Quantum lower bound for sorting ∗

Needless to say, sorting, especially sorting by comparisons, has been a classical problem in computer science. Despite its enormous importance, the complexity, in terms of the number of comparisons needed, is simple to analyze. Straightforward information theoretical argument gives the tight lower bound of log2(n!), which is achieved by several simple sorting algorithms, for example, Insertion Sort, with an O(n) additive term. However, when quantum computer is deployed to sort, it is not obvious how fast it can be. The sorting by comparison problem neatly fits in the framework of decision tree model. In this probably the simplest model of computation, the computer needs to make queries to an oracle which knows the input, once for a bit. The complexity measurement is the number of queries made in order to compute the desired task. For sorting by comparisons, the oracle is just the comparison matrix of the numbers being sorted. Decision tree model has been studied intensively for classical deterministic and probabilistic algorithms, and lately, for quantum decision tree algorithms. Several problems have been identified such that quantum computers can make asymptotically fewer queries than any classical one. One of the most well known problems is finding the location of the only 1 in a n bit binary string. Clearly, any classical algorithm would require Ω(n) queries, while Grover[11] gives a striking quantum algorithm of O( √ n) queries. His algorithm has