Quantum Lower Bounds by Polynomials
<|reference_start|>Quantum Lower Bounds by Polynomials: We 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.<|reference_end|>
@article{beals1998quantum,
    title={Quantum Lower Bounds by Polynomials},
    author={Robert Beals (U of Arizona), Harry Buhrman (CWI), Richard Cleve (U of
  Calgary), Michele Mosca (U of Oxford), Ronald de Wolf (CWI and U of
  Amsterdam)},
    journal={arXiv preprint arXiv:quant-ph/9802049},
    year={1998},
    archivePrefix={arXiv},
    eprint={quant-ph/9802049},
    primaryClass={quant-ph cs.CC}
}