Oracles with Costs

TL;DR

This paper introduces a new quantum query model with multiple oracles of different costs, applies it to a search problem, and develops an asymptotically optimal algorithm supported by geometric analysis.

Contribution

It proposes a novel model for quantum query complexity with multiple, cost-differentiated oracles and provides an optimal algorithm for the Search with Two Oracles problem.

Findings

The algorithm is asymptotically optimal.

Evidence suggests the algorithm is exactly optimal.

The model captures more realistic problem complexities.

Abstract

While powerful tools have been developed to analyze quantum query complexity, there are still many natural problems that do not fit neatly into the black box model of oracles. We create a new model that allows multiple oracles with differing costs. This model captures more of the difficulty of certain natural problems. We test this model on a simple problem, Search with Two Oracles, for which we create a quantum algorithm that we prove is asymptotically optimal. We further give some evidence, using a geometric picture of Grover's algorithm, that our algorithm is exactly optimal.