量子和经典有序二元决策图的指数分离,重排序方法和层次

IF 1.7 4区 计算机科学 Q3 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE
Kamil Khadiev, Aliya Khadieva, Alexander Knop
{"title":"量子和经典有序二元决策图的指数分离,重排序方法和层次","authors":"Kamil Khadiev, Aliya Khadieva, Alexander Knop","doi":"10.1007/s11047-022-09904-3","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we study quantum Ordered Binary Decision Diagrams(<span>\\(\\mathrm {OBDD}\\)</span>) model; it is a restricted version of read-once quantum branching programs, with respect to “width” complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a new technique (“reordering”) for proving lower bounds and upper bounds for OBDD with an arbitrary order of input variables if we have similar bounds for the natural order. Using this transformation, we construct a total function <span>\\(\\mathrm {REQ}\\)</span> such that the deterministic <span>\\(\\mathrm {OBDD}\\)</span> complexity of it is at least <span>\\(2^{\\varOmega (n / \\log n)}\\)</span>, and the quantum <span>\\(\\mathrm {OBDD}\\)</span> complexity of it is at most <span>\\(O(n^2/\\log n)\\)</span>. It is the biggest known gap for explicit functions not representable by <span>\\(\\mathrm {OBDD}\\)</span>s of a linear width. Another function(shifted equality function) allows us to obtain a gap <span>\\(2^{\\varOmega (n)}\\)</span> vs <span>\\(O(n^2)\\)</span>. Moreover, we prove the bounded error quantum and probabilistic <span>\\(\\mathrm {OBDD}\\)</span> width hierarchies for complexity classes of Boolean functions. Additionally, using “reordering” method we extend a hierarchy for read-<i>k</i>-times Ordered Binary Decision Diagrams (<span>\\({\\textit{k}}\\text {-}\\mathrm {OBDD}\\)</span>) of polynomial width, for <span>\\(k = o(n / \\log ^3 n)\\)</span>. We prove a similar hierarchy for bounded error probabilistic <span>\\({\\textit{k}}\\text {-}\\mathrm {OBDD}\\)</span>s of polynomial, superpolynomial and subexponential width. The extended abstract of this work was presented on International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8 – 12, 2017 Khadiev and Khadieva (2017)</p>","PeriodicalId":49783,"journal":{"name":"Natural Computing","volume":"11 1","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Exponential separation between quantum and classical ordered binary decision diagrams, reordering method and hierarchies\",\"authors\":\"Kamil Khadiev, Aliya Khadieva, Alexander Knop\",\"doi\":\"10.1007/s11047-022-09904-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we study quantum Ordered Binary Decision Diagrams(<span>\\\\(\\\\mathrm {OBDD}\\\\)</span>) model; it is a restricted version of read-once quantum branching programs, with respect to “width” complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a new technique (“reordering”) for proving lower bounds and upper bounds for OBDD with an arbitrary order of input variables if we have similar bounds for the natural order. Using this transformation, we construct a total function <span>\\\\(\\\\mathrm {REQ}\\\\)</span> such that the deterministic <span>\\\\(\\\\mathrm {OBDD}\\\\)</span> complexity of it is at least <span>\\\\(2^{\\\\varOmega (n / \\\\log n)}\\\\)</span>, and the quantum <span>\\\\(\\\\mathrm {OBDD}\\\\)</span> complexity of it is at most <span>\\\\(O(n^2/\\\\log n)\\\\)</span>. It is the biggest known gap for explicit functions not representable by <span>\\\\(\\\\mathrm {OBDD}\\\\)</span>s of a linear width. Another function(shifted equality function) allows us to obtain a gap <span>\\\\(2^{\\\\varOmega (n)}\\\\)</span> vs <span>\\\\(O(n^2)\\\\)</span>. Moreover, we prove the bounded error quantum and probabilistic <span>\\\\(\\\\mathrm {OBDD}\\\\)</span> width hierarchies for complexity classes of Boolean functions. Additionally, using “reordering” method we extend a hierarchy for read-<i>k</i>-times Ordered Binary Decision Diagrams (<span>\\\\({\\\\textit{k}}\\\\text {-}\\\\mathrm {OBDD}\\\\)</span>) of polynomial width, for <span>\\\\(k = o(n / \\\\log ^3 n)\\\\)</span>. We prove a similar hierarchy for bounded error probabilistic <span>\\\\({\\\\textit{k}}\\\\text {-}\\\\mathrm {OBDD}\\\\)</span>s of polynomial, superpolynomial and subexponential width. The extended abstract of this work was presented on International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8 – 12, 2017 Khadiev and Khadieva (2017)</p>\",\"PeriodicalId\":49783,\"journal\":{\"name\":\"Natural Computing\",\"volume\":\"11 1\",\"pages\":\"\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2022-07-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Natural Computing\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1007/s11047-022-09904-3\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Natural Computing","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1007/s11047-022-09904-3","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE","Score":null,"Total":0}
引用次数: 4

