Near-Term Quantum Spin Simulation of the Spin-$\frac{1}{2}$ Square $J_{1}-J_{2}$ Heisenberg Model

Simulating complex spin systems, particularly those with high degrees of frustration and entanglement, presents significant challenges. These systems often defy traditional simulation techniques due to their complex energy landscapes and entanglement properties. We focus on the $J{1}-J{2}$ Heisenberg model, known for its rich phase behavior on the square lattice. The model serves to study magnetic states, including phases that might be linked to high-temperature superconductivity. We carry out 16-qubit experiments on the 127-qubit IBM Rensselear Eagle processor to perform ground state simulation using the Variational Quantum Eigensolver (VQE) algorithm, enhanced through classical warm-starting. Our results are qualitatively consistent with established theoretical predictions, underscoring the viability of VQE for ground-state estimation in the noisy intermediate-scale quantum (NISQ) era. We utilize existing error mitigation strategies, introduce a novel Classically-Reinforced VQE error mitigation scheme, and compare its performance with the Quantum Moments algorithm. Additionally, we explore an experimental implementation of the Quantum Lanczos (QLanczos) algorithm using Variational-Fast Forwarding (VFF) on a 4 qubit site. Our study demonstrates the capability of near-term quantum devices to both identify trends and predict phase transitions within the $J1-J2$ Heisenberg model.