Near-Term Quantum Spin Simulation of the Spin-$\frac{1}{2}$ Square $J_{1}-J_{2}$ 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 $J_{1}-J_{2}$ 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 $10^{3}$ iterations, significantly fewer than the $10^{4}-10^{5}$ 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 $\approx 20\%$ 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 $\approx 97\%$ 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 $J_1-J_2$ Heisenberg model, transitioning focus from theoretical to experimental, and suggesting general improvements to VQE-based methods.