A Quantum Monte Carlo Algorithm for arbitrary spin-1/2 Hamiltonians

Almost all physical Hamiltonians can be expressed in Permutation Matrix Representation (PMR) form, decomposing them into a summation of permutation matrices multiplied by their corresponding diagonal counterparts. This allows for the generation of quantum Monte Carlo (QMC) updates through modular linear algebra over the set of permutation matrices existent in the PMR representation. This feat guarantees the ergodicity of our QMC algorithm.

Our algorithm enables the simulation of virtually any condensed matter Hamiltonian. We have designed an automated protocol that, when presented with a Hamiltonian, automatically generates the requisite QMC updates while upholding detailed balance and ergodicity, thus ensuring the convergence of the Markov chain to equilibrium. The applicability and adaptability of our approach are illustrated through numerous examples. We have successfully replicated systems ranging from the XY model on a triangular lattice to intricate configurations like the Toric code, Fermi-Hubbard model, high spin systems, and a plethora of other complex structures. Our algorithm opens the door to the limitless possibilities of condensed matter physics simulations.