How significant are the known collision and element distinctness quantum algorithms

Authors:
Lov Grover;Terry Rudolph
Affiliations:
Bell Labs, Murray Hill, NJ;Bell Labs, Murray Hill, NJ
Venue:
Quantum Information & Computation
Year:
2004

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Abstract

Quantum search is a technique for searching N possibilities for a desired target in O(√N)steps. It has been applied in the design of quantum algorithms fur several structuredproblems. Many of these algorithms require significant amount of quantum hardware.In this paper we propose the criterion that an algorithm width requires O(S) hardwareshould be considered significant if it produces a speedup of better than O(√S) over asimple quantum search algorithm. This is because a speedup of O (√S) can be triviallyobtained by dividing the search space into S separate parts and handing the problem to Sindependent processors that do a quantum search (in this paper we drop all logarithmicfactors when discussing time/space complexity). Known algorithms for collision andelement distinctness exactly saturate the criterion.