Positive-definite parametrization of mixed quantum states with deep neural networks

TL;DR

This paper presents the GHDO, a deep neural network architecture for efficiently representing and optimizing mixed quantum states, with demonstrated advantages in simulating dissipative quantum systems.

Contribution

Introduction of the Gram-Hadamard Density Operator (GHDO), a neural network architecture that encodes positive semi-definite density operators with polynomial resources and enables direct sampling.

Findings

Significant improvements in estimating local observables.

Effective simulation of steady states in dissipative quantum models.

Enhanced variational optimization of mixed quantum states.

Abstract

We introduce the Gram-Hadamard Density Operator (GHDO), a new deep neural-network architecture that can encode positive semi-definite density operators of exponential rank with polynomial resources. We then show how to embed an autoregressive structure in the GHDO to allow direct sampling of the probability distribution. These properties are especially important when representing and variationally optimizing the mixed quantum state of a system interacting with an environment. Finally, we benchmark this architecture by simulating the steady state of the dissipative transverse-field Ising model. Estimating local observables and the R\'enyi entropy, we show significant improvements over previous state-of-the-art variational approaches.