Quantum 10, 2138 (2026). https://doi.org/10.22331/q-2026-06-15-2138 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 c