One qubit as a Universal Approximant

TL;DR

This paper demonstrates that a single-qubit quantum circuit with variable re-uploading can approximate any bounded complex function, with accuracy improving as the circuit depth increases, supported by proofs, benchmarks, and real-device implementation.

Contribution

It introduces a novel single-qubit approximant using re-uploading of the independent variable, with proofs and experimental validation of its universal approximation capabilities.

Findings

Quantum circuit accuracy improves with more re-uploaded layers.

The method is supported by Fourier series and Universal Approximation Theorem proofs.

Experimental implementation on a superconducting qubit device confirms the approach's effectiveness.

Abstract

A single-qubit circuit can approximate any bounded complex function stored in the degrees of freedom defining its quantum gates. The single-qubit approximant presented in this work is operated through a series of gates that take as their parameterization the independent variable of the target function and an additional set of adjustable parameters. The independent variable is re-uploaded in every gate while the parameters are optimized for each target function. The output state of this quantum circuit becomes more accurate as the number of re-uploadings of the independent variable increases, i. e., as more layers of gates parameterized with the independent variable are applied. In this work, we provide two different proofs of this claim related to both the Fourier series and the Universal Approximation Theorem for Neural Networks, and we benchmark both methods against their classical counterparts. We further implement a single-qubit approximant in a real superconducting qubit device, demonstrating how the ability to describe a set of functions improves with the depth of the quantum circuit. This work shows the robustness of the re-uploading technique on Quantum Machine Learning.