The Complexity of Quantum Disjointness
We introduce the communication problem QNDISJ, short for Quantum (Unique) Non-Disjointness, and study its complexity under different modes of communication complexity. The main motivation for the problem is that it is a candidate for the separation of the quantum communication complexity classes QMA and QCMA. The problem generalizes the Vector-in-Subspace and Non-Disjointness problems. We give tight bounds for the QMA, quantum, randomized communication complexities of the problem. We show polynomially related upper and lower bounds for the MA complexity. We also show an upper bound for QCMA protocols, and show that the bound is tight for a natural class of QCMA protocols for the problem. The latter lower bound is based on a geometric lemma, that states that every subset of the n-dimensional sphere of measure 2^-p must contain an ortho-normal set of points of size Omega(n/p).
We also study a "small-spaces" version of the problem, and give upper and lower bounds for its randomized complexity that show that the QNDISJ problem is harder than Non-disjointness for randomized protocols. Interestingly, for quantum modes the complexity depends only on the dimension of the smaller space, whereas for classical modes the dimension of the larger space matters.
Communication Complexity
Quantum Proof Systems
15:1-15:13
Regular Paper
Hartmut
Klauck
Hartmut Klauck
10.4230/LIPIcs.MFCS.2017.15
S. Aaronson. Qma/qpoly ⊆ pspace/poly: De-merlinizing quantum protocols. In Proceedings of 21st IEEE Conference on Computational Complexity, 2006.
S. Aaronson and A. Ambainis. Quantum search of spatial regions. In Proceedings of 44th IEEE FOCS, pages 200-209, 2003.
S. Aaronson and G. Kuperberg. Quantum versus classical proofs and advice. Theory of Computing, 3(1):129-157, 2007.
S. Aaronson and A. Wigderson. Algebrization: A New Barrier in Complexity Theory. ACM Transactions on Computation Theory, 1(1), 2009.
D. Aharonov and T. Naveh. Quantum np - a survey. quant-ph/0210077, 2002.
L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In Proceedings of 27th IEEE FOCS, pages 337-347, 1986.
Z. Bar-Yossef, T. S. Jayram, R. Kumar, and D. Sivakumar. Information theory methods in communication complexity. In Proceedings of 17th IEEE Conference on Computational Complexity, pages 93-102, 2002.
G. Brassard, P. Høyer, M. Mosca, and A. Tapp. Quantum amplitude amplification and estimation. In Quantum Computation and Quantum Information: A Millennium Volume, volume 305 of AMS Contemporary Mathematics Series, pages 53-74. AMS, 2002. quant-ph/0005055.
M. Braverman, A. Garg, Young K.K., J. Mao, and D. Touchette. Near-optimal bounds on bounded-round quantum communication complexity of disjointness. In IEEE 56th Annual Symposium on Foundations of Computer Science, pages 773-791, 2015.
A. Chakrabarti, G. Cormode, A. McGregor, J. Thaler, and S. Venkatasubramanian. Verifiable stream computation and arthur-merlin communication. In 30th Conference on Computational Complexity, pages 217-243, 2015.
S. Dasgupta and A. Gupta. An elementary proof of a theorem of johnson and lindenstrauss. Random Structures &Algorithms, 22(1):60-65, 2003.
M. Göös, T. Pitassi, and T. Watson. Zero-information protocols and unambiguity in arthur-merlin communication. Algorithmica, 76(3):684-719, 2016.
L. K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of 28th ACM STOC, pages 212-219, 1996.
B. Kalyanasundaram and G. Schnitger. The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics, 5(4):545-557, 1992. Earlier version in Structures'87.
A. Yu. Kitaev. Quantum NP, January 1999. Talk given at AQIP'99, DePaul University, Chicago.
B. Klartag and O. Regev. Quantum one-way communication is exponentially stronger than classical communication. In Proceedings of 43rd ACM STOC, 2011.
H. Klauck. On quantum and probabilistic communication: Las Vegas and one-way protocols. In Proceedings of 32nd ACM STOC, pages 644-651, 2000.
H. Klauck. Rectangle size bounds and threshold covers in communication complexity. In 18th Annual IEEE Conference on Computational Complexity, pages 118-134, 2003.
H. Klauck. A strong direct product theorem for disjointness. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC, pages 77-86, 2010.
H. Klauck. On arthur merlin games in communication complexity. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity, pages 189-199, 2011.
H. Klauck and S. Podder. Two Results about Quantum Messages. In Proceedings of MFCS, 2014.
I. Kremer. Quantum communication. Master’s thesis, Hebrew University, Computer Science Department, 1995.
E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge University Press, 1997.
R. Raz. Exponential separation of quantum and classical communication complexity. In Proceedings of 31st ACM STOC, pages 358-367, 1999.
R. Raz and A. Shpilka. On the power of quantum proofs. In 19th Annual IEEE Conference on Computational Complexity, pages 260-274, 2004.
A. Razborov. On the distributional complexity of disjointness. Theoretical Computer Science, 106(2):385-390, 1992.
A. Razborov. Quantum communication complexity of symmetric predicates. Izvestiya of the Russian Academy of Sciences, mathematics, 67(1):159-176, 2003. quant-ph/0204025.
A. Sherstov. The pattern matrix method for lower bounds on quantum communication. In Proceedings of 40th ACM STOC, pages 85-94, 2008.
R. de Wolf. Quantum communication and complexity. Theoretical Computer Science, 287(1):337-353, 2002.
A. C-C. Yao. Some complexity questions related to distributive computing. In Proceedings of 11th ACM STOC, pages 209-213, 1979.
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