2005/07/31 by Shahar Dolev, Itamar Pitowsky, Boaz Tamir
Computer Science · Physics and Astronomy · #Computability, Logic, AI Algorithms #Logic, programming, and type systems #Quantum Computing Algorithms and Architecture #cs.CC #quant-ph
paper · pdf · doi:10.1103/physreva.73.022308
published as Physical Review A 73 022308 (2006) · 8 pages, revtex, Title changed
openalex publication_date 2006/02/03 · arxiv created 2006/02/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can find a single marked object in a database of size N by using only O(√(N)) queries of the oracle that identifies the object. His result was generalized to the case of finding one object in a subset of marked elements. We consider the following computational problem: A subset of marked elements is given whose number of elements is either M or K, our task is to determine which is the case. We show how to solve this problem with a high probability of success using iterations of Grover's basic step only, and no other algorithm. Let m be the required number of iterations; we prove that under certain restrictions on the sizes of M and K the estimation m<2√(N)∕(√(K)\ensuremath-√(M)) obtains. This bound reproduces previous results based on more elaborate algorithms, and is known to be optimal up to a constant factor. Our method involves simultaneous Diophantine approximations, so that Grover's algorithm is conceptualized as an orbit of an ergodic automorphism of the torus. We comment on situations where the algorithm may be slow, and note the similarity between these cases and the problem of small divisors in classical mechanics.