Quantum Algorithms for Finding Extrema with Unary Predicates

We study the problem of finding the maximum or the minimum of a given set S = {x0, x1,... xn−1}, each element xi drawn from some finite universe U of real numbers. We assume that the inputs are abstracted within an oracle O where we can only gain information through unary comparisons in the form ”Is xi greater than, equal to, or less than some constant k? ” Classically, this problem is solved optimally with a runtime of Θ(n + lg |U |). In the setting of Quantum Computing, we show that at least Ω ( √ n+lg |U |) queries are required to solve the problem even with bounded error. Combining variants of the Grover’s search [1, 2] algorithm and the optimal classical unary extrema finding algorithm, we have derived a series of new quantum algorithms, some running in time as fast as O ( √ nlg ∗ n+lg |U |). This shows that quantum computers can accelerate the speed in the unary comparison model asymptotically. Inspecting our tools, we find convincing arguments that our lower bound is most probably tight, but we may need an entirely new approach to solve the problem optimally. The technique used in our algorithm can also be extended to solve variations of quantum statistics problems. For instance, our result can be directly extended to approximation of extrema of real numbers, similar to that of [3]. Moreover, we can also solve the quantum k-select problem optimally in time O ( √ kn) with constant success probability. We hope that our ideas and tools will prove to be useful in other areas. i