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arXiv:2506.12391

Simulating the Antiferromagnetic Heisenberg Model on a Spin-Frustrated Kagome Lattice with the Contextual Subspace Variational Quantum Eigensolver

Tim Weaving, Alexis Ralli, Vinul Wimalaweera, Peter J. Love, Peter V. Coveney

Published Jun 14, 2025• 0 citations
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Abstract

In this work we investigate the ground state properties of a candidate quantum spin liquid using a superconducting Noisy Intermediate-Scale Quantum (NISQ) device. Specifically, we study the antiferromagnetic Heisenberg model on a Kagome lattice, a geometrically frustrated structure that gives rise to a highly degenerate energy spectrum. To successfully simulate this system, we employ a qubit reduction strategy leveraging the Contextual Subspace methodology, significantly reducing the problem size prior to execution on the quantum device. We improve the quality of these subspaces by using the wavefunctions obtained from low bond dimension Density Matrix Renormalization Group (DMRG) calculations to bias the subspace stabilizers through a symplectic approximate symmetry generator extraction algorithm. Reducing the Hamiltonian size allows us to implement tiled circuit ensembles and deploy the Variational Quantum Eigensolver (VQE) to estimate the ground state energy. We adopt a hybrid quantum error mitigation strategy combining Readout Error Mitigation (REM), Symmetry Verification (SV) and Zero Noise Extrapolation (ZNE). This approach yields high-accuracy energy estimates, achieving error rates on the order of 0.01% and thus demonstrating the potential of near-term quantum devices for probing frustrated quantum materials.

