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Quantum and Stochastic Branching Programs of Bounded Width

Farid Ablayev·Cristopher Moore·Chris Pollett
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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.

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Farid Ablayev, Cristopher Moore, Chris Pollett. 2004-01-03. Quantum and Stochastic Branching Programs of Bounded Width. https://arxiv.org/abs/quant-ph/0201139

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