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arXiv:2108.08086 · Physical review. B./Physical review. B

Probing ground-state properties of the kagome antiferromagnetic Heisenberg model using the variational quantum eigensolver

Jan Lukas Bosse, Ashley Montanaro

Published Mar 7, 2022• 32 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.105.094409

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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.
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.
We present experimental quantum computation of the ground-state energy in a 103-site flat Kagome lattice under the antiferromagnetic Heisenberg model (KAFH), with IBM's Heron r1 and Heron r2 quantum processors. For spin-1/2 KAFH, our per-site ground-state energy estimate is −0.417 J-0.417\,J−0.417J, which, under open-boundary corrections, matches the energy in the thermodynamic limit, i.e., −0.4386 J-0.4386\,J−0.4386J. To achieve this, we used a hybrid approach that splits the conventional Variational Quantum Eigensolver (VQE) into local (classical) and global (quantum) components for efficient hardware utilization. More importantly, we introduce a Hamiltonian engineering strategy that increases coupling on defect triangles to mimic loop-flip dynamics, allowing us to simplify the ansatz while retaining computational accuracy. Using a single-repetition, hardware-efficient ansatz, we entangle up to 103 qubits with high fidelity to determine the Hamiltonian's lowest eigenvalue. This work demonstrates the scalability of VQE for frustrated 2D systems and lays the foundation for future studies using deeper ansatz circuits and larger lattices on utility quantum processors.
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.
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.
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{J}_{1}J1​ and J2{J}_{2}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.
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.