Quantum Algorithm of the GLMY Homology on Digraphs

Quantum algorithms for topological data analysis provide significant advantage over the best classical algorithm. Different from the previous simplical complex on points cloud, the GLMY homology introduced by Alexander Grigor'yan, Yong Lin, Yuri Muranov and Shing-Tung Yau, is defined on digraph and is a arising realm in Topological Data Analysis (TDA), which attracts more and more attention recently. We propose a quantum algorithm for the GLMY homology with significant advantage over the best classical algorithm. We design a universal encoding protocol for the quantum states and boundary operators of GLMY homology on digraphs. And a property of the GLMY homology is proved for the theoretical guarantee of the quantum algorithm. The quantum algorithm for GLMY homology provides a cubic speedup in general cases, and it can provide an exponential quantum advantage in the case of the input data is given as a specification of paths.