Quantum Walks on Simplicial Complexes and Harmonic Homology: Application to Topological Data Analysis with Superpolynomial Speedups
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 object reflecting the topology of the simplicial complex. We construct a unitary encoding projecting onto the kernel of the Laplacian, representing the harmonic cycles in the complex's homology. Our efficient construction of quantum walk unitaries for clique complexes paves the way for exploring higher-order interactions within topological structures. Our construction requires $O(n^3\log(1/ε)/λ_k)$ gates, where $n$ is the number of vertices, $λ_k$ is the smallest non-zero eigenvalue of the Laplacian, and $ε$ is the projection error. Our results indicate apparent superpolynomial quantum speedup with quantum walks, without quantum oracles, provided the spectral gap of the Laplacian is inverse-polynomially bounded and efficient simplex sampling is available. 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 is our major technical contribution. We also extend the framework by constructing variant quantum walks that enable us to: (1) estimate normalized persistent Betti numbers throughout a deformation process, (2) verify a specific QMA$_1$-hard problem related to clique complex homology, showcasing potential applications in computational complexity theory, and (3) solve the high-dimensional discrete Dirichlet problem (HDDP), generalizing the classical discrete Dirichlet problem on graphs to simplicial complexes, with an apparent superpolynomial speedup over the best known classical algorithm.