On the computational power of probabilistic and quantum branching program

In this paper we show that one-qubit polynomial time computations are as powerful as NC circuits. More generally, we define syntactic models for quantum and stochastic branching programs of bounded width and prove upper and lower bounds on their power. We show that any NC 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 NC = 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 NC. For read-once quantum branching programs (QBPs), we give a symmetric Boolean function which is computable by a read-once QBP with O(log n) width, but not by a deterministic read-once BP with o(n) width, or by a classical randomized read-once BP with o(n) width which is “stable” in the sense that its transitions depend on the value of the queried variable but do not vary from step to step. Finally, we present a general lower bound on the width of read-once QBPs, showing that our O(log n) upper bound for this symmetric function is almost tight. ∗Dept. of Theoretical Cybernetics, Kazan State University. [email protected] ; work done in part while visiting Institute of Advanced Study and Max-Planck Institute for Mathematics, supported in part by Russia Fund for Basic Research grant 03-01-00769 †Dept. of Theoretical Cybernetics, Kazan State University. [email protected]; supported in part by Russia Fund for Basic Research grant 03-01-00769 ‡Dept. of Computer Science, University of Bonn. [email protected]; supported in part by DFG grants, and IST grant 14036 (RAND-APX) §Computer Science Department, University of New Mexico, Albuquerque and the Santa Fe Institute [email protected]; supported by NSF grants PHY-0200909, CCR-0220070, and EIA-0218563 ¶Dept. of Computer Science San Jose State University. [email protected]