19 research outputs found
A Note About Claw Function with a Small Range
In the claw detection problem we are given two functions f:D ? R and g:D ? R (|D| = n, |R| = k), and we have to determine if there is exist x,y ? D such that f(x) = g(y). We show that the quantum query complexity of this problem is between ?(n^{1/2}k^{1/6}) and O(n^{1/2+?}k^{1/4}) when 2 ? k < n
Separations in Query Complexity Based on Pointer Functions
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 on bits defined by a complete
binary tree of NAND gates of depth , which achieves . We show this is false by giving an example of a total
boolean function on bits whose deterministic query complexity is
while its zero-error randomized query complexity is . We further show that the quantum query complexity of the same
function is , 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 on variables that has
zero-error randomized query complexity and bounded-error
randomized query complexity . This is the first
super-linear separation between these two complexity measures. The exact
quantum query complexity of the same function is .
These two functions show that the relations and are optimal, up to poly-logarithmic factors. Further
variations of these functions give additional separations between other query
complexity measures: a cubic separation between and , a -power
separation between and , 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
\goos, 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.Comment: 25 pages, 6 figures. Version 3 improves separation between Q_E and
R_0 and updates reference
Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language
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 ?(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 O?(?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 ?(n^{1.5-?}) for the directed 2D grid and ?(n^{2-?}) 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