Quantum Time-Space Tradeoffs for Sorting
<|reference_start|>Quantum Time-Space Tradeoffs for Sorting: We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that in general a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We observe that for all storage bounds n/\log n\ge S\ge \log^3 n, one can devise a quantum algorithm that sorts n numbers (using comparisons only) in time T=O(n^{3/2}\log^{3/2} n/\sqrt S). We then show the following lower bound on the time-space tradeoff for sorting $n$ numbers from a polynomial size range in a general sorting algorithm (not necessarily based on comparisons): TS=\Omega(n^{3/2}). Hence for small values of S the upper bound is almost tight. Classically the time-space tradeoff for sorting is TS=\Theta(n^2).<|reference_end|>
@article{klauck2002quantum,
    title={Quantum Time-Space Tradeoffs for Sorting},
    author={Hartmut Klauck},
    journal={arXiv preprint arXiv:quant-ph/0211174},
    year={2002},
    archivePrefix={arXiv},
    eprint={quant-ph/0211174},
    primaryClass={quant-ph cs.CC}
}