Universal approximation of continuous functions with minimal quantum circuits

TL;DR

This paper demonstrates that universal quantum circuits can approximate multivariate continuous functions using minimal qubits, with two methods achieving fixed encoding and sub-linear qubit scaling, advancing quantum function approximation capabilities.

Contribution

The authors establish that universal approximation of multivariate functions is possible with fixed encoding and sub-linear qubits, introducing two novel quantum circuit methods.

Findings

Single-qubit circuit with independent coordinate input

Multi-qubit approach with logarithmic qubit scaling

Universal approximation achieved with minimal qubits

Abstract

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 parameterized 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 on 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 sub-linearly 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: (i) a single-qubit circuit where each coordinate of the arguments to the function to represent is input independently, and (ii) a multi-qubit 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 towards representing functions whose arguments are densely encoded in a unitary operation, possibly encoding for instance quantum processes.