We study quantum algorithms on search trees of unknown structure, in a model where the tree can be discovered by local exploration. That is, we are given the root of the tree and access to a black box which, given a vertex v, outputs the children of v.We construct a quantum algorithm which, given such access to a search tree of depth at most n, estimates the size of the tree T within a factor of 1 ± δ inÕ( √ nT ) steps. More generally, the same algorithm can be used to estimate size of directed acyclic graphs (DAGs) in a similar model.We then show two applications of this result:• We show how to transform a classical backtracking search algorithm which examines T nodes of a search tree into anÕ( √ T n 3/2 ) time quantum algorithm, improving over an earlier quantum backtracking algorithm of Montanaro [12].• We give a quantum algorithm for evaluating AND-OR formulas in a model where the formula can be discovered by local exploration (modeling position trees in 2-player games). We show that, in this setting, formulas of size T and depth T o(1) can be evaluated in quantum time O(T 1/2+o(1) ). Thus, the quantum speedup is essentially the same as in the case when the formula is known in advance.

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 ).

A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk, resolving a question that had been open for 15 years. CCS CONCEPTS• Theory of computation → Quantum computation theory; Algorithm design techniques; Random walks and Markov chains.

A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk.

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