Quantum Pattern Matching Fast on Average
Quantum Pattern Matching Fast on Average
复制标题
量子模式匹配平均速度快
作者：
A. Montanaro
D维模式匹配问题是在长度为n×⋯×nDocumentClass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amssymb}usepackage{amsbsy}usepackage{amsbsy}usepackage{upgreek}setlong{oddsidemarin}{-69pt}例如{Document}$$m imes imes m$$end{DocumentClass[12pt]{Minimum}usepackage{amsackage{wa syysym}usepack{amsackage{amsbsy}usepackage{amsssy}usepackage{madsidemarin}(例如{Document}$$m imes in m$$End{DocumentClass[12pt]{Minimum}usepackage{amsackage{wa syysym}usepackage{amsbsy}usesackage{amssidb}usepackage{mamssidememin}中{-69pt}例如{文档}$$n个点，n$$个结尾{文档}，With n≥mDocumentclass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$$n ge m$$End{Document}。此任务模拟文本和图像处理以及其他应用领域中的各种问题。描述了一种量子算法，该算法解决了Time O~((n/m)d/22O(d3/2logm))documentclass[12pt]{minimal}中随机模式和文本的模式匹配问题。该算法解决了随机模式和文本的模式匹配问题。对于较大的m，这比最好的经典算法要快超多项式，后者需要时间Ω~(nd/2+(n/m)d)Documentclass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如in{Document}$$widetilde{Omega}(n^{d/2}+(n/m)^d)$$end{Document}。该算法基于使用量子子例程来寻找d维的隐藏移位，这是Kuperberg提出的算法的变体。
The d-dimensional pattern matching problem is to find an occurrence of a pattern of length m×⋯×mdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$m imes dots imes m$$end{document} within a text of length n×⋯×ndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$n imes dots imes n$$end{document}, with n≥mdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$n ge m$$end{document}. This task models various problems in text and image processing, among other application areas. This work describes a quantum algorithm which solves the pattern matching problem for random patterns and texts in time O~((n/m)d/22O(d3/2logm))documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$widetilde{O}((n/m)^{d/2} 2^{O(d^{3/2}sqrt{log m})})$$end{document}. For large m this is super-polynomially faster than the best possible classical algorithm, which requires time Ω~(nd/2+(n/m)d)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$widetilde{Omega }( n^{d/2} + (n/m)^d)$$end{document}. The algorithm is based on the use of a quantum subroutine for finding hidden shifts in d dimensions, which is a variant of algorithms proposed by Kuperberg.