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Journal of Applied Physics

Evolution of a two-dimensional quantum cellular neural network driven by an external field

Bi Qiao, Harry E. Ruda

Published Mar 1, 1999• 8 citations
Publisher

Abstract

A model of a two-dimensional quantum cellular neural network (QCNN) is presented in this article. The eigenvalues and eigenvectors for the Hamiltonian of a cell (neuron) are obtained, and we confirm that the ground or memory states are approximately two polarization states of 16 possible states in a cell (neuron) only when electron tunneling is relatively weak compared with the Coulomb repulsion. The evolution of the QCNN driven by a local external magnetic field is studied by solving the Liouville equation of the corresponding two-dimensional Ising model. The formula for the evolution of the density operator is given by using a subdynamics approach. We show that the local external magnetic field can drive the system to a global polarization state and induce a dynamical response in the original QCNN. This dynamical response can be interpreted as a computable function and measured by the system output.

DOI
10.1063/1.369060

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A model of a two-dimensional quantum cellular neural network (QCNN) is presented in this article. The eigenvalues and eigenvectors for the Hamiltonian of a cell (neuron) are obtained, and we confirm that the ground or memory states are approximately two polarization states of 16 possible states in a cell (neuron) only when electron tunneling is relatively weak compared with the Coulomb repulsion. The evolution of the QCNN driven by a local external magnetic field is studied by solving the Liouville equation of the corresponding two-dimensional Ising model. The formula for the evolution of the density operator is given by using a subdynamics approach. We show that the local external magnetic field can drive the system to a global polarization state and induce a dynamical response in the original QCNN. This dynamical response can be interpreted as a computable function and measured by the system output.
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Capturing the dynamics of quantum many-body systems under time-dependent driving protocols is a central challenge for numerical simulations. Existing methods such as tensor networks and time-dependent neural quantum states, however, must be re-run for every protocol. In this work, we introduce the Neural Operator Quantum State (NOQS) as a foundation model for quantum dynamics. Rather than solving the Schrödinger equation for individual trajectories, our approach aims to \emph{learn the solution operator} that maps entire driving protocols to time-evolved quantum states. Once trained, the NOQS predicts time evolution under unseen protocols in a single forward pass, requiring no additional optimization. We validate NOQS on the two-dimensional Ising model with time-dependent longitudinal and transverse fields, demonstrating accurate prediction not only for unseen in-distribution protocols, but also for qualitatively different, out-of-distribution functional forms of driving. Further, a single NOQS model can be transferred between different temporal resolutions, and can be efficiently fine-tuned with sparse experimental measurements to improve predictions across all observables at negligible cost. Our work introduces a new paradigm for quantum dynamics simulation and provides a practical computational-experimental interface for driven quantum systems.
We present a new class of quantum neural networks (QNNs) whose states are solutions of ppp-adic Schrödinger equations with a non-local potential that controls the interaction between the neurons. These equations are obtained as Wick rotations of the state equations of ppp-adic cellular neural networks (CNNs). The CNNs are continuous limits of discrete hierarchical neural networks (NNs). The CNNs are bio-inspired by the Wilson-Cowan model, which describes the macroscopic dynamics of large populations of neurons. We provide a detailed study of the discretization of the new ppp-adic Schrödinger equations, which allows the construction of new QNNs on simple graphs. We also conduct detailed numerical simulations, offering a clear insight into the functioning of the new QNNs. At a mathematical level, we show the existence of local solutions for the new ppp -adic Schrödinger equations.
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It has been known a variety of optimization problems can be solved with a neural network, and a quantum computer executes real parallel computation. A quantum neural network has been proposed in order to incorporate quantum dynamics. In this paper, we test the possibility of real implementation of a quantum neural network with a nuclear spin as a qubit. First, we introduce the relation between spin and neuron, then describe the adiabatic Hamiltonian evolution applied for the state change. Next, we describe a real spin quantum system and show the simulation results. A nuclear spin system proposed by Kane behaves as a neuron with inhibitory interactions as expected in analogy to a Hopfield network.