Quantum and Stochastic Branching Programs of Bounded Width

TL;DR

This paper demonstrates that quantum branching programs of width 2 can recognize all $ ext{NC}^1$ languages with polynomial length, showing a significant computational advantage over classical stochastic programs of the same width.

Contribution

It introduces syntactic models for quantum and stochastic branching programs of bounded width and establishes their computational power, including exact recognition of $ ext{NC}^1$ languages by width-2 quantum programs.

Findings

Width-2 quantum branching programs can recognize all $ ext{NC}^1$ languages.

Classical doubly stochastic programs cannot compute the middle bit of multiplication.

Quantum and stochastic programs of bounded width can be simulated by classical programs in $ ext{NC}^1$.

Abstract

In this paper we show that one qubit polynomial time computations are at least as powerful as $\NC^1$ circuits. More precisely, we define syntactic models for quantum and stochastic branching programs of bounded width and prove upper and lower bounds on their power. We show 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 $\NC^1$. The change in the revised version is the addition of the syntactic condition.