A lower bound for quantum search of an ordered list

Andrew M. Childs, Troy Lee

Peter Høyer, Jan Neerbek, Yaoyun Shi

We consider the quantum complexities of searching an ordered list and sorting an un-ordered list. For searching an ordered list of N elements, we prove a lower bound of \\frac{1}{\\pi}(\\ln(N)-1) on the number of oracle queries that access the list elements. This improves the previously best lower bound of ({1/12}\\log_2(N) - O(1)) due to Ambainis. For sorting N numbers, we prove a lower bound of \\frac{N}{2\\pi}(\\ln(N)-1) on the number of binary comparisons. The previously best lower bound is \\Omega(N). Our proofs are based on a weighted all-pairs inner product argument, and our results generalize to bounded error quantum algorithms. Both results are proven in the so-called quantum black box model, a quantum analogue of classical decision trees. In addition to our lower bound results, we give an exact quantum algorithm for ordered searching using (\\log_3(N) + O(1)) queries, which is roughly 0.631 \\log_2(N). Although our algorithm is worse than that of Farhi, Goldstone, Gutmann and Sipser, which makes 0.526 \\log_2(N) queries, its philosophy is completely different. Our algorithm is a quantum version of the classical binary search algorithm, and it uses a quantum routine for traversing through a binary search tree faster than classically.

The goal of the ordered search problem is to find a particular item in an ordered list of n items. Using the adversary method, Høyer, Neerbek, and Shi proved a quantum lower bound for this problem of 1 π lnn + Θ(1). Here, we find the exact value of the best possible quantum adversary lower bound for a symmetrized version of ordered search (whose query complexity differs from that of the original problem by at most 1). Thus we show that the best lower bound for ordered search that can be proved by the adversary method is 1 π lnn + O(1). Furthermore, we show that this remains true for the generalized adversary method allowing negative weights. 1

Tathagat Tulsi, Lov K. Grover, Apoorva D. Patel

V. E. Korepin, Lov K. Grover

Andris Ambainis

We show that any quantum algorithm searching an ordered list of n elements needs to examine at least (log,n)/12-O(1) of them. Classically, log/sub 2/ n queries are both necessary and sufficient. This shows that quantum algorithms can achieve only a constant speedup for this problem.