The variational study of the ground state of the spin
−1/2 anisotropic Heisenberg antiferromagnet has been revisited on a square lattice by improving and correcting past numerical results found in Sol. State. Comm. 165, 33 (2013). The Hamiltonian has been implemented on a square lattice with antiferromagnetic interactions between nearest- and next-nearest neighbors. The nearest-neighbor couplings have different strengths, namely,
J1 and
J1′, for the x and y directions, respectively. These couplings compete with the next-nearest ones denoted by
J2. We obtained a new phase diagram in the
λ−α plane, where
λ=J1′/J1 and
α=J2/J1, whose topology is slightly different of that previously found. There is no direct frontier dividing the collinear (CAF) and the antiferromagnetic order (AF), rather, the quantum paramagnetic phase (QP) separates these two phases for all positive values of
λ and
α. The true nature of the frontiers has been obtained by scanning rigorously the relevant points of the
λ−α plane.
Abstract The variational quantum eigensolver (VQE) has recently been demonstrated for solving the challenging Heisenberg Antiferromagnet (HAFM) models. Apart from the ground state energy, many important issues such as excited states and general frustration for HAFM are worth investigating, which have only been partially solved by classical methods and rarely by quantum approaches. Here, VQE is applied to the GPU quantum simulator to calculate the excited states of a ‐ HAFM model on both square and kagome lattices. The invariant subspace property is analyzed during the process of VQE and hence the even‐fold degeneracy is explained that is difficult to interpret using classical methods. Moreover, the VQE results for different / ratios show a more efficient way to observe the phase transition in this model. The advantageous properties of VQE are demonstrated for exploring fundamental physical mechanisms.
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.
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.
We propose a divide-and-conquer method for the quantum-classical hybrid algorithm to solve larger problems with small-scale quantum computers.Specifically, we concatenate a variational quantum eigensolver (VQE) with a reduction in the system dimension, where the interactions between divided subsystems are taken as an effective Hamiltonian expanded by the reduced basis.Then the effective Hamiltonian is further solved by the VQE, which we call deep VQE.Deep VQE allows us to apply quantum-classical hybrid algorithms on small-scale quantum computers to large systems with strong intrasubsystem interactions and weak intersubsystem interactions, or strongly correlated spin models on large regular lattices.As proof-of-principle numerical demonstrations, we use the proposed method for quasi-one-dimensional models, including one-dimensionally coupled 12-qubit Heisenberg antiferromagnetic models on kagome lattices as well as two-dimensional Heisenberg antiferromagnetic models on square lattices.The largest problem size of 64 qubits is solved by simulating 20-qubit quantum computers with a reasonably good accuracy approximately a few %.The proposed scheme enables us to handle the problems of > 1000 qubits by concatenating VQEs with a few tens of qubits.While it is unclear how accurate ground-state energy can be obtained for such a large system, our numerical results on a 64-qubit system suggest that deep VQE provides a good approximation (discrepancy within a few percent) and has room for further improvement.Therefore, deep VQE provides us a promising pathway to solve practically important problems on noisy intermediate-scale quantum computers.
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.