Quantum Walks on Simplicial Complexes and Harmonic Homology: Application to Topological Data Analysis with Superpolynomial Speedups
arxiv(2024)
摘要
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.
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