Improved bounds for the randomized decision tree complexity of recursive majority

We consider the randomized decision tree complexity of the recursive 3-majority function. We prove a lower bound of $(1/2-delta) cdot 2.57143^h$ for the two-sided-error randomized decision tree complexity of evaluating height $h$ formulae with error $delta in [0,1/2)$. This improves the lower bound of $(1-2delta)(7/3)^h$ given by Jayram, Kumar, and Sivakumar (STOC03), and the one of $(1-2delta) cdot 2.55^h$ given by Leonardos (ICALP13). Second, we improve the upper bound by giving a new zero-error randomized decision tree algorithm that has complexity at most $(1.007) cdot 2.64944^h$. The previous best known algorithm achieved complexity $(1.004) cdot 2.65622^h$. The new lower bound follows from a better analysis of the base case of the recursion of Jayram et al. The new algorithm uses a novel interleaving of two recursive algorithms.

On the hitting times of quantum versus random walks

In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Further, we prove that for any reversible ergodic Markov chain $P$, the quantum hitting time of the quantum analogue of $P$ has the same order as the square root of the classical hitting time of $P$. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. Finally, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem for the 2D grid. Extending his result, we show that for any state-transitive Markov chain with unique marked state, the quantum hitting time is of the same order for both the detection and finding problems.