Complexity of Simon's problem in classical sense

TL;DR

This paper explores the classical complexity of Simon's problem when the function is represented by circuits or BDDs, revealing NP-hardness in general but polynomial-time solutions for BDDs.

Contribution

It demonstrates the NP-hardness of checking the non-zero element presence in the vector space for circuit representations and provides a polynomial-time algorithm for BDD representations.

Findings

Checking vector space membership is NP-hard for circuit representations.

A basis of the vector space can be computed in polynomial time for BDD representations.

The classical complexity of Simon's problem varies significantly with the function representation.

Abstract

Simon's problem is a standard example of a problem that is exponential in classical sense, while it admits a polynomial solution in quantum computing. It is about a function $f$ for which it is given that a unique non-zero vector $s$ exists for which $f(x) = f(x \oplus s)$ for all $x$, where $\oplus$ is the exclusive or operator. The goal is to find $s$. The exponential lower bound for the classical sense assumes that $f$ only admits black box access. In this paper we investigate classical complexity when $f$ is given by a standard representation like a circuit. We focus on finding the vector space of all vectors $s$ for which $f(x) = f(x \oplus s)$ for all $x$, for any given $f$. Two main results are: (1) if $f$ is given by any circuit, then checking whether this vector space contains a non-zero element is NP-hard, and (2) if $f$ is given by any ordered BDD, then a basis of this vector space can be computed in polynomial time.