Universal approximation of continuous functions with minimal quantum circuits

The conventional paradigm of quantum computing is discrete: It utilizes discrete sets of gates to realize bitstring-to-bitstring mappings, some of them arguably intractable for classical computers. In parametrized quantum approaches, the input becomes continuous and the output represents real-valued functions. While the universality of discrete quantum computers is well understood, basic questions remained open in the continuous case. We focus on universality of multivariate functions. Current approaches require either a number of qubits scaling linearly with the dimension of the input for fixed encodings or a tunable encoding procedure in single-qubit circuits. The question of whether universality can be reached with a fixed encoding and sublinearly many qubits remained open for the last five years. In this paper, we answer this question in the affirmative for arbitrary multivariate functions. We provide two methods: (1) a single-qubit circuit where each coordinate of the arguments to the function to represent is input independently and (2) a multiqubit approach where all coordinates are input in one step, with number of qubits scaling logarithmically with the dimension of the argument of the function of interest. We view the first result of inherent and fundamental interest, whereas the second result opens the path toward representing functions whose arguments are densely encoded in a unitary operation, possibly encoding, for instance, quantum processes.