Gradient descent reliably finds depth- and gate-optimal circuits for generic unitaries

When the gate set has continuous parameters, synthesizing a unitary operator as a quantum circuit is always possible using exact methods, but finding minimal circuits efficiently remains a challenging problem. The landscape is very different for compiled unitaries, which arise from programming and typically have short circuits, as compared with generic unitaries, which use all parameters and typically require circuits of maximal size. We show that simple gradient descent reliably finds depth- and gate-optimal circuits for generic unitaries, including in the presence of restricted chip connectivity. This runs counter to earlier evidence that optimal synthesis required combinatorial search, and we show that this discrepancy can be explained by avoiding the random selection of certain parameter-deficient circuit skeletons.

Generative Learning of Continuous Data by Tensor Networks

Beyond their origin in modeling many-body quantum systems, tensor networks have emerged as a promising class of models for solving machine learning problems, notably in unsupervised generative learning. While possessing many desirable features arising from their quantum-inspired nature, tensor network generative models have previously been largely restricted to binary or categorical data, limiting their utility in real-world modeling problems. We overcome this by introducing a new family of tensor network generative models for continuous data, which are capable of learning from distributions containing continuous random variables. We develop our method in the setting of matrix product states, first deriving a universal expressivity theorem proving the ability of this model family to approximate any reasonably smooth probability density function with arbitrary precision. We then benchmark the performance of this model on several synthetic and real-world datasets, finding that the model learns and generalizes well on distributions of continuous and discrete variables. We develop methods for modeling different data domains, and introduce a trainable compression layer which is found to increase model performance given limited memory or computational resources. Overall, our methods give important theoretical and empirical evidence of the efficacy of quantum-inspired methods for the rapidly growing field of generative learning.