A universal quantum circuit design for periodical functions

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 = ⌈log2 N⌉ for control operations. The desired output will be measured in the last qubit q N with a time complexity of the computation of \(O({N}^{2}{\lceil {\mathrm{log}}_{2}\enspace N\rceil }^{2})\), which leads to polynomial speedup under certain circumstances. 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.