Constrained Optimization of Higher-Order Cluster-Expansion Hamiltonians for Alloys Using Simulated Bifurcation
Abstract
Identifying ground-state and low-energy atomic configurations is a central problem in alloy design. The cluster-expansion (CE) method represents configurational energetics on a fixed lattice as an effective Hamiltonian; for binary alloys, higher-order CE models become polynomial Ising Hamiltonians. Using Au-Cu as a model binary alloy, we formulate cubic and quartic cluster-expansion Hamiltonians as penalty-augmented polynomial unconstrained binary optimization (PUBO) problems under fixed-composition constraints. We optimize these PUBO problems using SQBM+, a simulated-bifurcation-based solver that can treat higher-order polynomial binary objectives directly. This direct PUBO treatment avoids the need to construct an explicit quadratic reformulation with auxiliary variables. Composition constraints are imposed through quadratic penalty terms, whose weights are estimated from derivative coefficients of the continuous relaxation of the CE objective. Benchmark calculations for systems up to 2048 atoms show that SQBM+ robustly obtains low-energy feasible configurations for cubic CE models and remains effective for many quartic instances. Formation-energy convex hulls constructed from the optimized configurations recover the CuAu and Cu3Au ordering trends and reveal finite-size effects at off-stoichiometric compositions. These results demonstrate simulated bifurcation as a practical route to constrained higher-order CE optimization for alloy configuration search.
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BibTeX
@misc{ichikawa2026constrained,
title = {{Constrained Optimization of Higher-Order Cluster-Expansion Hamiltonians for Alloys Using Simulated Bifurcation}},
author = {Kazuhide Ichikawa and Satoru Ohuchi and Tomoyasu Yokoyama and Takuma Saito and Yoshiki Matsuda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10310},
eprint = {2609.10310},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10310},
}
RIS
TY - UNPB
TI - Constrained Optimization of Higher-Order Cluster-Expansion Hamiltonians for Alloys Using Simulated Bifurcation
AU - Kazuhide Ichikawa
AU - Satoru Ohuchi
AU - Tomoyasu Yokoyama
AU - Takuma Saito
AU - Yoshiki Matsuda
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10310
UR - https://arxiv.org/abs/2609.10310
AB - Identifying ground-state and low-energy atomic configurations is a central problem in alloy design. The cluster-expansion (CE) method represents configurational energetics on a fixed lattice as an effective Hamiltonian; for binary alloys, higher-order CE models become polynomial Ising Hamiltonians. Using Au-Cu as a model binary alloy, we formulate cubic and quartic cluster-expansion Hamiltonians as penalty-augmented polynomial unconstrained binary optimization (PUBO) problems under fixed-composition constraints. We optimize these PUBO problems using SQBM+, a simulated-bifurcation-based solver that can treat higher-order polynomial binary objectives directly. This direct PUBO treatment avoids the need to construct an explicit quadratic reformulation with auxiliary variables. Composition constraints are imposed through quadratic penalty terms, whose weights are estimated from derivative coefficients of the continuous relaxation of the CE objective. Benchmark calculations for systems up to 2048 atoms show that SQBM+ robustly obtains low-energy feasible configurations for cubic CE models and remains effective for many quartic instances. Formation-energy convex hulls constructed from the optimized configurations recover the CuAu and Cu3Au ordering trends and reveal finite-size effects at off-stoichiometric compositions. These results demonstrate simulated bifurcation as a practical route to constrained higher-order CE optimization for alloy configuration search.
ER -