Konrad Anand

Konrad Anand, Zongchen Chen, Mary Cryan et al.

We show that a distribution related to Gaussian Boson Sampling (GBS) on graphs can be sampled classically in polynomial time. Graphical applications of GBS typically sample from this distribution, and thus quantum algorithms do not provide exponential speedup for these applications. We also show that another distribution related to Boson sampling can be sampled classically in polynomial time.

Konrad Anand, Luc Devroye

This thesis presents analysis of the properties and run-time of the Rapidly-exploring Random Tree (RRT) algorithm. It is shown that the time for the RRT with stepsize $ε$ to grow close to every point in the $d$-dimensional unit cube is $Θ\left(\frac1{ε^d} \log \left(\frac1ε\right)\right)$. Also, the time it takes for the tree to reach a region of positive probability is $O\left(ε^{-\frac32}\right)$. Finally, a relationship is shown to the Nearest Neighbour Tree (NNT). This relationship shows that the total Euclidean path length after $n$ steps is $O(\sqrt n)$ and the expected height of the tree is bounded above by $(e + o(1)) \log n$.