Abstract. We present several results on comparative complexity for different variants of OBDD models.-We present some results on comparative complexity of classical and quantum OBDDs. We consider a partial function depending on parameter k such that for any k > 0 this function is computed by an exact quantum OBDD of width 2 but any classical OBDD (deterministic or stable bounded error probabilistic) needs width 2 k+1 . -We consider quantum and classical nondeterminism. We show that quantum nondeterminism can be more efficient than classical one. In particular, an explicit function is presented which is computed by a quantum nondeterministic OBDD with constant width but any classical nondeterministic OBDD for this function needs non-constant width. -We also present new hierarchies on widths of deterministic and nondeterministic OBDDs. We focus both on small and large widths.

This is a review of quantum methods for machine learning problems that consists of two parts. The first part, "quantum tools", presents the fundamentals of qubits, quantum registers, and quantum states, introduces important quantum tools based on known quantum search algorithms and SWAP-test, and discusses the basic quantum procedures used for quantum search methods. The second part, "quantum classification algorithms", introduces several classification problems that can be accelerated by using quantum subroutines and discusses the quantum methods used for classification.

Abstract. The paper examines hierarchies for nondeterministic and deterministic ordered read-k-times Branching programs. The currently known hierarchies for deterministic k-OBDD models of Branching programs for k = o(n 1/2 / log 3/2 n) are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic k-OBDD it is known that, if k is constant then polynomial size k-OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read k-times Branching programs for k = o( √ log n/ log log n) are proved by Okolnishnikova in 1997, and for probabilistic read k-times Branching programs for k ≤ log n/3 are proved by Hromkovic and Saurhoff in 2003.We show that increasing k for polynomial size nodeterministic k-OBDD makes model more powerful if k is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic k-OBDDs for k = o(n/ log n). These results extends hierarchies for read k-times Branching programs, but k-OBDD has more regular structure. The lower bound techniques we propose are a "functional description" of Boolean function presented by nondeterministic k-OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic k-OBDDs.Additionally we expand the hierarchies for deterministic k-OBDDs using our lower bounds for k = o(n/ log n). We also analyze similar hierarchies for superpolynomial and subexponential width k-OBDDs.

Online algorithm is a well-known computational model. We introduce quantum online algorithms and investigate them with respect to a competitive ratio in two points of view: space complexity and advice complexity. We start with exploring a model with restricted memory and show that quantum online algorithms can be better than classical ones (deterministic or randomized) for sublogarithmic space (memory), and they can be better than deterministic online algorithms without restriction for memory. Additionally, we consider polylogarithmic space case and show that in this case, quantum online algorithms can be better than deterministic ones as well. Another point of view to the online algorithms model is advice complexity. So, we introduce quantum online algorithms with a quantum channel with an adviser. Firstly, we show that quantum algorithms have at least the same computational power as classical ones have. And we give some examples of quantum online algorithms with advice. Secondly, we show that if we allow to use shared entangled qubits (EPR-pairs), then quantum online algorithm can use two times less advise qubits comparing to a classical one. We apply this approach to the well-known Paging Problem.

Abstract. We consider Quantum OBDD model. It is restricted version of read-once Quantum Branching Programs, with respect to "width" complexity. It is known that maximal complexity gap between deterministic and quantum model is exponential. But there are few examples of such functions. We present method (called "reordering"), which allows to build Boolean function g from Boolean Function f , such that if for f we have gap between quantum and deterministic OBDD complexity for natural order of variables, then we have almost the same gap for function g, but for any order. Using it we construct the total function REQ which deterministic OBDD complexity is 2 Ω(n/ log n) and present quantum OBDD of width O(n 2 ). It is bigger gap for explicit function that was known before for OBDD of width more than linear. Using this result we prove the width hierarchy for complexity classes of Boolean functions for quantum OBDDs. Additionally, we prove the width hierarchy for complexity classes of Boolean functions for bounded error probabilistic OBDDs. And using "reordering" method we extend a hierarchy for k-OBDD of polynomial size, for k = o(n/ log 3 n). Moreover, we proved a similar hierarchy for bounded error probabilistic k-OBDD. And for deterministic and probabilistic k-OBDDs of superpolynomial and subexponential size.

Automata and branching programs are known models of computation with restricted memory. These models of computation were in focus of a large number of researchers during the last decades. Streaming algorithms are a modern model of computation with restricted memory. In this paper, we present recent results on the comparative computational power of quantum and classical models of branching programs and streaming algorithms. In addition to comparative complexity results, we present a quantum branching program for computing a practically important quantum function (quantum hash function) and prove optimality of this algorithm.

We consider online algorithms with respect to the competitive ratio. Here, we investigate quantum and classical one-way automata with non-constant size of memory (streaming algorithms) as a model for online algorithms. We construct problems that can be solved by quantum online streaming algorithms better than by classical ones in a case of logarithmic or sublogarithmic size of memory.

In this paper was explored well known model k-OBDD. There are proven width based hierarchy of classes of boolean functions which computed by k-OBDD. The proof of hierarchy is based on sufficient condition of Boolean function's non representation as k-OBDD and complexity properties of Boolean function SAF. This function is modification of known Pointer Jumping (PJ) and Indirect Storage Access (ISA) functions. PreliminariesThe k-OBDD and OBDD models are well known models of branching programs. Good source for a different models of branching programs is the book by Ingo Wegener [13].The branching program P over a set X of n Boolean variables is a directed acyclic graph with a source node and sink nodes. Sink nodes are labeled by 1 (Accept) or 0 (Reject). Each inner node v is associated with a variable x ∈ X and has two outgoing edges labeled x = 0 and x = 1 respectively. An input ν ∈ {0, 1} n determines a computation (consistent) path of from the source node of P to a one of the sink nodes of P . We denote P (ν) the label of sink finally reached by P on the input ν. The input ν is accepted or rejected if P (ν) = 1 or P (ν) = 0 respectively.Program P computes (presents) Boolean function f (X) (f :A branching program is leveled if the nodes can be partitioned into levels V 1 , . . . , V and a level V +1 such that the nodes in V +1 are the sink nodes, 2000 Mathematical Subject Classification. .