Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B./Physical review. B

Statistical aspects of a quantum Hopfield model

Shuohang Wu, Zi Cai

Published May 24, 2023• 0 citations
Publisher

Abstract

In this paper, we study the eigenstate properties of a quantum Hopfield model by the exact diagonalization method. The local permutational symmetry in this model organizes the spins into clusters, which can each be considered a large quantum spin interacting with others. It is shown that such a quantum Hopfield model, even though without dissipation, is interesting in its own right as an example of quantum frustrated magnetism and quantum spin glass. It exhibits three distinct phases: a low-energy spin-glass phase at a low transverse field, a thermal paramagnetic phase at a high transverse field, and a nonthermal high-energy paramagnetic phase. The dynamics of the revival probability starting from a memory pattern in such a closed quantum many-body model has also been studied.

DOI
10.1103/physrevb.107.184310

Related papers

Nonergodic delocalized paramagnetic states in quantum neural networks

Shuohang Wu, Zi Cai

Typically, it is assumed that a high-energy eigenstate of a generic interacting quantum many-body Hamiltonian is thermal and obeys the eigenstate thermalization hypothesis. In this work, we show that the paramagnetic phase of a quantum Hopfield neural network model is delocalized but nonergodic. The combination of permutational symmetry and frustration in this model organize its high-energy eigenstates into clusters, which can each be considered a large quantum spin and has no correlation with others. This model provides another ergodicity-breaking mechanism in quantum many-body systems.

arXiv:2205.00623•May 2, 2022•0 citations

Quantum Hopfield model with dilute memories

Rongfeng Xie, Alex Kamenev

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.

arXiv:2405.13240•Physical Review A•Sep 13, 2024•2 citations

Clustering of Nonergodic Eigenstates in Quantum Spin Glasses

Christopher L. Baldwin, Chris R. Laumann, Arijeet Pal +1 more

The two primary categories for eigenstate phases of matter at a finite temperature are many-body localization (MBL) and the eigenstate thermalization hypothesis (ETH). We show that, in the paradigmatic quantum p-spin models of the spin-glass theory, eigenstates violate the ETH yet are not MBL either. A mobility edge, which we locate using the forward-scattering approximation and replica techniques, separates the nonergodic phase at a small transverse field from an ergodic phase at a large transverse field. The nonergodic phase is also bounded from above in temperature, by a transition in configuration-space statistics reminiscent of the clustering transition in the spin-glass theory. We show that the nonergodic eigenstates are organized in clusters which exhibit distinct magnetization patterns, as characterized by an eigenstate variant of the Edwards-Anderson order parameter.

arXiv:1611.02296•Physical Review Letters•Mar 23, 2017•55 citations

Hopfield spin-glass model in a transverse field

Yu‐qiang Ma, Chang-De Gong

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.

Physical review. B, Condensed matter•Nov 1, 1993•12 citations

Statistical mechanics of a Hopfield neural-network model in a transverse field

Yu‐qiang Ma, Yueming Zhang, Y. G. +1 more

We construct a soluble model to describe the retrieval properties of the quantum transverse Hopfield model in neural networks. A set of equations for the order parameters describing retrieval and spin-glass phases of the Hopfield model in a transverse field are obtained. Based on these equations, phase diagrams are examined and the memory storage capacity of the networks is analyzed as a function of the temperature and transverse field.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 1, 1993•10 citations

Solvable model of the quantum spin glass in a transverse field

Yadin Y. Goldschmidt

The spin-glass model with p-spin interactions in the presence of a transverse field is solved in the limit p\ensuremath{\rightarrow}\ensuremath{\infty}. The phase diagram is obtained and consists of three phases: A spin-glass phase and two paramagnetic phases. The paramagnetic phases are distinguished by transverse ordering. The spin-glass phase is similar to that of the random-energy model.

Physical review. B, Condensed matter•Mar 1, 1990•110 citations
In this paper, we study the eigenstate properties of a quantum Hopfield model by the exact diagonalization method. The local permutational symmetry in this model organizes the spins into clusters, which can each be considered a large quantum spin interacting with others. It is shown that such a quantum Hopfield model, even though without dissipation, is interesting in its own right as an example of quantum frustrated magnetism and quantum spin glass. It exhibits three distinct phases: a low-energy spin-glass phase at a low transverse field, a thermal paramagnetic phase at a high transverse field, and a nonthermal high-energy paramagnetic phase. The dynamics of the revival probability starting from a memory pattern in such a closed quantum many-body model has also been studied.
Typically, it is assumed that a high-energy eigenstate of a generic interacting quantum many-body Hamiltonian is thermal and obeys the eigenstate thermalization hypothesis. In this work, we show that the paramagnetic phase of a quantum Hopfield neural network model is delocalized but nonergodic. The combination of permutational symmetry and frustration in this model organize its high-energy eigenstates into clusters, which can each be considered a large quantum spin and has no correlation with others. This model provides another ergodicity-breaking mechanism in quantum many-body systems.
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.
The two primary categories for eigenstate phases of matter at a finite temperature are many-body localization (MBL) and the eigenstate thermalization hypothesis (ETH). We show that, in the paradigmatic quantum p-spin models of the spin-glass theory, eigenstates violate the ETH yet are not MBL either. A mobility edge, which we locate using the forward-scattering approximation and replica techniques, separates the nonergodic phase at a small transverse field from an ergodic phase at a large transverse field. The nonergodic phase is also bounded from above in temperature, by a transition in configuration-space statistics reminiscent of the clustering transition in the spin-glass theory. We show that the nonergodic eigenstates are organized in clusters which exhibit distinct magnetization patterns, as characterized by an eigenstate variant of the Edwards-Anderson order parameter.
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.