Given a function <i>f</i> as an oracle, the collision problem is to find two distinct indexes <i>i</i> and <i>j</i> such that <i>f</i>(<i>i</i>) = <i>f</i>(<i>j</i>), under the promise that such… (More)

Abstract. We consider the quantum complexities of the following three problems: searching an ordered list, sorting an un-ordered list, and deciding whether the numbers in a list are all distinct.… (More)

A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a total Boolean function in the twoparty… (More)

Given copies of the same problem, does it take times the amount of resources to solve these problems? This is the direct sum problem, a fundamental question that has been studied in many… (More)

The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with T gates whose underlying graph has treewidth d can be simulated… (More)

We investigate the randomized and quantum communication complexity of the HAMMING DISTANCE problem, which is to determine if the Hamming distance between two n-bit strings is no less than a threshold… (More)

We prove that, to compute a Boolean function f : {0, 1}N → {−1, 1} with error probability ǫ, any quantum black-box algorithm has to query at least 1−2 √ ǫ 2 ρfN = 1−2√ǫ 2 S̄f times, where ρf is the… (More)

We call F : {0, 1}n×{0, 1}n → {0, 1} a symmetric XOR function if for a function S : {0, 1, ..., n} → {0, 1}, F (x, y) = S(|x⊕ y|), for any x, y ∈ {0, 1}n, where |x⊕ y| is the Hamming weight of the… (More)

The parity decision tree model extends the decision tree model by allowing the computation of a parity function in one step. We prove that the deterministic parity decision tree complexity of any… (More)