Sample complexity of hidden subgroup problem

TL;DR

This paper investigates the classical sample complexity of the hidden subgroup problem (HSP), providing bounds and specific results for generalized Simon's problem, thereby advancing understanding of its learnability.

Contribution

It offers the first comprehensive bounds on the classical sample complexity of HSP and characterizes the complexity for generalized Simon's problem.

Findings

Established upper and lower bounds for HSP sample complexity.

Derived the exact sample complexity for generalized Simon's problem.

Provided insights into the classical learnability of quantum-inspired problems.

Abstract

The hidden subgroup problem ($\mathsf{HSP}$) has been attracting much attention in quantum computing, since several well-known quantum algorithms including Shor algorithm can be described in a uniform framework as quantum methods to address different instances of it. One of the central issues about $\mathsf{HSP}$ is to characterize its quantum/classical complexity. For example, from the viewpoint of learning theory, sample complexity is a crucial concept. However, while the quantum sample complexity of the problem has been studied, a full characterization of the classical sample complexity of $\mathsf{HSP}$ seems to be absent, which will thus be the topic in this paper. $\mathsf{HSP}$ over a finite group is defined as follows: For a finite group $G$ and a finite set $V$, given a function $f:G \to V$ and the promise that for any $x, y \in G, f(x) = f(xy)$ iff $y \in H$ for a subgroup $H \in \mathcal{H}$, where $\mathcal{H}$ is a set of candidate subgroups of $G$, the goal is to identify $H$. Our contributions are as follows: For $\mathsf{HSP}$, we give the upper and lower bounds on the sample complexity of $\mathsf{HSP}$. Furthermore, we have applied the result to obtain the sample complexity of some concrete instances of hidden subgroup problem. Particularly, we discuss generalized Simon's problem ($\mathsf{GSP}$), a special case of $\mathsf{HSP}$, and show that the sample complexity of $\mathsf{GSP}$ is $\Theta\left(\max\left\{k,\sqrt{k\cdot p^{n-k}}\right\}\right)$. Thus we obtain a complete characterization of the sample complexity of $\mathsf{GSP}$.