Exact distributed quantum algorithm for generalized Simon’s problem

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Acta Informatica Pub Date : 2024-03-10 DOI:10.1007/s00236-024-00455-x
Hao Li, Daowen Qiu, Le Luo, Paulo Mateus
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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.

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广义西蒙问题的精确分布式量子算法
西蒙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延伸,也是一个特殊的隐藏子群问题:给定一个函数 \(f:\秩为k的隐藏子群(Sle \mathbb {Z}_2^n\) ,这样对于任意的\(x, y\in {0, 1\}^n\), \(f(x) = f(y)\) iff \(x \oplus y \in S\).广义西蒙问题的目标是找到隐藏子群 S。首先,我们描述了分布式场景下广义西蒙问题的结构,并引入了相应的分布式量子算法。其次,我们对算法进行了改进,以确保量子振幅放大技术的精确性。与分布式经典算法相比,我们的算法具有指数级的速度提升。与广义西蒙问题的量子算法相比,我们算法的神谕所需的量子比特更少,因此更易于物理实现。特别是,我们为广义西蒙问题开发的精确分布式量子算法在普适性和精确性方面都优于之前为西蒙问题提出的最佳分布式量子算法。
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来源期刊
Acta Informatica
Acta Informatica 工程技术-计算机:信息系统
CiteScore
2.40
自引率
16.70%
发文量
24
审稿时长
>12 weeks
期刊介绍: Acta Informatica provides international dissemination of articles on formal methods for the design and analysis of programs, computing systems and information structures, as well as related fields of Theoretical Computer Science such as Automata Theory, Logic in Computer Science, and Algorithmics. Topics of interest include: • semantics of programming languages • models and modeling languages for concurrent, distributed, reactive and mobile systems • models and modeling languages for timed, hybrid and probabilistic systems • specification, program analysis and verification • model checking and theorem proving • modal, temporal, first- and higher-order logics, and their variants • constraint logic, SAT/SMT-solving techniques • theoretical aspects of databases, semi-structured data and finite model theory • theoretical aspects of artificial intelligence, knowledge representation, description logic • automata theory, formal languages, term and graph rewriting • game-based models, synthesis • type theory, typed calculi • algebraic, coalgebraic and categorical methods • formal aspects of performance, dependability and reliability analysis • foundations of information and network security • parallel, distributed and randomized algorithms • design and analysis of algorithms • foundations of network and communication protocols.
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