Heger Arfaoui
Abstract
The main objective of this paper is to provide illustrative examples of distributed computing problems for which it is possible to design tight lower bounds for quantum algorithms without having to manipulate concepts from quantum mechanics, at all. As a case study, we address the following class of 2-player problems. Alice (resp., Bob) receives a boolean x (resp., y) as input, and must return a boolean a (resp., b) as output. A game between Alice and Bob is defined by a pair (?, f) of boolean functions. The objective of Alice and Bob playing game (?, f) is, for every pair (x, y) of inputs, to output values a and b, respectively, satisfying ?(a, b) = f(x, y), in absence of any communication between the two players, but in presence of shared resources. The ability of the two players to solve the game then depends on the type of resources they share. It is known that, for the so-called CHSH game, i.e., for the game a ? b = x ? y, the ability for the players to use entangled quantum bits (qubits) helps. We show that, apart from the CHSH game, quantum correlations do not help, in the sense that, for every game not equivalent to the CHSH game, there exists a classical protocol (using shared randomness) whose probability of success is at least as large as the one of any protocol using quantum resources. This result holds for both worst case and average case analysis. It is achieved by considering a model stronger than quantum correlations, the non-signaling model, which subsumes quantum mechanics, but is far easier to handle.
- N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms 7(4):567--583, 1986. Google Scholar
Digital Library
- H. Arfaoui and P. Fraigniaud. What can be computed without communications? In proc. 19th Int. Colloquium on Structural Information and Communication Complexity (SIROCCO), pages 135--146, 2012. Google ScholarDigital Library
- H. Arfaoui, P. Fraigniaud, and A. Pelc. Local Decision and Verification with Bounded--Size Outputs. In proc. 15th Int. Symp. on Stabilization, Safety, and Security of Distributed Systems (SSS), 2013.Google Scholar
- L. Barenboim and M. Elkin. Distributed (? + 1)-coloring in linear (in delta) time. In Proc. 41st ACM Symp. on Theory of computing (STOC), pages 111--120, 2009. Google ScholarDigital Library
- L. Barenboim, M. Elkin, S. Pettie, and J. Schneider. The Locality of Distributed Symmetry Breaking. In Proc. 53rd IEEE Symp. on Foundations of Computer Science (FOCS), pages 321--330, 2012. Google ScholarDigital Library
- J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, and D. Roberts. Nonlocal correlations as an information-theoretic resource. Physical Review A 71(2):1--11, 2005.Google ScholarCross Ref
- J. Barrett and S. Pironio. Popescu--Rohrlich correlations as a unit of nonlocality. Phys. Rev. Lett. 95(14), 2005.Google ScholarCross Ref
- J. S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics 1(3):195--200, 1964.Google ScholarCross Ref
- G. Brassard, A. Broadbent, and A. Tapp. Quantum pseudo-telepathy. Foundations of Physics 5:18771907, 2005.Google Scholar
- A. Broadbent, A. Tapp. Can quantum mechanics help distributed computing? SIGACT News 39(3): 67--76 (2008) Google ScholarDigital Library
- H. Buhrman, R. Cleve, S. Massar, and R. deWolf. Non-locality and communication complexity. Reviews of Modern Physics 82:665--698, 2010.Google ScholarCross Ref
- H. Buhrman and H. Röhrig. Distributed Quantum Computing. In proc 28th International Symposium on Mathematical Foundations of Computer Science (MFCS), LNCS 2747, pp. 120, 2003.Google Scholar
- B. S. Cirel'son. Quantum generalizations of bell's inequality. Letters in Math. Phys. 4(2):93--100, 1980.Google ScholarCross Ref
- J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. Proposed experiment to test local Hidden-variable theories. Physical Review Letters 23(15):880--884, 1969.Google ScholarCross Ref
- W. van Dam. Implausible Consequences of Superstrong Nonlocality. Natural Computing 12(1): 9--12 (2013) Google ScholarDigital Library
