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 ).

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function f on n = 2 k bits defined by a complete binary tree of NAND gates of depth k, which achieves R 0 (f ) = O(D(f ) 0.7537... ). We show this is false by giving an example of a total boolean function f on n bits whose deterministic query complexity is Ω(n/ log(n)) while its zero-error randomized query complexity is O( √ n). We further show that the quantum query complexity of the same function is O(n 1/4 ), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities.We also construct a total boolean function g on n variables that has zero-error randomized query complexity Ω(n/ log(n)) and bounded-error randomized query complexity R(g) = O( √ n). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is Q E (g) = O( √ n). These two functions show that the relations D(f ) = O(R 1 (f ) 2 ) and R 0 (f ) = O(R(f ) 2 ) are optimal, up to poly-logarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R 0 , a 3/2-power separation between Q E and R, and a 4th power separation between approximate degree and bounded-error randomized query complexity.All of these examples are variants of a function recently introduced by Göös, Pitassi, and Watson which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.

We initiate a study of random instances of nonlocal games. We show that quantum strategies are better than classical for almost any 2-player XOR game. More precisely, for large n, the entangled value of a random 2-player XOR game with n questions to every player is at least 1.21... times the classical value, for 1 − o(1) fraction of all 2-player XOR games.

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Intent detection is one of the main tasks of a dialogue system. In this paper, we present our intent detection system that is based on fastText word embeddings and a neural network classifier. We find an improvement in fastText sentence vectorization, which, in some cases, shows a significant increase in intent detection accuracy. We evaluate the system on languages commonly spoken in Baltic countries—Estonian, Latvian, Lithuanian, English, and Russian. The results show that our intent detection system provides state-of-the-art results on three previously published datasets, outperforming many popular services. In addition to this, for Latvian, we explore how the accuracy of intent detection is affected if we normalize the text in advance.