Synthesizing Quantum Circuits for Simple Periodic Functions

TL;DR

This paper presents a new method for constructing quantum circuits for simple periodic functions, crucial in quantum algorithms like Shor's, and conjectures an upper bound on the gate complexity based on the period size.

Contribution

It introduces a circuit construction method for monoperiodic functions and proposes a conjecture on the maximum number of Toffoli gates needed for period p.

Findings

Proposed a circuit construction for monoperiodic functions.

Conjectured an upper bound of n Toffoli gates for period p.

Enhanced understanding of quantum circuit complexity for periodic functions.

Abstract

Periodic functions are of special importance in quantum computing, particularly in applications of Shor's algorithm. We explore methods of creating circuits for periodic functions to better understand their properties. We introduce a method for constructing the circuit for a simple monoperiodic function, that is one-to-one within a single period, of a given period p. We conjecture that to create a simples periodic function of period p, where p is an n-bit number, one needs at most n Toffoli gates.