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.
Variational quantum eigensolver (VQE) is an efficient classical-quantum hybrid method to take advantage of quantum computers in the Noisy Intermediate-Scale Quantum (NISQ) era. In this work we test the performance of VQE by studying the
J1-
J2 anti-ferromagnetic Heisenberg model on the kagome lattice, which is found to display a first order phase transition at
J2/J1≈0.01. By comparing the VQE states with the exact diagonalization results, we find VQE energies agree well with the exact values in most region of parameters for the 18-site system we studied. However, near the phase transition point, VQE tends to converge to the excited states when the number of variational parameters is not large enough. For the system studied in this work, this issue can be solved by either increasing the number of parameters or by initializing the parameters with converged values for
J2/J1 away from the phase transition point. Our results provide useful guidance for the practical application of VQE on real quantum computers to study strongly correlated quantum many-body systems.
The ground state properties of the two-dimensional ${J}_{1}\ensuremath{-}{J}_{2}$-model are very challenging to analyze via classical numerical methods due to the high level of frustration. This makes the model a promising candidate for which quantum computers could be helpful to possibly explore regimes that classical computers cannot reach. The ${J}_{1}\ensuremath{-}{J}_{2}$-model is a quantum spin model composed of Heisenberg interactions along the rectangular lattice edges and along diagonal edges between next-nearest-neighbor spins. We propose an ansatz for the variational quantum eigensolver to approximate the ground state of an antiferromagnetic ${J}_{1}\ensuremath{-}{J}_{2}$ Hamiltonian for different lattice sizes and different ratios of
J1 and
J2. Moreover, we demonstrate that this ansatz can work without the need for gates along the diagonal next-nearest-neighbor interactions. This simplification is of great importance for solid-state-based hardware with qubits on a rectangular grid, where it eliminates the need for swap gates. In addition, we provide an extrapolation for the number of gates and parameters needed for larger lattice sizes, showing that these are expected to grow linearly in the qubit number up to lattice sizes which eventually can no longer be treated with classical computers.
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.
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.
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−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% ground state energy approximation) and a newly developed ansatz (
+9.31% improvement compared to prior best), we improved ground state approximation accuracy on the 16-site variant, achieving usable results with approximately
103 iterations, significantly fewer than the
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% ground state accuracy, compared to
83.7% with the Quantum Moments algorithm and
60% with standard error mitigation. These strategies reduced the average error of observable prediction from
≈20% to
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% 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−J2 Heisenberg model, transitioning focus from theoretical to experimental, and suggesting general improvements to VQE-based methods.