Optimal quantum adversary lower bounds for ordered search

TL;DR

This paper determines the exact quantum adversary lower bound for a symmetrized ordered search problem, establishing it as (1/pi) ln n + O(1), and shows this bound holds even with generalized adversary methods.

Contribution

It precisely calculates the optimal quantum adversary lower bound for symmetrized ordered search, refining previous bounds and demonstrating its robustness with generalized methods.

Findings

The lower bound for symmetrized ordered search is (1/pi) ln n + O(1).

The bound applies even with negative weights in the adversary method.

The result tightens the understanding of quantum query complexity for ordered search.

Abstract

The goal of the ordered search problem is to find a particular item in an ordered list of n items. Using the adversary method, Hoyer, Neerbek, and Shi proved a quantum lower bound for this problem of (1/pi) ln n + Theta(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/pi) ln n + O(1). Furthermore, we show that this remains true for the generalized adversary method allowing negative weights.