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Design of Magnetic Lattices with Quantum Optimization Algorithms

Zekeriya Ender Eğer, Waris Khan, Priyabrata Maharana, Kandula Eswara Sai Kumar, Udbhav Sharma, Abhishek Chopra, Rut Lineswala, Pınar Acar

TL;DR

The paper addresses identifying magnetic spin configurations on 2D lattices by minimizing the Ising energy $H = - ω_j sum_{<i,j>} σ_i σ_j - h sum_i σ_i$ under uncertainty, incorporating Gaussian long-range couplings and a temperature-/field-induced variability. It introduces quantum-inspired optimization via the BQPhy/QIEO framework to navigate the $2^{n^2}$ design space, contrasting with a classical genetic algorithm and showing comparable optima with substantially reduced compute time, scalable to lattices up to $50 \times 50$. By leveraging Latin Hypercube Sampling of 3,000 uncertainty realizations and a probabilistic qubit representation, the approach demonstrates significant speedups while preserving solution quality, highlighting the practicality of quantum-inspired methods for complex Hamiltonian landscapes. The work suggests hybrid quantum-inspired strategies as viable intermediate solutions before wide-access to quantum hardware, with implications for scalable magnetic materials design under uncertainty.

Abstract

This article investigates the identification of magnetic spin distributions in ferromagnetic materials by minimizing the system's free energy. Magnetic lattices of varying sizes are constructed, and the free energy is computed using an Ising model that accounts for spin-to-spin neighbor interactions and the influence of an external magnetic field. The problem reduces to determining the state of each spin, either up or down, leading to an optimization problem with $2^{n \times n}$ design variables for an $n \times n$ lattice. To address the high-dimensional and computationally intractable nature of this problem, particularly for large domains, we employ a quantum optimization algorithm, BQP. The BQP results are first validated against solutions obtained using a genetic algorithm for smaller lattices. Finally, the approach is extended to large-scale systems, including $50 \times 50$ lattices, where conventional methods become impractical.

Design of Magnetic Lattices with Quantum Optimization Algorithms

TL;DR

The paper addresses identifying magnetic spin configurations on 2D lattices by minimizing the Ising energy under uncertainty, incorporating Gaussian long-range couplings and a temperature-/field-induced variability. It introduces quantum-inspired optimization via the BQPhy/QIEO framework to navigate the design space, contrasting with a classical genetic algorithm and showing comparable optima with substantially reduced compute time, scalable to lattices up to . By leveraging Latin Hypercube Sampling of 3,000 uncertainty realizations and a probabilistic qubit representation, the approach demonstrates significant speedups while preserving solution quality, highlighting the practicality of quantum-inspired methods for complex Hamiltonian landscapes. The work suggests hybrid quantum-inspired strategies as viable intermediate solutions before wide-access to quantum hardware, with implications for scalable magnetic materials design under uncertainty.

Abstract

This article investigates the identification of magnetic spin distributions in ferromagnetic materials by minimizing the system's free energy. Magnetic lattices of varying sizes are constructed, and the free energy is computed using an Ising model that accounts for spin-to-spin neighbor interactions and the influence of an external magnetic field. The problem reduces to determining the state of each spin, either up or down, leading to an optimization problem with design variables for an lattice. To address the high-dimensional and computationally intractable nature of this problem, particularly for large domains, we employ a quantum optimization algorithm, BQP. The BQP results are first validated against solutions obtained using a genetic algorithm for smaller lattices. Finally, the approach is extended to large-scale systems, including lattices, where conventional methods become impractical.
Paper Structure (10 sections, 7 equations, 3 figures, 1 table)

This paper contains 10 sections, 7 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Ising Model Formulation
  3. Mathematical Formulation
  4. Investigation of Spin-to-Spin Interactions
  5. Modeling the Uncertainty of Temperature and External Field
  6. Optimization of Magnetic Spin States under Uncertainty
  7. Application of Quantum Optimization Algorithm
  8. Quantum-Inspired Evolutionary Optimization Algorithm
  9. Performance Comparison: Optimization using BQPhy's QIEO vs. Optimization using Traditional Genetic Algorithm on a Classical Computer
  10. Discussions and Future Work

Figures (3)

  • Figure 1: Solution of the first optimization problem identifying the optimum spin distributions that minimize the expected value of the system's free magnetic energy under the effects of the uncertainty of temperature and external magnetic field using the genetic algorithm.
  • Figure 2: Exemplary solutions of the second optimization problem identifying the optimum spin distributions that minimize the standard deviation of the magnetic free energy under the effects of the uncertainty of temperature and external magnetic field using the genetic algorithm.
  • Figure 3: For comparison, the optimum solutions obtained using both BQP and GA are illustrated for the $15 \times 15$, $25 \times 25$, and $50 \times 50$ domains.