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.
Dylan Sheils Trevor David Rhone