A Limit on the Speed of Quantum Computation in Determining Parity

TL;DR

This work addresses whether quantum computation can outperform classical methods for determining the parity of a function $f:\{1,\dots,N\}\to\{\pm1\}$. By formulating the problem in the quantum oracle model with the unitary $U_f$ and analyzing the query complexity, it proves a tight lower bound of $k\ge N/2$ uses of $U_f$ for any algorithm to determine par$(f)$ with nontrivial success on all $f$. The authors also construct an explicit $N/2$-query algorithm that perfectly computes parity and connect the parity problem to evaluating the $N$-th iterate of a related function, thereby strengthening existing bounds on quantum speedups in iterated-function contexts. Together, these results delineate a concrete limit on quantum advantage for parity-type, black-box problems and illuminate the relationship between parity determination and iteration-based computation in quantum algorithms.

Abstract

Consider a function f which is defined on the integers from 1 to N and takes the values -1 and +1. The parity of f is the product over all x from 1 to N of f(x). With no further information about f, to classically determine the parity of f requires N calls of the function f. We show that any quantum algorithm capable of determining the parity of f contains at least N/2 applications of the unitary operator which evaluates f. Thus for this problem, quantum computers cannot outperform classical computers.