Multidimensional Quantum Walks, with Application to $k$-Distinctness
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866912258145124352 |
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| author | Jeffery, Stacey Zur, Sebastian |
| author_facet | Jeffery, Stacey Zur, Sebastian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_13492 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Multidimensional Quantum Walks, with Application to $k$-Distinctness Jeffery, Stacey Zur, Sebastian Quantum Physics Discrete Mathematics 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. |
| title | Multidimensional Quantum Walks, with Application to $k$-Distinctness |
| topic | Quantum Physics Discrete Mathematics |
| url | https://arxiv.org/abs/2208.13492 |