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

• URL: http://arxiv.org/abs/2208.13492v3
• Date: Tue, 27 Aug 2024 10:49:10 GMT
• Title: Multidimensional Quantum Walks, with Application to $k$-Distinctness
• Authors: Stacey Jeffery, Sebastian Zur,
• Abstract summary: We give a new upper bound of $widetildeOleft(n3/4-1/4(2k-1)right)$ on the time complexity. We show how to solve the welded trees problem in $O(n)$ queries and $O(n2)$ time using this new technique.
• Score: 0.5064404027153093
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: While the quantum query complexity of $k$-distinctness is known to be $O\left(n^{3/4-1/4(2^k-1)}\right)$ for any constant $k \geq 4$, the best previous upper bound on the time complexity was $\widetilde{O}\left(n^{1-1/k}\right)$. We give a new upper bound of $\widetilde{O}\left(n^{3/4-1/4(2^k-1)}\right)$ 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 $O(n^2)$ time using this new technique, showing that the new quantum walk framework can achieve exponential speedups.

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