Multidimensional Quantum Walks, with Application to k-Distinctness

While the quantum query complexity of k-distinctness is known to be O(n^{3/4−1/4 21/k−1}) for any constant k ≥ 4 [Belovs, FOCS 2012], the best previous upper bound on the time complexity was Õ(n^1−1/k). We give a new upper bound of Õ(n^{3/4−1/4 21/k−1}) 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(n²) time using this new technique, showing that the new quantum walk framework can achieve exponential speedups.