We present a novel framework to learn a representation of quantum circuits based on the phase space formulation of many-body systems. Given that $N$-qubit states can be represented as probability distributions on a smooth $2N$-dimensional manifold, we show how unitary gates can be parameterized as normalizing flows between these distributions, opening a new avenue for quantum circuit simulation using generative models. We evaluate our method on three common families of quantum circuits, and show that its accuracy exceeds that of current state-of-the-art neural network methods for large system sizes.

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