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We introduce the Gram-Hadamard Density Operator (GHDO), a new deepneural-network architecture that can encode positive semi-definite densityoperators of exponential rank with polynomial resources. We then show how toembed an autoregressive structure in the GHDO to allow direct sampling of theprobability distribution. These properties are especially important whenrepresenting and variationally optimizing the mixed quantum state of a systeminteracting with an environment. Finally, we benchmark this architecture bysimulating the steady state of the dissipative transverse-field Ising model.Estimating local observables and the R\'enyi entropy, we show significantimprovements over previous state-of-the-art variational approaches.

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