Quantum and Stochastic Branching Programs of Bounded Width
We prove upper and lower bounds on the power of quantum and stochastic branching programs of bounded width. We show any NC1 language can be accepted exactly by a width-2 quantum branching program of polynomial length, in contrast to the classical case where width 5 is necessary unless NC1 = ACC. This separates width-2 quantum programs from width-2 doubly stochastic programs as we show the latter cannot compute the middle bit of multiplication. Finally, we show that bounded-width quantum and stochastic programs can be simulated by classical programs of larger but bounded width, and thus are in NC1.
KeywordsBoolean Function Sink Node Stochastic Program Binary Decision Diagram Stochastic Matrice
Unable to display preview. Download preview PDF.
- 1.F. Ablayev, A. Gainutdinova, and M. Karpinski. On computational Power of quantum branching programs. Proc. FCT 2001, Lecture Notes in Computer Science 2138: 59–70, 2001.Google Scholar
- 2.F. Ablayev and M. Karpinski. A lower bound for integer multiplication on randomized read-once branching programs. Electronic Colloquium on Computational Complexity TR 98-011, 1998. http://www.eccc.uni-trier.de/eccc
- 3.P. Alexandrov. Introduction to set theory and general topology. Berlin, 1984.Google Scholar
- 4.A. Ambainis and R. Freivalds. 1-way quantum finite automata: strengths, weakness, and generalizations. Proc. 39th IEEE Symp. on Foundations of Computer Science (FOCS), 332–342, 1998.Google Scholar
- 5.A. Ambainis, L. Schulman, and U. Vazirani. Computing with Highly Mixed States. Proc. 32nd Annual ACM Symp. on Theory of Computing (STOC), 697–704, 2000.Google Scholar
- 8.A. Kondacs and J. Watrous On the power of quantum finite automata. Proc. of the 38th IEEE Symp. on Foundations of Computer Science (FOCS), 66–75, 1997.Google Scholar
- 11.M. Nakanishi, K. Hamaguchi, and T. Kashiwabara. Ordered quantum branching programs are more powerful than ordered probabilistic branching programs under a bounded-width restriction. Proc. 6th Annual International Conference on Computing and Combinatorics (COCOON) Lecture Notes in Computer Science 1858: 467–476, 2000.Google Scholar
- 12.M.A. Nielson and I.L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press. 2000.Google Scholar
- 15.Ingo Wegener. Branching Programs and Binary Decision Diagrams. SIAM Monographs on Discrete Mathematics and Applications. 2000.Google Scholar
- 16.A.C. Yao. Lower Bounds by Probabilistic Arguments Proc. of the 24th IEEE Symp. on Foundations of Computer Science (FOCS), 420–428, 1983.Google Scholar