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arXiv · 2503.21090

Translation-Invariant Quantum Algorithms for Ordered Search are Optimal

Joseph Carolan·Andrew M. Childs·Matt Kovacs-Deak·Luke Schaeffer
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Abstract

Ordered search is the task of finding an item in an ordered list using comparison queries. The best exact classical algorithm for this fundamental problem uses $\lceil \log_{2}{n}\rceil$ queries for a list of length $n$. Quantum computers can achieve a constant-factor speedup, but the best possible coefficient of $\log_{2}{n}$ for exact quantum algorithms is only known to lie between $(\ln{2})/\pi \approx 0.221$ and $4/\log_{2}{605} \approx 0.433$. We consider a special class of translation-invariant algorithms with no workspace, introduced by Farhi, Goldstone, Gutmann, and Sipser, that has been used to find the best known upper bounds. First, we show that any bounded-error, $k$-query quantum algorithm for ordered search can be implemented by a $k$-query algorithm in this special class. Second, we use linear programming to show that the best exact $5$-query quantum algorithm can search a list of length $7265$, giving an ordered search algorithm that asymptotically uses $5 \log_{7265}{n} \approx 0.390 \log_{2}{n}$ quantum queries.

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Joseph Carolan, Andrew M. Childs, Matt Kovacs-Deak, Luke Schaeffer. 2025-03-27. Translation-Invariant Quantum Algorithms for Ordered Search are Optimal. https://arxiv.org/abs/2503.21090

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