Exact distributed quantum algorithm for generalized Simon’s problem

Abstract

Simon’s problem is one of the most important problems demonstrating the power of quantum algorithms, as it greatly inspired the proposal of Shor’s algorithm. The generalized Simon’s problem is a natural extension of Simon’s problem and also a special hidden subgroup problem: Given a function \(f:\{0,1\}^n \rightarrow \{0,1\}^m\), it is promised that there exists a hidden subgroup \(S\le \mathbb {Z}_2^n\) of rank k such that for any \(x, y\in {\{0, 1\}}^n\), \(f(x) = f(y)\) iff \(x \oplus y \in S\). The goal of generalized Simon’s problem is to find the hidden subgroup S. In this paper, we present two key contributions. Firstly, we characterize the structure of the generalized Simon’s problem in distributed scenario and introduce a corresponding distributed quantum algorithm. Secondly, we refine the algorithm to ensure exactness due to the application of quantum amplitude amplification technique. Our algorithm offers exponential speedup compared to the distributed classical algorithm. When contrasted with the quantum algorithm for the generalized Simon’s problem, our algorithm’s oracle requires fewer qubits, thus making it easier to be physically implemented. Particularly, the exact distributed quantum algorithm we develop for the generalized Simon’s problem outperforms the best previously proposed distributed quantum algorithm for Simon’s problem in terms of generalizability and exactness.