Quantum Walks on Simplicial Complexes and Harmonic Homology: Application to Topological Data Analysis with Superpolynomial Speedups

Abstract: Incorporating higher-order interactions in information processing enables us to build more accurate models, gain deeper insights into complex systems, and address real-world challenges more effectively. However, existing methods, such as random walks on oriented simplices and homology, which capture these interactions, are not known to be efficient. This work investigates whether quantum walks on simplicial complexes exhibit quantum advantages. We introduce a novel quantum walk that encodes the combinatorial Laplacian, a key mathematical object whose spectral properties reflect the topology of the underlying simplicial complex. Furthermore, we construct a unitary encoding that projects onto the kernel of the Laplacian, representing the space of harmonic cycles in the complex's homology. Combined with the efficient construction of quantum walk unitaries for clique complexes that we present, this paves the way for utilizing quantum walks to explore higher-order interactions within topological structures. Our results achieve superpolynomial quantum speedup with quantum walks without relying on quantum oracles for large datasets. Crucially, the walk operates on a state space encompassing both positively and negatively oriented simplices, effectively doubling its size compared to unoriented approaches. Through coherent interference of these paired simplices, we are able to successfully encode the combinatorial Laplacian, which would otherwise be impossible. This observation constitutes our major technical contribution. We also extend the framework by constructing variant quantum walks. These variants enable us to: (1) estimate the normalized persistent Betti numbers, capturing topological information throughout a deformation process, and (2) verify a specific QMA$_1$-hard problem, showcasing potential applications in computational complexity theory.