DOI
10.48550/arxiv.2506.12391

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In this work we investigate the ground state properties of a candidate quantum spin liquid using a superconducting Noisy Intermediate-Scale Quantum (NISQ) device. Specifically, we study the antiferromagnetic Heisenberg model on a Kagome lattice, a geometrically frustrated structure that gives rise to a highly degenerate energy spectrum. To successfully simulate this system, we employ a qubit reduction strategy leveraging the Contextual Subspace methodology, significantly reducing the problem size prior to execution on the quantum device. We improve the quality of these subspaces by using the wavefunctions obtained from low bond dimension Density Matrix Renormalization Group (DMRG) calculations to bias the subspace stabilizers through a symplectic approximate symmetry generator extraction algorithm. Reducing the Hamiltonian size allows us to implement tiled circuit ensembles and deploy the Variational Quantum Eigensolver (VQE) to estimate the ground state energy. We adopt a hybrid quantum error mitigation strategy combining Readout Error Mitigation (REM), Symmetry Verification (SV) and Zero Noise Extrapolation (ZNE). This approach yields high-accuracy energy estimates, achieving error rates on the order of 0.01% and thus demonstrating the potential of near-term quantum devices for probing frustrated quantum materials.
The Variational Quantum Eigensolver (VQE) algorithm, as applied to finding the ground state of a Hamiltonian, is particularly well-suited for deployment on noisy intermediate-scale quantum (NISQ) devices. Here, we utilize the VQE algorithm with a quantum circuit ansatz inspired by the Density Matrix Renormalization Group (DMRG) algorithm. To ameliorate the impact of realistic noise on the performance of the method, we employ zero-noise extrapolation. We find that, with realistic error rates, our DMRG–VQE hybrid algorithm delivers good results for strongly correlated systems. We illustrate our approach with the Heisenberg model on a Kagome lattice patch and demonstrate that DMRG–VQE hybrid methods can locate and faithfully represent the physics of the ground state of such systems. Moreover, the parameterized ansatz circuit used in this work is low depth and requires a reasonably small number of parameters, so it is efficient for NISQ devices.
Establishing the nature of the ground state of the Heisenberg antiferromagnet (HAFM) on the kagome lattice is well-known to be a prohibitively difficult problem for classical computers. Here, we give a detailed proposal for a variational quantum eigensolver (VQE) intending to solve this physical problem on a quantum computer. At the same time, this VQE constitutes an explicit experimental proposal for showing a useful quantum advantage on noisy intermediate-scale quantum devices because of its natural hardware compatibility. We classically emulate noiseless and noisy quantum computers with either 2D-grid or all-to-all connectivity and simulate patches of the kagome HAFM of up to 20 sites. In the noiseless case, the ground-state energy, as found by the VQE, approaches the true ground-state energy exponentially as a function of the circuit depth. Furthermore, VQEs for the HAFM on any graph can inherently perform their quantum computations in a decoherence-free subspace that protects against collective longitudinal and collective transversal noise, adding to the noise resilience of these algorithms. Nevertheless, the extent of the effects of other noise types suggests the need for error mitigation and performance targets alternative to high-fidelity ground-state preparation, even for essentially hardware-native VQEs.
Quantum spin simulations of frustrated lattices remain challenging for both classical and quantum algorithms, particularly in parameter regimes relevant to quantum spin liquid (QSL) phases. In this work, we apply Sample-based Krylov Quantum Diagonalization (SKQD) to estimate the ground state of the antiferromagnetic XXZ Heisenberg model on the J1J_1J1​--J2J_2J2​ square lattice, the Kagome lattice, and a 1D chain, studying system sizes from 12 to 72 spins. In our application of SKQD, we identify a ZZ deformation of Δ=2Δ=2Δ=2 as a sufficiently sparse Hamiltonian and introduce two modifications to the SKQD framework tailored to spin models: a canonical bitstring compression scheme that preserves the effectiveness of configuration recovery under spin-flip degeneracy, and the use of multiple Krylov subspaces to improve ground state coverage without any increase in quantum resources. For the 1D chain and Kagome lattice, SKQD achieves sub-percent ground-state energy errors at system sizes up to 24 spins, including a relative error of 0.002%0.002\%0.002% on the 12-site Kagome lattice, surpassing the best prior VQE result of 0.01%0.01\%0.01% on the same system while requiring no variational optimization. SKQD further extends to system sizes well beyond the reach of prior quantum algorithm studies, reaching 72 spins across all three geometries. Beyond 24 spins, accuracy degrades to relative errors of 19%19\%19%--36%36\%36% at 72 sites, but the gradual scaling of error with system size suggests these limits are set by available shot budgets and circuit depth rather than fundamental algorithmic constraints. Although classical tensor network methods remain state-of-the-art for these models, this work establishes a new benchmark for quantum simulation of the frustrated Heisenberg model and demonstrates SKQD as a scalable, hardware-compatible approach for studying strongly correlated spin systems.
This work investigates the nature of the ground state of the antiferromagnetic Heisenberg model on fundamental kagome cells, a triangle and a star, using the variational quantum eigensolver algorithm on real quantum hardware. We demonstrate that the ground-state preparation is achievable using a shallow hardware-efficient quantum circuit with a naturally Euclidean parameter space. Our custom ansatz is capable of accurately recovering meaningful properties of the ground state such as the spin-spin correlation terms and static structure factor without explicit error mitigation. These features are found to be resilient to noise. We exploited the Fubini-Study metric in constructing the ansatz, ensuring a singularity-free parameter space. With this ansatz design, the adaptive gradient descent optimizer achieves a faster convergence in terms of the number of iterations compared to simultaneous perturbation stochastic approximation. We further apply error mitigation techniques, including zero-noise extrapolation (ZNE) and qubitwise readout-error mitigation (REM). While ZNE does not obey the Rayleigh-Ritz variational principle, the conditions under which REM preserves it are discussed.
Finding and probing the ground states of spin lattices, such as the antiferromagnetic Heisenberg model on the kagome lattice (KAFH), is a very challenging problem on classical computers and only possible for relatively small systems. We propose using the variational quantum eigensolver (VQE) to find the ground state of the KAFH on a quantum computer. We find efficient ansatz circuits and show how physically interesting observables can be measured efficiently. To investigate the expressiveness and scaling of our ansatz circuits we used classical, exact simulations of VQE for the KAFH for different lattices ranging from 8 to 24 qubits. We find that the fidelity with the ground state approaches one exponentially in the circuit depth for all lattices considered, except for a 24-qubit lattice with an almost degenerate ground state. We conclude that VQE circuits that are able to represent the ground state of the KAFH on lattices inaccessible to exact diagonalization techniques may be achievable on near-term quantum hardware. However, for large systems, circuits with many variational parameters are needed to achieve high fidelity with the ground state.
Near-Term Quantum Spin Simulation of the Spin-12\frac{1}{2}21​ Square J1−J2J_{1}-J_{2}J1​−J2​ Heisenberg Model
Simulating complex spin systems, known for high frustration and entanglement, presents significant challenges due to their intricate energy landscapes. This study focuses on the J1−J2J_{1}-J_{2}J1​−J2​ Heisenberg model, renowned for its rich phase behavior on the square lattice, to investigate strongly correlated spin systems. We conducted the first experimental quantum computing study of this model using the 127-qubit IBM Rensselear Eagle processor and the Variational Quantum Eigensolver (VQE) algorithm. By employing classical warm-starting (+40%+40\%+40% ground state energy approximation) and a newly developed ansatz (+9.31%+9.31\%+9.31% improvement compared to prior best), we improved ground state approximation accuracy on the 16-site variant, achieving usable results with approximately 10310^{3}103 iterations, significantly fewer than the 104−10510^{4}-10^{5}104−105 steps proposed by previous theoretical studies. We utilized existing error mitigation strategies and introduced a novel Classically-Reinforced VQE error mitigation scheme, achieving 93%93\%93% ground state accuracy, compared to 83.7%83.7\%83.7% with the Quantum Moments algorithm and 60%60\%60% with standard error mitigation. These strategies reduced the average error of observable prediction from ≈20%\approx 20\%≈20% to 5%5\%5%, enhancing phase prediction from qualitative to quantitative alignment. Additionally, we explored an experimental implementation of the Quantum Lanczos (QLanczos) algorithm using Variational-Fast Forwarding (VFF) on a 4-qubit site, achieving ≈97%\approx 97\%≈97% ground state approximation. Theoretical simulations indicated that Krylov-based methods outperform VQE, with the Lanczos basis converging faster than the real-time basis. Our study demonstrates that near-term quantum devices can predict phase-relevant observables for the J1−J2J_1-J_2J1​−J2​ Heisenberg model, transitioning focus from theoretical to experimental, and suggesting general improvements to VQE-based methods.