{"title":"全息算法的基础崩溃","authors":"Mingji Xia","doi":"10.1145/2897518.2897560","DOIUrl":null,"url":null,"abstract":"A holographic algorithm solves a problem in a domain of size n, by reducing it to counting perfect matchings in planar graphs. It may simulate a n-value variable by a bunch of t matchgate bits, which has 2t values. The transformation in the simulation can be expressed as a n × 2t matrix M, called the base of the holographic algorithm. We wonder whether more matchgate bits bring us more powerful holographic algorithms. In another word, whether we can solve the same original problem, with a collapsed base of size n × 2r, where r<t. Base collapse is discovered for small domain n=2,3,4. For n=3, 4, the base collapse is proved under the condition that there is a full rank generator. We prove for any n, the base collapse to a r≤ ⌊ logn ⌋, under some similar conditions. One of the conditions is that the original problem is defined by one symmetric function. In the proof, we utilize elementary matchgate transformations instead of matchgate identities.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"60 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2015-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"Base collapse of holographic algorithms\",\"authors\":\"Mingji Xia\",\"doi\":\"10.1145/2897518.2897560\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A holographic algorithm solves a problem in a domain of size n, by reducing it to counting perfect matchings in planar graphs. It may simulate a n-value variable by a bunch of t matchgate bits, which has 2t values. The transformation in the simulation can be expressed as a n × 2t matrix M, called the base of the holographic algorithm. We wonder whether more matchgate bits bring us more powerful holographic algorithms. In another word, whether we can solve the same original problem, with a collapsed base of size n × 2r, where r<t. Base collapse is discovered for small domain n=2,3,4. For n=3, 4, the base collapse is proved under the condition that there is a full rank generator. We prove for any n, the base collapse to a r≤ ⌊ logn ⌋, under some similar conditions. One of the conditions is that the original problem is defined by one symmetric function. In the proof, we utilize elementary matchgate transformations instead of matchgate identities.\",\"PeriodicalId\":442965,\"journal\":{\"name\":\"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing\",\"volume\":\"60 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2015-11-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/2897518.2897560\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/2897518.2897560","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A holographic algorithm solves a problem in a domain of size n, by reducing it to counting perfect matchings in planar graphs. It may simulate a n-value variable by a bunch of t matchgate bits, which has 2t values. The transformation in the simulation can be expressed as a n × 2t matrix M, called the base of the holographic algorithm. We wonder whether more matchgate bits bring us more powerful holographic algorithms. In another word, whether we can solve the same original problem, with a collapsed base of size n × 2r, where r
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