{"title":"多项式的通信复杂度下界","authors":"H. Buhrman, R. D. Wolf","doi":"10.1109/CCC.2001.933879","DOIUrl":null,"url":null,"abstract":"The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication complexity, but except for the inner product function, no bounds are known for the model with unlimited prior entanglement. We show that the \"log rank\" lower bound extends to the strongest variant of quantum communication complexity (qubit communication+unlimited prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the \"log rank conjecture\" and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for bounded-error quantum protocols.","PeriodicalId":240268,"journal":{"name":"Proceedings 16th Annual IEEE Conference on Computational Complexity","volume":"13 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1999-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"134","resultStr":"{\"title\":\"Communication complexity lower bounds by polynomials\",\"authors\":\"H. Buhrman, R. D. Wolf\",\"doi\":\"10.1109/CCC.2001.933879\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication complexity, but except for the inner product function, no bounds are known for the model with unlimited prior entanglement. We show that the \\\"log rank\\\" lower bound extends to the strongest variant of quantum communication complexity (qubit communication+unlimited prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the \\\"log rank conjecture\\\" and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for bounded-error quantum protocols.\",\"PeriodicalId\":240268,\"journal\":{\"name\":\"Proceedings 16th Annual IEEE Conference on Computational Complexity\",\"volume\":\"13 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1999-10-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"134\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings 16th Annual IEEE Conference on Computational Complexity\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/CCC.2001.933879\",\"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 16th Annual IEEE Conference on Computational Complexity","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/CCC.2001.933879","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Communication complexity lower bounds by polynomials
The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication complexity, but except for the inner product function, no bounds are known for the model with unlimited prior entanglement. We show that the "log rank" lower bound extends to the strongest variant of quantum communication complexity (qubit communication+unlimited prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the "log rank conjecture" and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for bounded-error quantum protocols.