Quantum query algorithms improve graph collision problem efficiency
Breaking the quantum barriers for Graph Collision
Computational ComplexityData Structures and Algorithms
Summary
The graph collision problem asks if a graph has an edge connecting two marked points, which is important in computer science. The authors found better quantum algorithms that use fewer queries to solve this problem than previously thought possible. They also discovered tighter limits on how efficient any quantum algorithm can be for this task. Their work improves understanding of quantum computing's potential for solving search problems in graphs.
What this means in practice
- •For quantum algorithm designers: Develop more query-efficient quantum algorithms for detecting collisions in data networks or database search tasks.
- •For network security engineers: Explore enhanced quantum methods for identifying collision patterns in communication graphs, improving anomaly detection approaches.
A theory result. No direct application yet.
Authors
Arjan Cornelissen, Amin Shiraz Gilani, Shiva Kintali, François Le Gall, Subhasree Patro, Xingyu Zhou
Abstract
In the graph collision problem, the task is to decide whether a known graph on $n$ vertices contains an edge with both endpoints marked, given query access to the input bits specifying the marked vertices. We first refute the conjectured $Ω(n^{2/3})$ query lower bound with a bounded error quantum algorithm using $\widetilde O(n^{92/139})$ queries. We then refine this approach to obtain a $\widetilde O(n^{11/17})$ query algorithm. We also obtain a $\widetilde O((nq)^{11/17})$ query bound for all (partial) Boolean output functions with positive certificate complexity at most 2 and alphabet size $q$, contrasting a known $Ω(n^{2/3})$ lower bound for such problems with large alphabet sizes. We complement these algorithms with an $Ω(\sqrt{n\log n})$ quantum query lower bound for graph collision, exceeding the $O(\sqrt n)$ certificate complexity barrier for the positive adversary method.