Dimensionality Reduction of the Complete Bipartite Graph with K Edges Removed for Quantum Walks

Abstract

Systematic dimensionality reduction allows for the optimization of quantum search and transport problems on particular graphs. In the past, the Lanczos Algorithm has been used to perform systematic dimensionality reduction on matrices of graphs including the Complete Graph (CG), the CG with symmetry broken, and Complete Multipartite Graphs (CMPGs), including the Complete Bipartite Graph (CBG). We focus on expanding the scope of these reductions to the CBG with symmetry broken in order to allow the optimization of Quantum Walks on this type of graph.We show that similarly to the CG, the Lanczos Algorithm can be expanded to the CBG with broken symmetry, which has k random edges removed with the constraints that no more than one edge per node is removed and that no edges that connect to the solution node are removed. Unlike the CG with broken edges, which, after reduction, has 3 types of nodes and a resulting 3×3 matrix, the CBG with broken edges reduces to a graph with 5 types of nodes, resulting in a reduction from an NxN matrix to a 5×5 matrix. From these results, it may be further explored whether or not the more general CMPG reduction may also be expanded by breaking the graph’s symmetry, and if so, how the dimensions of the reduced matrices will be affected as the number of partitions grows.