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arXiv:2507.07457

Enhanced Quantum behavior on frustrated Ising model: Quantum Approximate Optimization Algorithm study

Seunghan Lee, Hunpyo Lee

Published Jul 10, 2025• 0 citations
arXivHTMLPDF

Abstract

We investigated the quantum effects of a frustrated Ising model on a two-dimensional square lattice using the Quantum Approximate Optimization Algorithm (QAOA). While strong spin frustration is known to induce quantum fluctuations at low temperatures, previous classical approaches restricted to binary (up or down) spin configurations have been insufficient to fully capture the quantum contributions of frustration. In this study, we introduced a quantitative metric to evaluate the quantum effects arising from frustration and employed QAOA to differentiate between classical and quantum regimes. Notably, we found that in the weakly frustrated region, QAOA measurements rarely capture first excited states, as they are energetically well separated from the ground state. In contrast, near the quantum phase transition point, excited states appear more frequently in QAOA measurements, highlighting the increased role of quantum fluctuations.

DOI
10.48550/arxiv.2507.07457

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We investigated the quantum effects of a frustrated Ising model on a two-dimensional square lattice using the Quantum Approximate Optimization Algorithm (QAOA). While strong spin frustration is known to induce quantum fluctuations at low temperatures, previous classical approaches restricted to binary (up or down) spin configurations have been insufficient to fully capture the quantum contributions of frustration. In this study, we introduced a quantitative metric to evaluate the quantum effects arising from frustration and employed QAOA to differentiate between classical and quantum regimes. Notably, we found that in the weakly frustrated region, QAOA measurements rarely capture first excited states, as they are energetically well separated from the ground state. In contrast, near the quantum phase transition point, excited states appear more frequently in QAOA measurements, highlighting the increased role of quantum fluctuations.
Novel magnetic materials are important for future technological advances. Theoretical and numerical calculations of ground-state properties are essential in understanding these materials, however, computational complexity limits conventional methods for studying these states. Here we investigate an alternative approach to preparing materials ground states using the quantum approximate optimization algorithm (QAOA) on near-term quantum computers. We study classical Ising spin models on unit cells of square, Shastry-Sutherland and triangular lattices, with varying field amplitudes and couplings in the material Hamiltonian. We find relationships between the theoretical QAOA success probability and the structure of the ground state, indicating that only a modest number of measurements ([Formula: see text]) are needed to find the ground state of our nine-spin Hamiltonians, even for parameters leading to frustrated magnetism. We further demonstrate the approach in calculations on a trapped-ion quantum computer and succeed in recovering each ground state of the Shastry-Sutherland unit cell with probabilities close to ideal theoretical values. The results demonstrate the viability of QAOA for materials ground state preparation in the frustrated Ising limit, giving important first steps towards larger sizes and more complex Hamiltonians where quantum computational advantage may prove essential in developing a systematic understanding of novel materials. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
Geometric frustration, arising from competing interactions that prevent simultaneous energy minimization, presents a fundamental challenge for variational quantum algorithms applied to quantum many-body systems. We investigate the transverse-field Ising model on a square lattice with frustrated diagonal coupling and show that geometric frustration leads to strongly inhomogeneous correlations that are difficult to capture using standard Hamiltonian-inspired ansätze with global parameters. As a result, the required circuit depth increases significantly in the intermediate-field regime. We demonstrate that this limitation is not caused by optimization difficulties such as barren plateaus, but instead arises from insufficient expressibility of the ansatz. By introducing bond-resolved variational parameters, we recover accurate results at reduced circuit depth. We also study low-energy excitations and find that near-degenerate spectra in the frustrated regime further challenge variational methods. Our results provide a clear physical explanation for the limitations of variational quantum algorithms in frustrated systems and suggest improved ansatz design strategies for quantum simulation.
The one-dimensional quantum Ising model is used to describe many physical systems, therefore the calculation of ground and excited state energies of Ising model that has many applications in statistical mechanics, has been paid attention. We obtain the ground and excited states of this model by using the Quantum Approximate Optimization Algorithm(QAOA), which is a kind of hybrid quantum-classical algorithms. For this purpose, first, we find the ground state with quantum approximate optimization algorithm, then we generalize this algorithm by adding an overlap term to obtain excited states. We calculated ground state energy and excited states energies for different coupling coefficients which is compatible with exact calculation.
The frustrated Ising ring is one of the simplest models exhibiting exponential closing spectral gaps, making it a paradigmatic and challenging benchmark for quantum annealing (QA). Ground-state preparation for this model has therefore been studied extensively in both continuous-time QA and digitized protocols such as the Quantum Approximate Optimization Algorithm (QAOA). Here, we use the frustrated Ising ring to investigate how the parity mapping affects the performance of both QA and QAOA. For QA, finite-size calculations show that the parity mapping increases the minimum spectral gap under the energy normalization used in this work, thereby enabling faster continuous-time ground state preparation protocols. An ideal implementation of Parity-QA, with a single global constraint, shows no evidence of exponential gap closing over the accessible system sizes, whereas a hardware-motivated decomposition into local constraints restores the exponential decrease, albeit with a smaller fitted exponent than conventional QA. For the digitized protocol, we find that the number of Parity-QAOA layers required to prepare the exact ground state remains constant over the simulated sizes, improving upon the quadratic scaling required by conventional QAOA. To investigate the role of constraints in Parity-QAOA, we further consider a modified Ising ring instance in which the constraint term is essential for preparing the target ground state. We then compare the corresponding resource requirements with those of conventional QAOA.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
The quantum antiferromagnetic spin-1/2 Ising model on a triangular lattice and analogous fully frustrated Ising model on a square lattice with quantum fluctuation induced by the application of the transverse magnetic field are studied at finite temperatures by constructing an exact mapping onto a purely classical model with a more complex interaction. It is shown that in weak fields the temperatures of the phase transitions separating the critical phase from the ordered and disordered phases in both models are proportional to the magnitude of the field.