  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.
