Calculating Renyi Entropies with Neural Autoregressive Quantum States

Entanglement entropy is an essential metric for characterizing quantum many-body systems, but its numerical evaluation for neural network representations of quantum states has so far been inefficient and only demonstrated for the restricted Boltzmann machine architecture. We estimate generalized Renyi entropies S_n of an autoregressive neural quantum state using quantum Monte Carlo methods. A naive “direct sampling” approach performs well for small Renyi order n but fails for larger orders when benchmarked on a 1D Heisenberg model. We therefore propose an improved “conditional sampling” method exploiting the autoregressive structure of the network ansatz, which outperforms direct sampling in both 1D and 2D Heisenberg models. Conditional sampling facilities calculations of high-order Renyi entropies up to at least n > 30, which allows for a polynomial approximation of the von Neumann entropy as well as extraction of the largest eigenvalue of the reduced density matrix, and thus the single copy entanglement. By demonstrating good convergence even up to high Renyi order, our methods elucidate the potential of neural network quantum states in quantum Monte Carlo studies of entanglement entropy for many-body systems.