摘要

本文研究了量子有序二元决策图(\(\mathrm {OBDD}\))模型;就“宽度”复杂度而言,它是只读一次量子分支程序的限制版本。已知确定性复杂度和量子复杂度之间的最大差距是指数级的。但是很少有函数有这样的差距。我们提出了一种新的技术(“重新排序”)来证明具有任意阶输入变量的OBDD的下界和上界,如果我们有相似的自然阶的边界。利用这种变换,我们构造了一个总函数\(\mathrm {REQ}\),使它的确定性\(\mathrm {OBDD}\)复杂度至少为\(2^{\varOmega (n / \log n)}\),量子\(\mathrm {OBDD}\)复杂度最多为\(O(n^2/\log n)\)。对于不能用线性宽度的\(\mathrm {OBDD}\) s表示的显式函数,这是已知的最大间隙。另一个函数(移位相等函数)允许我们获得\(2^{\varOmega (n)}\) vs \(O(n^2)\)的差距。此外,我们还证明了布尔函数复杂度类的有界误差量子和概率\(\mathrm {OBDD}\)宽度层次结构。此外,使用“重新排序”方法,我们扩展了一个层次结构,用于读取k次有序二进制决策图(\({\textit{k}}\text {-}\mathrm {OBDD}\))的多项式宽度,为\(k = o(n / \log ^3 n)\)。我们证明了多项式、上多项式和次指数宽度的有界误差概率\({\textit{k}}\text {-}\mathrm {OBDD}\) s的类似层次结构。这项工作的扩展摘要在俄罗斯国际计算机科学研讨会上发表,CSR 2017,喀山,俄罗斯,2017年6月8日至12日
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Exponential separation between quantum and classical ordered binary decision diagrams, reordering method and hierarchies

Exponential separation between quantum and classical ordered binary decision diagrams, reordering method and hierarchies

In this paper, we study quantum Ordered Binary Decision Diagrams(\(\mathrm {OBDD}\)) model; it is a restricted version of read-once quantum branching programs, with respect to “width” complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a new technique (“reordering”) for proving lower bounds and upper bounds for OBDD with an arbitrary order of input variables if we have similar bounds for the natural order. Using this transformation, we construct a total function \(\mathrm {REQ}\) such that the deterministic \(\mathrm {OBDD}\) complexity of it is at least \(2^{\varOmega (n / \log n)}\), and the quantum \(\mathrm {OBDD}\) complexity of it is at most \(O(n^2/\log n)\). It is the biggest known gap for explicit functions not representable by \(\mathrm {OBDD}\)s of a linear width. Another function(shifted equality function) allows us to obtain a gap \(2^{\varOmega (n)}\) vs \(O(n^2)\). Moreover, we prove the bounded error quantum and probabilistic \(\mathrm {OBDD}\) width hierarchies for complexity classes of Boolean functions. Additionally, using “reordering” method we extend a hierarchy for read-k-times Ordered Binary Decision Diagrams (\({\textit{k}}\text {-}\mathrm {OBDD}\)) of polynomial width, for \(k = o(n / \log ^3 n)\). We prove a similar hierarchy for bounded error probabilistic \({\textit{k}}\text {-}\mathrm {OBDD}\)s of polynomial, superpolynomial and subexponential width. The extended abstract of this work was presented on International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8 – 12, 2017 Khadiev and Khadieva (2017)

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
Natural Computing
Natural Computing Computer Science-Computer Science Applications
CiteScore
4.40
自引率
4.80%
发文量
49
审稿时长
3 months
期刊介绍: The journal is soliciting papers on all aspects of natural computing. Because of the interdisciplinary character of the journal a special effort will be made to solicit survey, review, and tutorial papers which would make research trends in a given subarea more accessible to the broad audience of the journal.
×
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