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On the Computational Power of Probabilistic and Quantum Branching Programs (Revised Version)
"... In this paper we show that onequbit 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 accepte ..."
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In this paper we show that onequbit 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 width2 quantum branching program of polynomial length, in contrast to the classical case where width 5 is necessary unless NC 1 = ACC. This separates width2 quantum programs from width2 doubly stochastic programs as we show the latter cannot compute the middle bit of multiplication. Finally, we show that boundedwidth quantum and stochastic programs can be simulated by classical programs of larger but bounded width, and thus are in NC 1. For readonce quantum branching programs (QBPs), we give a symmetric Boolean function which is computable by a readonce QBP with O(log n) width, but not by a deterministic readonce BP with o(n) width, or by a classical randomized readonce 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 readonce QBPs, showing that our O(log n) upper bound for this symmetric function is almost tight.
Quantum vs. classical readonce branching programs
, 504
"... Abstract. The paper presents the first nontrivial upper and lower bounds for (nonoblivious) quantum readonce branching programs. It is shown that the computational power of quantum and classical readonce branching programs is incomparable in the following sense: (i) A simple, explicit boolean func ..."
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Abstract. The paper presents the first nontrivial upper and lower bounds for (nonoblivious) quantum readonce branching programs. It is shown that the computational power of quantum and classical readonce branching programs is incomparable in the following sense: (i) A simple, explicit boolean function on 2n input bits is presented that is computable by errorfree quantum readonce branching programs of size O � n 3 � , while each classical randomized readonce branching program and each quantum OBDD for this function with bounded twosided error requires size 2 Ω(n). (ii) Quantum branching programs reading each input variable exactly once are shown to require size 2 Ω(n) for computing the setdisjointness function DISJn from communication complexity theory with twosided error bounded by a constant smaller than 1/2−2 √ 3/7. This function is trivially computable even by deterministic OBDDs of linear size. The technically most involved part is the proof of the lower bound in (ii). For this, a new model of quantum multipartition communication protocols is introduced and a suitable extension of the information cost technique of Jain, Radhakrishnan, and Sen (2003) to this model is presented. 1.
Abstract
, 2005
"... Quantum branching programs (quantum binary decision diagrams, respectively) are a convenient tool for examining quantum computations using only a logarithmic amount of space. Recently several types of restricted quantum branching programs have been considered, e. g. read–once quantum branching progr ..."
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Quantum branching programs (quantum binary decision diagrams, respectively) are a convenient tool for examining quantum computations using only a logarithmic amount of space. Recently several types of restricted quantum branching programs have been considered, e. g. read–once quantum branching programs. This paper considers quantum ordered binary decision diagrams (QOBDDs) and answers the question: How does the computational power of QOBDDs increase, if we allow repeated tests. Additionally it is described how to synthesize QOBDDs according to Boolean operations.