Abstract. In this paper, the space complexity of nonuniform quantum computations is investigated. The model chosen for this are quantum branching programs (QBPs), which provide a graphic description of sequential quantum algorithms. In the first part of the paper, simulations between QBPs and nonuniform quantum Turing machines are presented, which allow to transfer lower and upper bound results between the two models. Using additional insights about the connection between running time and the precision of amplitudes, it is shown that nonuniform quantum Turing machines with algebraic amplitudes and QBPs with a suitable analogous set of amplitudes are equivalent in computational power if both models work with bounded or unbounded error. Quantum ordered binary decision diagrams (QOBDDs) are a restricted variant of QBPs, which can be considered as nonuniform analog of one-way quantum finite automata. In the second part of the paper, lower and upper bound results for QOBDDs are presented in order to compare variants of QOBDDs with their deterministic and randomized counterparts. In the third part QBPs are considered where the performed unitary operation may depend on the result of a previous measurement. For this model a simulation of randomized OBDDs and exponential lower bounds are presented.

This paper deals with the number of monochromatic combinatorial rectangles required to approximate a Boolean function on a constant fraction of all inputs, where each rectangle may be defined with respect to its own partition of the input variables. The main result of the paper is that the number of rectangles required for the approximation of Boolean functions in this model is very sensitive to the allowed error: There is an explicitly defined sequence of functions f n : {0, 1} n # {0, 1} such that f n has rectangle approximations with a constant number of rectangles and one-sided error 1/3+o(1) or two-sided error 1/4+2 -#(n) , but, on the other hand, f n requires exponentially many rectangles if the error bounds are decreased by an arbitrarily small constant. Rectangle partitions and rectangle approximations with the same partition of the input variables for all rectangles have been thoroughly investigated in communication complexity theory. The complexity measures where each r...

Abstract. The paper presents the first nontrivial upper and lower bounds for (nonoblivious) quantum read-once branching programs. It is shown that the computational power of quantum and classical read-once branching programs is incomparable in the following sense: (i) A simple, explicit boolean function on 2n input bits is presented that is computable by error-free quantum read-once branching programs of size O � n 3 � , while each classical randomized read-once branching program and each quantum OBDD for this function with bounded two-sided 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 set-disjointness function DISJn from communication complexity theory with two-sided 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 multi-partition 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.

. It is well-known that an arbitrary nondeterministic Turing machine can be simulated with polynomial overhead by a so-called guess-and-verify machine. It is an open question whether an analogous simulation exists in the context of space-bounded computation. In this paper, a negative answer to this question is given for nondeterministic OBDDs. If we require that all nondeterministic variables are tested at the top of the OBDD, i. e., at the beginning of the computation, this may blow-up the size exponentially. This is a consequence of the following main result of the paper. There is a sequence of Boolean functions fn : {0, 1} n # {0, 1} such that fn has nondeterministic OBDDs of polynomial size with O(n 1/3 log n) nondeterministic variables, but fn requires exponential size if only at most O(log n) nondeterministic variables may be used. 1 Introduction and Definitions So far, there are only few models of computation for which it has been possible to analyze the power of nondete...

We study k-partition communication protocols, an extension of the standard two-party best-partition model to k input partitions. The main results are as follows.

2 Scientific Work