0000000000461598
AUTHOR
Troy Lee
showing 3 related works from this author
Quadratically Tight Relations for Randomized Query Complexity
2020
In this work we investigate the problem of quadratically tightly approximating the randomized query complexity of Boolean functions R(f). The certificate complexity C(f) is such a complexity measure for the zero-error randomized query complexity R0(f): C(f) ≤R0(f) ≤C(f)2. In the first part of the paper we introduce a new complexity measure, expectational certificate complexity EC(f), which is also a quadratically tight bound on R0(f): EC(f) ≤R0(f) = O(EC(f)2). For R(f), we prove that EC2/3 ≤R(f). We then prove that EC(f) ≤C(f) ≤EC(f)2 and show that there is a quadratic separation between the two, thus EC(f) gives a tighter upper bound for R0(f). The measure is also related to the fractional…
Separations in Query Complexity Based on Pointer Functions
2015
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 $R_0(f) = O(D(f)^{0.7537\ldots})$. We show this is false by giving an example of a total boolean function $f$ on $n$ bits whose deterministic query complexity is $\Omega(n/\log(n))$ while its zero-error randomized query complexity is $\tilde O(\sqrt{n})$. We further show that the quantum query complexity of the same function is $\tilde O(n^{1/4})$, giving the first example of a total function with a super-quadra…
Quantum Algorithm for k-distinctness with Prior Knowledge on the Input
2011
It is known that the dual of the general adversary bound can be used to build quantum query algorithms with optimal complexity. Despite this result, not many quantum algorithms have been designed this way. This paper shows another example of such algorithm. We use the learning graph technique from arXiv:1105.4024 to give a quantum algorithm for $k$-distinctness problem that runs in $o(n^{3/4})$ queries, for a fixed $k$, given some prior knowledge on the structure of the input. The best known quantum algorithm for the unconditional problem uses $O(n^{k/(k+1)})$ queries.