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29 March 2024
 
  » arxiv » quant-ph/9802049

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Quantum Lower Bounds by Polynomials
Robert Beals ; Harry Buhrman ; Richard Cleve ; Michele Mosca ; Ronald de Wolf ;
Date 18 Feb 1998
Subject Quantum Physics; Computational Complexity | quant-ph cs.CC
AffiliationU of Arizona), Harry Buhrman (CWI), Richard Cleve (U of Calgary), Michele Mosca (U of Oxford), Ronald de Wolf (CWI and U of Amsterdam
AbstractWe examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T^6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.
Source arXiv, quant-ph/9802049
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