Quantum simulation of the classical XY Hamiltonian using a quantum Gibbs sampler

We first combine the Fourier pseudo-spectral techniques of arXiv:2210.08104, and the quantum singular value thresholding of arXiv:2505.05301 to obtain a high-precision first-order quantum Gibbs sampler for periodic and analytic real-valued functions without a warm start. We then present respective classical query complexity lower bounds that reveal a rigorous exponential quantum advantage in precision for the quantum algorithm while enjoying a quadratic speedup in the Poincare constant. Periodic analytic functions are ubiquitous in physics, and therefore, as a first application, we focus on the classical XY Hamiltonian. We bound the Poincare constant of the XY Hamiltonian in the low temperature regime and in the presence of an external field by studying the spectrum of the Hessian at the critical loci of the Hamiltonian, and show that our results reduce to a degree-2.9 quantum advantage in the number of lattice sites besides the exponentially superior precision compared to Riemmannian Langevin Algorithm arXiv:2010.11176.