The topology of data hides in quantum thermal states

We provide a quantum protocol to perform topological data analysis (TDA) via the distillation of quantum thermal states. Recent developments of quantum thermal state preparation algorithms reveal their characteristic scaling defined by properties of dissipative Lindbladians. This contrasts with protocols based on unitary evolution which have a scaling depending on the properties of the combinatorial Laplacian. To leverage quantum thermal state preparation algorithms, we translate quantum TDA from a real-time to an imaginary-time picture, shifting the paradigm from a unitary approach to a dissipative one. Starting from an initial state overlapping with the ground state of the system, one can dissipate its energy via channels unique to the dataset, naturally distilling its information. Therefore calculating Betti numbers translates into a purity estimation. Alternatively, this can be interpreted as the evaluation of the R\'{e}nyi 2-entropy, Uhlmann fidelity or Hilbert-Schmidt distance relative to thermal states with the embedded topology of simplicial complexes. Our work opens the field of TDA toward a more physical interpretation of the topology of data.