A Universal Quantum Circuit Design for Periodical Functions

TL;DR

This paper introduces a universal quantum circuit capable of estimating any one-dimensional periodic function using Fourier expansion, with demonstrated accuracy on a square wave function via IBM-QASM simulations.

Contribution

It presents a general quantum circuit design for periodic functions based on Fourier components, applicable to any such function.

Findings

Successfully constructed a quantum circuit for the square wave function.

Achieved accurate results through IBM-QASM simulations.

Complexity scales as O(N^2 log^2 N).

Abstract

We propose a universal quantum circuit design that can estimate any arbitrary one-dimensional periodic functions based on the corresponding Fourier expansion. The quantum circuit contains N-qubits to store the information on the different N-Fourier components and $M+2$ auxiliary qubits with $M = \lceil{\log_2{N}}\rceil$ for control operations. The desired output will be measured in the last qubit $q_N$ with a time complexity of the computation of $O(N^2\lceil \log_2N\rceil^2)$. We illustrate the approach by constructing the quantum circuit for the square wave function with accurate results obtained by direct simulations using the IBM-QASM simulator. The approach is general and can be applied to any arbitrary periodic function.