Improved Algorithm and Lower Bound for Variable Time Quantum Search

We study variable time search, a form of quantum search where queries to different items take different time. Our first result is a new quantum algorithm that performs variable time search with complexity $O(\sqrt{T}\log n)$ where $T={\sum }_{i=1}^n{t}_i^2$ with $t_i$ denoting the time to check the $i$-th item. Our second result is a quantum lower bound of $\mathrm{\Omega }(\sqrt{T\log T})$. Both the algorithm and the lower bound improve over previously known results by a factor of $\sqrt{\log T}$ but the algorithm is also substantially simpler than the previously known quantum algorithms.