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.

西蒙的问题是展示量子算法威力的最重要问题之一，因为它极大地启发了肖尔算法的提出。广义西蒙问题是西蒙问题的自然扩展，也是一个特殊的隐藏子群问题：给定一个函数\(f:\{0,1\}^n \rightarrow \{0,1\}^m\)，它承诺存在一个秩为k的隐藏子群\(S\le \mathbb {Z}_2^n\)，使得对于任何\(x, y\in {\{0, 1\}}^n\) , \(f(x) = f(y)\) iff \(x \oplus y \in S\)。广义西蒙问题的目标是找到隐藏子群S。在本文中，我们提出了两个关键贡献。首先，我们描述了分布式场景下广义西蒙问题的结构，并介绍了相应的分布式量子算法。其次，由于量子振幅放大技术的应用，我们改进了算法以确保准确性。与分布式经典算法相比，我们的算法提供了指数级的加速。与广义西蒙问题的量子算法相比，我们算法的预言机需要更少的量子位，从而更容易在物理上实现。特别是，我们针对广义西蒙问题开发的精确分布式量子算法在泛化性和准确性方面优于先前提出的针对西蒙问题的最佳分布式量子算法。