Universal approximation of continuous functions with minimal quantum circuits

TL;DR

This paper addresses universal approximation of continuous multivariate functions by parameterized quantum circuits under fixed input encoding. It introduces two universal schemes: (1) a single-qubit circuit with fixed encodings that input each coordinate independently, and (2) a multi-qubit circuit with a fixed, densely encoded input using $\mathcal{O}(\log m)$ qubits. The authors show that both constructions achieve universality for all continuous functions on $[0,1]^m$ (under the $\|\cdot\|_\infty$ norm), by leveraging quantum signal processing and re-uploading concepts to approximate exponentials and compose multivariate gates. They further extend the framework to a diagonal encoding in SU$(2m)$ and prove a universal model with only poly$(m)$ parameters and qubits that scale logarithmically in the input dimension. These results close the gap between fixed-encoding and tunable-encoding universality and suggest efficient, densely-encoded quantum-function representations with potential impact on quantum machine learning and quantum-process modeling.

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.

Figures (4)

• Figure 1: Quantum circuits with parameters and fixed encoding gates can represent multivariate continuous functions with multiple qubits. In the figures $x_i$ is a shorthand for $\operatorname{exp}(i \sigma_z x_i)$ and $\theta_{i, j}$ for an arbitrary parameterized gate. The top figure corresponds to our main result, in \ref{['th.multivariate_fixed']}, while bottom figure is a common approach in re-uploading models, e. g. schuld2021effectyu2024nonasymptotic. This figure serves as a summary for the paper: we show that single-qubit quantum circuits with fixed encoding gates retain universal representation capabilities for multivariate functions.
• Figure 2: High level illustration for the basic concepts of the proof for universality. Consider the curved blob to be a representation of a set of functions. Triangles are components specified by different gates in the circuit. The re-uploading model with tunable parameters (left) is represented by triangles with tunable sizes and shapes, while fixed encoding gates (right) are represented by fixed-shaped triangles, but tunable in size. We can approximate the large tunable triangles with small fixed-shaped triangles. In the limit, any curved blob can be approximated by many small triangles.
• Figure 3: Quantum circuit allowing the universality of quantum re-uploading models, with fixed encoding gates (in this figure represented by $\{x_1, x_2, \ldots, x_m$). By \ref{['le.one_gate']}, we show that gates of the form $e^{i w x\hat{\sigma}_z}$ can be approximated to arbitrary accuracy by a re-uploading circuit with fixed encoding gates. The successive application of these approximations allow for an approximation of $e^{i \bm w \cdot \bm x\hat{\sigma}_z}$, where $\bm x$ is now multidimensional.
• Figure :

Theorems & Definitions (26)

• Definition 1: Universality
• Definition 2: Multivariate fixed re-uploading
• Theorem 1
• Corollary 1
• Definition 3: Re-uploading: fixed encoding gates and arbitrary dimensions
• Theorem 2
• Theorem 3: Fourier theorem
• Theorem 4: Universal approximation theorem, adapted from cybenko1989approximation
• Theorem 5: Re-uploading: fixed encoding gates, adapted from silva2022fourierbasedmotlagh2024generalized
• proof