# Applications of the quantum algorithm for st-connectivity

## TL;DR

This paper introduces quantum algorithms for graph connectivity problems, including cycle detection and bipartiteness, with optimal query complexity and improved performance under specific graph conditions.

## Contribution

It presents new quantum algorithms for multiple graph connectivity problems, optimizing query complexity and extending capabilities to estimate circuit rank.

## Key findings

- Query-optimal algorithms for cycle detection and bipartiteness
- Improved performance with large circuit rank or few edges
- Algorithms operate with logarithmic space complexity

## Abstract

We present quantum algorithms for various problems related to graph connectivity. We give simple and query-optimal algorithms for cycle detection and odd-length cycle detection (bipartiteness) using a reduction to st-connectivity. Furthermore, we show that our algorithm for cycle detection has improved performance under the promise of large circuit rank or a small number of edges. We also provide algorithms for detecting even-length cycles and for estimating the circuit rank of a graph. All of our algorithms have logarithmic space complexity.