We study the quantum query complexity of two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most k. We call this the Dyck_{k,n} problem. We prove a lower bound of Ω(c^k √n), showing that the complexity of this problem increases exponentially in k. Here n is the length of the word. When k is a constant, this is interesting as a representative example of star-free languages for which a surprising Õ(√n) query quantum algorithm was recently constructed by Aaronson et al. [Scott Aaronson et al., 2018]. Their proof does not give rise to a general algorithm. When k is not a constant, Dyck_{k,n} is not context-free. We give an algorithm with O(√n(log n)^{0.5k}) quantum queries for Dyck_{k,n} for all k. This is better than the trival upper bound n for k = o({log(n)}/{log log n}). Second, we consider connectivity problems on grid graphs in 2 dimensions, if some of the edges of the grid may be missing. By embedding the "balanced parentheses" problem into the grid, we show a lower bound of Ω(n^{1.5-ε}) for the directed 2D grid and Ω(n^{2-ε}) for the undirected 2D grid. The directed problem is interesting as a black-box model for a class of classical dynamic programming strategies including the one that is usually used for the well-known edit distance problem. We also show a generalization of this result to more than 2 dimensions.

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 that this is false by giving an example of a total Boolean function f on n bits whose deterministic query complexity is Ω( n ) while its zero-error randomized query complexity is Õ(√ n ). We further show that the quantum query complexity of the same function is Õ( 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 ) = Õ(√ 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 ) = Õ(√ n ). These functions show that the relations D ( f ) = O ( R 1 ( f ) 2 ) and R 0 ( f ) = Õ( R ( f ) 2 ) are optimal, up to polylogarithmic 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 show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials. Namely, a partial Boolean function f is computable by a 1-query quantum algorithm with error bounded by epsilon<1/2 iff f can be approximated by a degree-2 polynomial with error bounded by epsilon'<1/2. This result holds for two different notions of approximation by a polynomial: the standard definition of Nisan and Szegedy and the approximation by block-multilinear polynomials recently introduced by Aaronson and Ambainis [Aaronson/Ambainis, STOC 2015]. The proof uses Grothendieck's inequality to relate two matrix norms, with one norm corresponding to polynomial approximations and the other norm corresponding to quantum algorithms. We also show two results for polynomials of higher degree. First, there is a total Boolean function which requires ~Omega(n) quantum queries but can be represented by a block-multilinear polynomial of degree ~O(sqrt(n)). Thus, in the general case (for an arbitrary number of queries), block-multilinear polynomials are not equivalent to quantum algorithms. Second, for any constant degree k, the two notions of approximation by a polynomial (the standard and the block-multilinear) are equivalent. As a consequence, we solve an open problem from [Aaronson/Ambainis, STOC 2015], showing that one can estimate the value of any bounded degree-k polynomial p:{0,1}^n -> [-1,1] with O(n^{1-1/(2k)) queries.

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=2k bits defined by a complete binary tree of NAND gates of depth k, which achieves R0(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 Õ(√n). We further show that the quantum query complexity of the same function is Õ(n1/4), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities.

We show that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms. This improves over an earlier n/4 lower bound of Ambainis (A. Ambainis, 1999), and shows that van Dam's oracle interrogation (W. van Dam, 1998) is essentially optimal for almost all functions. Our proof uses the fact that the acceptance probability of a T-query algorithm can be written as the sum of squares of degree-T polynomials.