Multidimensional Quantum Walks, with Application to $k$-Distinctness

Abstract: While the quantum query complexity of $k$-distinctness is known to be $O\left(n<sup>{3/4-1/4(2<sup>k-1)}\right)$ for any constant $k \geq 4$, the best previous upper bound on the time complexity was $\widetilde{O}\left(n<sup>{1-1/k}\right)$. We give a new upper bound of $$\widetilde{O}\left(n<sup>{3/4-1/4(2<sup>k-1)}\right)$$ on the time complexity, matching the query complexity up to polylogarithmic factors. In order to achieve this upper bound, we give a new technique for designing quantum walk search algorithms, which is an extension of the electric network framework. We also show how to solve the welded trees problem in $O(n)$ queries and $O(n<sup>2)$ time using this new technique, showing that the new quantum walk framework can achieve exponential speedups.