for their phase diagrams. Next we discuss the question of replica symmetry restoration in quantum spin glasses (due to tunnelling possibility through the barriers). Then we discuss the quantum annealing technique and indicate its relationship with replica symmetry restoration in quantum spin glasses. We have also breifly discussed the possibility of Quantum Annealing in context of kinetically constrained systems. Mean-field calculation for BCS superconductivity, Real-space RG calculation for one dimensional transverse Ising system, scattering amplitude calculation for tunneling through asymmetric barrier (useful for quantum kinetically constrained system) are given in the appendices.