Strong direct product theorems for quantum communication and query complexity

Alexander A. Sherstov
2011 Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11  
A strong direct product theorem (SDPT) states that solving n instances of a problem requires Ω(n) times the resources for a single instance, even to achieve success probability 2 − n for a small enough constant > 0. We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by the generalized discrepancy method, the strongest technique in that model. We prove that quantum query complexity obeys an SDPT whenever the query
more » ... henever the query lower bound for a single instance is proved by the polynomial method, one of the two main techniques in that model. In both models, we prove the corresponding XOR lemmas and threshold direct product theorems. ). ALEXANDER A. SHERSTOV The models of interest to us in this paper are quantum communication complexity and quantum query complexity, where the direct product phenomenon is understood quite poorly. Furthermore, work here has advanced much more slowly than in the classical case, a point conveyed by the following overview of the classical and quantum literature. Classical communication and query complexity. The direct sum problem in communication complexity was raised for the first time in the work of Karchmer, Raz, and Wigderson [30], who showed that its resolution for relations would prove that NC 1 is properly contained in NC 2 . Feder, Kushilevitz, Naor, and Nisan [20] established a direct sum theorem for nondeterministic communication complexity and inferred a weaker result for deterministic communication. Information-theoretic methods have enabled substantial progress [16, 7, 26, 27, 23, 8] on the direct sum question in the randomized model and its restrictions, including one-way communication and simultaneous message passing. In what generality randomized communication complexity obeys a direct sum theorem remains unknown; some counterexamples have been discovered for a careful choice of parameters [20] . It also remains unknown whether randomized communication complexity in general obeys a strong direct product theorem. A variety of results have been established, however, for concrete functions and some restrictions of the randomized model. Parnafes, Raz, and Wigderson [44] proved the first result of the kind, for "forests" of communication protocols. Shaltiel [49] proved an XOR lemma for uniform-distribution discrepancy, a well-studied communication complexity measure. Shaltiel's result has been generalized and strengthened in several ways [31, 10, 56, 40] . Jain, Klauck, and Nayak [24] obtained strong direct product theorems for an information-theoretic complexity measure called the subdistribution bound. Most recently, Klauck [33] proved the long-conjectured strong direct product theorem for the randomized communication complexity of the disjointness function. In classical query complexity, the direct product phenomenon is well understood. Strong direct product theorems have been obtained for "decision forests" by Nisan, Rudich, and Saks [43], for "fair" decision trees by Shaltiel [49], and for the randomized query complexity of symmetric functions by Klauck,Špalek, and de Wolf [34]. Very recently, Drucker [18] obtained strong direct product theorems for the randomized query complexity of arbitrary functions. Quantum communication and query complexity. Klauck,Špalek, and de Wolf [34] proved a strong direct product theorem for the quantum communication complexity of the disjointness function. Shaltiel [49] and Lee, Shraibman, andŠpalek [40] obtained an XOR lemma for correlation with low-cost protocols, which gives a strong direct product theorem for certain communication problems such as Hadamard matrices or random matrices. The only other results known to us are for the restricted models of one-way communication and simultaneous message passing [27, 12, 23] . The results for quantum query complexity are just as few in number. The first direct product result is due to Aaronson [1], who proved it for the problem of k-fold search. Aaronson's result was improved to optimal with respect to all parameters by Klauck,Špalek, and de Wolf [34], who established a strong direct product theorem for the quantum query complexity of the OR function. In follow-up work, Ambainis,Špalek, and de Wolf [5] obtained a strong direct product theorem for all other symmetric functions. The only other result prior to our paper is due toŠpalek [55], who developed a multiplicative adversary method for quantum query complexity and proved that that method obeys a strong direct product theorem. ALEXANDER A. SHERSTOV
doi:10.1145/1993636.1993643 dblp:conf/stoc/Sherstov11 fatcat:jipoiuuymjblvmndmjd4exxyyu