Query Complexity: Worst-Case Quantum Versus Average-Case Classical
<|reference_start|>Query Complexity: Worst-Case Quantum Versus Average-Case Classical: In this note we investigate the relationship between worst-case quantum query complexity and average-case classical query complexity. Specifically, we show that if a quantum computer can evaluate a total Boolean function f with bounded error using T queries in the worst case, then a deterministic classical computer can evaluate f using O(T^5) queries in the average case, under a uniform distribution of inputs. If f is monotone, we show furthermore that only O(T^3) queries are needed. Previously, Beals et al. (1998) showed that if a quantum computer can evaluate f with bounded error using T queries in the worst case, then a deterministic classical computer can evaluate f using O(T^6) queries in the worst case, or O(T^4) if f is monotone. The optimal bound is conjectured to be O(T^2), but improving on O(T^6) remains an open problem. Relating worst-case quantum complexity to average-case classical complexity may suggest new ways to reduce the polynomial gap in the ordinary worst-case versus worst-case setting.<|reference_end|>
@article{aaronson2000query,
    title={Query Complexity: Worst-Case Quantum Versus Average-Case Classical},
    author={Scott Aaronson},
    journal={arXiv preprint arXiv:cs/0001013},
    year={2000},
    archivePrefix={arXiv},
    eprint={cs/0001013},
    primaryClass={cs.CC quant-ph}
}