A. Ambainis, Andrew M. Childs, B. Reichardt, R. Spalek, Shengyu Zhang
{"title":"在量子计算机上，任何大小为N的与或公式都可以在N^{1/2 + o(1)}时间内求值","authors":"A. Ambainis, Andrew M. Childs, B. Reichardt, R. Spalek, Shengyu Zhang","doi":"10.1137/080712167","DOIUrl":null,"url":null,"abstract":"For any AND-OR formula of size N, there exists a bounded-error N1/2+o(1)-time quantum algorithm, based on a discrete-time quantum walk, that evaluates this formula on a black-box input. Balanced, or \"approximately balanced,\" formulas can be evaluated in O(radicN) queries, which is optimal. It follows that the (2-o(1))th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"203","resultStr":"{\"title\":\"Any AND-OR Formula of Size N can be Evaluated in time N^{1/2 + o(1)} on a Quantum Computer\",\"authors\":\"A. Ambainis, Andrew M. Childs, B. Reichardt, R. Spalek, Shengyu Zhang\",\"doi\":\"10.1137/080712167\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For any AND-OR formula of size N, there exists a bounded-error N1/2+o(1)-time quantum algorithm, based on a discrete-time quantum walk, that evaluates this formula on a black-box input. Balanced, or \\\"approximately balanced,\\\" formulas can be evaluated in O(radicN) queries, which is optimal. It follows that the (2-o(1))th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.\",\"PeriodicalId\":197431,\"journal\":{\"name\":\"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2007-10-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"203\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1137/080712167\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/080712167","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Any AND-OR Formula of Size N can be Evaluated in time N^{1/2 + o(1)} on a Quantum Computer
For any AND-OR formula of size N, there exists a bounded-error N1/2+o(1)-time quantum algorithm, based on a discrete-time quantum walk, that evaluates this formula on a black-box input. Balanced, or "approximately balanced," formulas can be evaluated in O(radicN) queries, which is optimal. It follows that the (2-o(1))th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.
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