On the computational power of probabilistic and quantum branching program

Abstract

In this paper, we show that one-qubit polynomial time computations are as powerful as NC 1 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 1 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 1 = 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 1 . 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.