Quantum annealing for lattice models with competing long-range interactions

TL;DR

This work addresses the challenge of determining ground states for lattice models with competing long-range interactions in the thermodynamic limit. It combines a unit-cell-based optimization framework (UCBOS) with state-of-the-art quantum annealing hardware (D-Wave Advantage, Pegasus connectivity) to solve effective unit-cell Ising problems whose couplings are resummed via Epstein zeta functions. Demonstrations on three paradigmatic LRIM settings—the triangular lattice with a devil's staircase, the Kagomé lattice ground state, and anisotropic Shastry–Sutherland-like models with added short-range terms—show that quantum annealing can reproduce classical optimization results and offer substantial run-time advantages, while highlighting embedding and reliability limits for larger unit cells. The results illustrate a tangible path to leveraging existing quantum annealing devices for complex lattice problems with long-range interactions and motivate future hardware improvements and algorithmic extensions to broader quantum-simulation scenarios.

Abstract

We use superconducting qubit quantum annealing devices to determine the ground state of Ising models with algebraically decaying competing long-range interactions in the thermodynamic limit. This is enabled by a unit-cell-based optimization scheme, in which the finite optimizations on each unit cell are performed using commercial quantum annealing hardware. To demonstrate the capabilities of the approach, we choose three exemplary problems relevant for other quantum simulation platforms and material science: (i) the calculation of devil's staircases of magnetization plateaux of the long-range Ising model in a longitudinal field on the triangular lattice, motivated by atomic and molecular quantum simulators; (ii) the evaluation of the ground state of the same model on the Kagome lattice in the absence of a field, motivated by artificial spin ice metamaterials; (iii) the study of models with additional few-nearest-neighbor interactions relevant for frustrated Ising compounds with potential long-range interactions. The approach discussed in this work provides a useful and realistic application of existing quantum annealing technology, applicable across many research areas in which lattice problems with resummable long-range interactions are relevant.

Figures (4)

• Figure 1: Schematic overview of the conceptual steps in the unit-cell-based optimization scheme (UCBOS). Left: The given long-range interacting model is converted into effective models on unit cells. Middle: The energy optimization can be done by quantum annealing or classical algorithms. Right: The optimal states of the effective models result in the optimal state of matter of the investigated long-range interacting model in the themodynamic limit.
• Figure 2: Devil's staircase of ground-state-magnetization plateaux of the dipolar ($\alpha=3$) antiferromagnetic LRIM in a longitudinal field on the triangular lattice calculated using the UCBOS with classical naive greedy optimization (blue line) and quantum annealing on the D-Wave Advantage™ system (orange line). The magnetizations are calculated on a grid of $10^{-2}$ in $h/J$.
• Figure 3: Magnetic configuration of the candidate ground state of the dipolar ($\alpha=3$) antiferromagnetic LRIM on the Kagomé lattice with no longitudinal field determined with the UCBOS with quantum annealing optimization on the D-Wave Advantage™ system. Blue circles refer to spins with $\sigma^z_i=+1$ and red circles to spins with $\sigma^z_i=-1$. The shaded region corresponds to the unit cell of the ordered structure.
• Figure 4: (a) Illustration of the geometry and additional anisotropic nearest-neighbor Ising couplings in the anisotropic Shastry-Sutherland lattice with realistic Ising coupling values $J_1$, $J_1^{\prime}$, $J_2$, and $J_2^{\prime}$ and interatomic distances $d_{J_1}$, $d_{J_1^{\prime}}$, $d_{J_2}$, and $d_{J_2^{\prime}}$ for the $\text{Er}_2\text{Be}_2\text{Ge}\text{O}_7$ compound Yadav2025. $J$ denotes the amplitude of the additional dipolar ($\alpha=3$) Ising interaction. (b) Magnetic configurations at magnetizations $m$ of $0$, $1/4$, and $1/2$ in units of the saturated magnetization determined with the UCBOS with quantum annealing optimization on the D-Wave Advantage™ system for the model with anisotropic nearest-neighbor and dipolar long-range interactions. Blue circles refer to spins with $\sigma^z_i=+1$ and red circles to spins with $\sigma^z_i=-1$. Shaded regions correspond to the unit cells of the ordered structures.