Papers › Translation-Invariant Quantum Algorithms for Ordered Search are Optimal
Translation-Invariant Quantum Algorithms for Ordered Search are Optimal
Joseph Carolan, Andrew M. Childs, Matt Kovacs-Deak, Luke Schaeffer
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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 ⌈log₂n⌉ queries for a list of length n. Quantum computers can achieve a constant-factor speedup, but the best possible coefficient of log₂n for exact quantum algorithms is only known to lie between (ln2)/π≈0.221 and 4/log₂605 ≈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 log₇₂₆₅n ≈0.390 log₂n quantum queries.
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