Quantum time-space tradeoffs for sorting
Hartmut Klauck · 2003
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 ≥ S ≥ log3n, one can devise a quantum algorithm that sorts n numbers (using comparisons only) in time T=O(n3/2 log3/2 n/√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=Ω(n3/2). Hence for small values of S the upper bound is almost tight. Classically the time-space tradeoff for sorting is TS=Θ(n2).