排列的量子复杂性

IF 0.5 4区 数学 Q3 MATHEMATICS
Andrew Yu
{"title":"排列的量子复杂性","authors":"Andrew Yu","doi":"10.4310/pamq.2023.v19.n2.a6","DOIUrl":null,"url":null,"abstract":"Let $S_n$ be the symmetric group of all permutations of $\\{1, \\cdots, n\\}$ with two generators: the transposition switching $1$ with $2$ and the cyclic permutation sending $k$ to $k+1$ for $1\\leq k\\leq n-1$ and $n$ to $1$ (denoted by $\\sigma$ and $\\tau$). In this article, we study quantum complexity of permutations in $S_n$ using $\\{\\sigma, \\tau, \\tau^{-1}\\}$ as logic gates. We give an explicit construction of permutations in $S_n$ with quadratic quantum complexity lower bound $\\frac{n^2-2n-7}{4}$. We also prove that all permutations in $S_n$ have quadratic quantum complexity upper bound $3(n-1)^2$. Finally, we show that almost all permutations in $S_n$ have quadratic quantum complexity lower bound when $n\\rightarrow \\infty$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2022-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Quantum complexity of permutations\",\"authors\":\"Andrew Yu\",\"doi\":\"10.4310/pamq.2023.v19.n2.a6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $S_n$ be the symmetric group of all permutations of $\\\\{1, \\\\cdots, n\\\\}$ with two generators: the transposition switching $1$ with $2$ and the cyclic permutation sending $k$ to $k+1$ for $1\\\\leq k\\\\leq n-1$ and $n$ to $1$ (denoted by $\\\\sigma$ and $\\\\tau$). In this article, we study quantum complexity of permutations in $S_n$ using $\\\\{\\\\sigma, \\\\tau, \\\\tau^{-1}\\\\}$ as logic gates. We give an explicit construction of permutations in $S_n$ with quadratic quantum complexity lower bound $\\\\frac{n^2-2n-7}{4}$. We also prove that all permutations in $S_n$ have quadratic quantum complexity upper bound $3(n-1)^2$. Finally, we show that almost all permutations in $S_n$ have quadratic quantum complexity lower bound when $n\\\\rightarrow \\\\infty$.\",\"PeriodicalId\":54526,\"journal\":{\"name\":\"Pure and Applied Mathematics Quarterly\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2022-07-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Pure and Applied Mathematics Quarterly\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/pamq.2023.v19.n2.a6\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Pure and Applied Mathematics Quarterly","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/pamq.2023.v19.n2.a6","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

让 $S_n$ 的所有排列的对称群 $\{1, \cdots, n\}$ 用两个发电机:换位开关 $1$ 有 $2$ 循环排列发送 $k$ 到 $k+1$ 为了 $1\leq k\leq n-1$ 和 $n$ 到 $1$ (表示为 $\sigma$ 和 $\tau$). 在本文中,我们研究了中排列的量子复杂性 $S_n$ 使用 $\{\sigma, \tau, \tau^{-1}\}$ 作为逻辑门。中置换的一个显式构造 $S_n$ 具有二次量子复杂度下界 $\frac{n^2-2n-7}{4}$。我们也证明了所有的排列 $S_n$ 二次量子复杂度有上界吗 $3(n-1)^2$。最后,我们证明了 $S_n$ 二次量子复杂度的下限是什么时候 $n\rightarrow \infty$.
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Quantum complexity of permutations
Let $S_n$ be the symmetric group of all permutations of $\{1, \cdots, n\}$ with two generators: the transposition switching $1$ with $2$ and the cyclic permutation sending $k$ to $k+1$ for $1\leq k\leq n-1$ and $n$ to $1$ (denoted by $\sigma$ and $\tau$). In this article, we study quantum complexity of permutations in $S_n$ using $\{\sigma, \tau, \tau^{-1}\}$ as logic gates. We give an explicit construction of permutations in $S_n$ with quadratic quantum complexity lower bound $\frac{n^2-2n-7}{4}$. We also prove that all permutations in $S_n$ have quadratic quantum complexity upper bound $3(n-1)^2$. Finally, we show that almost all permutations in $S_n$ have quadratic quantum complexity lower bound when $n\rightarrow \infty$.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
0.90
自引率
0.00%
发文量
30
审稿时长
>12 weeks
期刊介绍: Publishes high-quality, original papers on all fields of mathematics. To facilitate fruitful interchanges between mathematicians from different regions and specialties, and to effectively disseminate new breakthroughs in mathematics, the journal welcomes well-written submissions from all significant areas of mathematics. The editors are committed to promoting the highest quality of mathematical scholarship.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信