- V. Denchev and G. Pandurangan. Distributed quantum computing: a new frontier in distributed systems or science fiction? SIGACT News 39(3): 77--95 (2008) Google ScholarDigital Library
- B. Derbel, C. Gavoille, D. Peleg, and L. Viennot. On the locality of distributed sparse spanner construction. In proc. 27th ACM Symp. on Principles of Distributed Computing (PODC), pages 273--282, 2008. Google ScholarDigital Library
- F. Dupuis, N. Gisin, A. Hasidim, A. Allan Méthot, and H. Pilpel. No nonlocal box is universal. J. Math. Phys. 48(082107), 2007.Google Scholar
- A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Physical Review, 47(10):777--780, 1935.Google ScholarCross Ref
- M. Elkin. A near-optimal fully dynamic distributed algorithm for maintaining sparse spanners. In proc. 26th ACM Symp. on Principles of Distributed Computing (PODC), pages 195--204, 2007. Google ScholarDigital Library
- M. Elkin, H. Klauck, D. Nanongkai, and G. Pandurangan. Quantum Lower Bounds for Distributed Network Computing. Tech. Report arXiv:1207.5211 (2013)Google Scholar
- P. Fraigniaud, A. Korman, and D. Peleg. Local distributed decision. In proc. 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp 708--717, 2011. Google ScholarDigital Library
- P. Fraigniaud, S. Rajsbaum, and C. Travers. Locality and checkability in wait-free computing. 25th International Symposium on Distributed Computing (DISC), pp 333--347, 2011. Google ScholarDigital Library
- P. Fraigniaud, S. Rajsbaum, and C. Travers. An Impossibility Result for Run-Time Monitoring. Submitted, 2013.Google Scholar
- S. J. Freedman and J. F. Clauser Experimental Test of Local Hidden-Variable Theories. Phys. Rev. Lett. 28, 938--941 (1972)Google ScholarCross Ref
- C. Gavoille, R. Klasing, A. Kosowski, L. Kuszner, and A. Navarra. On the complexity of distributed graph coloring with local minimality constraints. Networks 54(1): 12--19 (2009) Google ScholarDigital Library
- C. Gavoille, A. Kosowski, and M. Markiewicz. What Can Be Observed Locally? In proc. 23rd Int. Symposium on Distributed Computing (DISC), pages 243--257, 2009. Google ScholarDigital Library
- Ghirardi, G. C. and Rimini, A. and Weber, T. A general argument against superluminal transmission through the quantum mechanical measurement process Lett. Nuovo Cimento 27:293--298, 1980.Google Scholar
- A. Korman, S. Kutten, and D. Peleg. Proof labeling schemes. Distributed Computing 22, (2010), 215--233.Google ScholarDigital Library
- F. Kuhn. Weak graph colorings: distributed algorithms and applications. In Proc. 21st ACM Symp. on Parallel Algorithms and Architectures (SPAA), pages 138--144, 2009. Google ScholarDigital Library
- F. Kuhn, T. Moscibroda, and R. Wattenhofer. What cannot be computed locally! In proc 23rd ACM Symp. on Principles of Distributed Computing (PODC), pages 300--309, 2004. Google ScholarDigital Library
- N. Linial. Locality in Distributed Graph Algorithms. SIAM J. Comput. 21(1): 193--201 (1992) Google ScholarDigital Library
- M. Luby. A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15:1036--1053 (1986). Google ScholarDigital Library
- M. Naor and L. Stockmeyer. What can be computed locally? SIAM J. Comput. 24(6): 1259--1277 (1995). Google ScholarDigital Library
- A. Panconesi and A. Srinivasan. On the Complexity of Distributed Network Decomposition. J. Algorithms 20(2): 356--374 (1996). Google ScholarDigital Library
- D. Peleg. Distributed computing: A locality-sensitive approach. SIAM, 2000. Google ScholarDigital Library
- S. Popescu and D. Rohrlich. Quantum nonlocality as an axiom. Foundations of Physics 24(3):379--385, 1994.Google ScholarCross Ref
- J. Schneider and R. Wattenhofer. A new technique for distributed symmetry breaking. In Proc. 29th ACM Symp. on Principles of Distributed Computing (PODC), 257--266, 2010. Google ScholarDigital Library
Index Terms
- What can be computed without communications?
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