In this paper we study quantum algorithms for NP-complete problems whose best classical algorithm is an exponential time application of dynamic programming. We introduce the path in the hypercube problem that models many of these dynamic programming algorithms. In this problem we are asked whether there is a path from 0 n to 1 n in a given subgraph of the Boolean hypercube, where the edges are all directed from smaller to larger Hamming weight. We give a quantum algorithm that solves path in the hypercube in time O * (1.817 n ). The technique combines Grover's search with computing a partial dynamic programming table. We use this approach to solve a variety of vertex ordering problems on graphs in the same time O * (1.817 n ), and graph bandwidth in time O * (2.946 n ). Then we use similar ideas to solve the travelling salesman problem and minimum set cover in time O * (1.728 n ).
Non-local games are widely studied as a model to investigate the properties of quantum mechanics as opposed to classical mechanics. In this paper, we consider a subset of non-local games: symmetric XOR games of n players with 0-1 valued questions. For this class of games, each player receives an input bit and responds with an output bit without communicating to the other players. The winning condition only depends on XOR of output bits and is constant w.r.t. permutation of players. We prove that for almost any n-player symmetric XOR game the entangled value of the game is Θ √ ln n n 1/4 adapting an old result by Salem and Zygmund on the asymptotics of random trigonometric polynomials. Consequently, we show that the classical-quantum gap is Θ( √ ln n) for almost any symmetric XOR game.
In the decision tree model, one's task is to compute a boolean function f : {0, 1} n → {0, 1} on an input x ∈ {0, 1} n that is accessible via queries to a black box (the black box hides the bits xi). In the quantum case, classical queries and computation are replaced by unitary transformations. A quantum algorithm is exact if it always outputs the correct value of f (in contrast to the standard model of quantum algorithms where the algorithm is allowed to be incorrect with a small probability). The minimum number of queries for an exact quantum algorithm computing the function f is denoted by QE(f ).We consider the following n bit function with 0 ≤ k ≤ l ≤ n:i.e. we want to give the answer 1 only when exactly k or l of the bits xi are 1. We construct a quantum query algorithm for this function and give lower bounds for it, with lower bounds matching the complexity of the algorithm in some cases (and almost matching it in other cases):• For all k, l: max{n − k, l} − 1 ≤ QE(EXACT n k,l ) ≤ max{n − k, l} + 1.
We consider representing of natural numbers by expressions using 1's, addition, multiplication and parentheses. n denotes the minimum number of 1's in the expressions representing n. The logarithmic complexity n log is defined as n /log 3 n. The values of n log are located in the segment [3, 4.755], but almost nothing is known with certainty about the structure of this "spectrum" (are the values dense somewhere in the segment etc.). We establish a connection between this problem and another difficult problem: the seemingly "almost random" behaviour of digits in the base 3 representations of the numbers 2 n . We consider also representing of natural numbers by expressions that include subtraction, and the so-called P -algorithms -a family of "deterministic" algorithms for building representations of numbers.
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