Lower Bounds for Quantum Search and Derandomization
<|reference_start|>Lower Bounds for Quantum Search and Derandomization: We prove lower bounds on the error probability of a quantum algorithm for searching through an unordered list of N items, as a function of the number T of queries it makes. In particular, if T=O(sqrt{N}) then the error is lower bounded by a constant. If we want error <1/2^N then we need T=Omega(N) queries. We apply this to show that a quantum computer cannot do much better than a classical computer when amplifying the success probability of an RP-machine. A classical computer can achieve error <=1/2^k using k applications of the RP-machine, a quantum computer still needs at least ck applications for this (when treating the machine as a black-box), where c>0 is a constant independent of k. Furthermore, we prove a lower bound of Omega(sqrt{log N}/loglog N) queries for quantum bounded-error search of an ordered list of N items.<|reference_end|>
@article{buhrman1998lower,
    title={Lower Bounds for Quantum Search and Derandomization},
    author={Harry Buhrman (CWI) and Ronald de Wolf (CWI and U Amsterdam)},
    journal={arXiv preprint arXiv:quant-ph/9811046},
    year={1998},
    archivePrefix={arXiv},
    eprint={quant-ph/9811046},
    primaryClass={quant-ph cs.CC}
}