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 ).

We examine an unexplored quantum phenomenon we call oscillatory localization, where a discretetime quantum walk with Grover's diffusion coin jumps back and forth between two vertices. We then connect it to the power dissipation of a related electric network. Namely, we show that there are only two kinds of oscillating states, called uniform states and flip states, and that the projection of an arbitrary state onto a flip state is bounded by the power dissipation of an electric circuit. By applying this framework to states along a single edge of a graph, we show that low effective resistance implies oscillatory localization of the quantum walk. This reveals that oscillatory localization occurs on a large variety of regular graphs, including edge-transitive, expander, and high degree graphs. As a corollary, high edge-connectivity also implies localization of these states, since it is closely related to electric resistance.

In typical discrete-time quantum walk algorithms, one measures the position of the walker while ignoring its internal spin/coin state. Rather than neglecting the information in this internal state, we show that additionally measuring it doubles the success probability of many quantum spatial search algorithms. For example, this allows Grover's unstructured search problem to be solved with certainty, rather than with probability 1/2 if only the walker's position is measured, so the additional measurement yields a search algorithm that is twice as fast as without it, on average. Thus the internal state of discrete-time quantum walks holds valuable information that can be utilized to improve algorithms. Furthermore, we determine conditions for which spatial search problems on regular graphs are amenable to this doubling of the success probability, and this involves diagrammatically analyzing search using degenerate perturbation theory and deriving a useful formula for how the quantum walk acts in its reduced subspace.

We study the structure of sets S ⊆ {0, 1} n with small sensitivity. The well-known Simon's lemma says that any S ⊆ {0, 1} n of sensitivity s must be of size at least 2 n−s . This result has been useful for proving lower bounds on the sensitivity of Boolean functions, with applications to the theory of parallel computing and the "sensitivity vs. block sensitivity" conjecture. In this paper we take a deeper look at the size of such sets and their structure. We show an unexpected "gap theorem": if S ⊆ {0, 1} n has sensitivity s, then we either have |S| = 2 n−s or |S| ≥ 3 2 2 n−s . This provides new insights into the structure of low sensitivity subsets of the Boolean hypercube {0, 1} n .

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