A note on quantum lower bounds for local search via congestion and expansion
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| Format: | Preprint |
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2024
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| _version_ | 1866913617000005632 |
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| author | Brânzei, Simina Recker, Nicholas J. |
| author_facet | Brânzei, Simina Recker, Nicholas J. |
| contents | We consider the quantum query complexity of local search as a function of graph geometry. Given a graph $G = (V,E)$ with $n$ vertices and black box access to a function $f : V \to \mathbb{R}$, the goal is find a vertex $v$ that is a local minimum, i.e. with $f(v) \leq f(u)$ for all $(u,v) \in E$, using as few oracle queries as possible.
We show that the quantum query complexity of local search on $G$ is $Ω\bigl( \frac{n^{\frac{3}{4}}}{\sqrt{g}} \bigr)$, where $g$ is the vertex congestion of the graph. For a $β$-expander with maximum degree $Δ$, this implies a lower bound of $ Ω\bigl(\frac{\sqrtβ \; n^{\frac{1}{4}}}{\sqrtΔ \; \log{n}} \bigr)$. We obtain these bounds by applying the strong weighted adversary method to a construction by Brânzei, Choo, and Recker (2024).
As a corollary, on constant degree expanders, we derive a lower bound of $Ω\bigl(\frac{n^{\frac{1}{4}}}{ \sqrt{\log{n}}} \bigr)$. This improves upon the best prior quantum lower bound of $Ω\bigl( \frac{n^{\frac{1}{8}}}{\log{n}}\bigr) $ by Santha and Szegedy (2004). In contrast to the classical setting, a gap remains in the quantum case between our lower bound and the best-known upper bound of $O\bigl( n^{\frac{1}{3}} \bigr)$ for such graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_13345 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on quantum lower bounds for local search via congestion and expansion Brânzei, Simina Recker, Nicholas J. Computational Complexity Quantum Physics We consider the quantum query complexity of local search as a function of graph geometry. Given a graph $G = (V,E)$ with $n$ vertices and black box access to a function $f : V \to \mathbb{R}$, the goal is find a vertex $v$ that is a local minimum, i.e. with $f(v) \leq f(u)$ for all $(u,v) \in E$, using as few oracle queries as possible. We show that the quantum query complexity of local search on $G$ is $Ω\bigl( \frac{n^{\frac{3}{4}}}{\sqrt{g}} \bigr)$, where $g$ is the vertex congestion of the graph. For a $β$-expander with maximum degree $Δ$, this implies a lower bound of $ Ω\bigl(\frac{\sqrtβ \; n^{\frac{1}{4}}}{\sqrtΔ \; \log{n}} \bigr)$. We obtain these bounds by applying the strong weighted adversary method to a construction by Brânzei, Choo, and Recker (2024). As a corollary, on constant degree expanders, we derive a lower bound of $Ω\bigl(\frac{n^{\frac{1}{4}}}{ \sqrt{\log{n}}} \bigr)$. This improves upon the best prior quantum lower bound of $Ω\bigl( \frac{n^{\frac{1}{8}}}{\log{n}}\bigr) $ by Santha and Szegedy (2004). In contrast to the classical setting, a gap remains in the quantum case between our lower bound and the best-known upper bound of $O\bigl( n^{\frac{1}{3}} \bigr)$ for such graphs. |
| title | A note on quantum lower bounds for local search via congestion and expansion |
| topic | Computational Complexity Quantum Physics |
| url | https://arxiv.org/abs/2412.13345 |