A quantum sort is any sorting algorithm that runs on a quantum computer. Any comparison-based quantum sorting algorithm would take at least Ω ( n log n ) {\displaystyle \Omega (n\log n)} steps, which is already achievable by classical algorithms. Thus, for this task, quantum computers are no better than classical ones, and should be disregarded when it comes to time complexity. However, in space-bounded sorts, quantum algorithms outperform their classical counterparts.
References
Høyer, P.; Neerbek, J.; Shi, Y. (2001). "Quantum complexities of ordered searching, sorting, and element distinctness". 28th International Colloquium on Automata, Languages, and Programming. Lecture Notes in Computer Science. Vol. 2076. pp. 62–73. arXiv:quant-ph/0102078. doi:10.1007/3-540-48224-5_29. ISBN 978-3-540-42287-7. 978-3-540-42287-7 ↩
Klauck, Hartmut (2003). "Quantum Time-Space Tradeoffs for Sorting". Proceedings of the thirty-fifth annual ACM symposium on Theory of computing. p. 69. arXiv:quant-ph/0211174. doi:10.1145/780542.780553. ISBN 1581136749. 1581136749 ↩