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arXiv · 2208.13492

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

Stacey Jeffery·Sebastian Zur
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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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Stacey Jeffery, Sebastian Zur. 2022-08-29. Multidimensional Quantum Walks, with Application to $k$-Distinctness. https://doi.org/10.46298/theoretics.25.7

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