  Topological data analysis (TDA) is a powerful technique for extracting
complex and valuable shape-related summaries of high-dimensional data. However,
the computational demands of classical algorithms for computing TDA are
exorbitant, and quickly become impractical for high-order characteristics.
Quantum computers offer the potential of achieving significant speedup for
certain computational problems. Indeed, TDA has been purported to be one such
problem, yet, quantum computing algorithms proposed for the problem, such as
the original Quantum TDA (QTDA) formulation by Lloyd, Garnerone and Zanardi,
require fault-tolerance qualifications that are currently unavailable. In this
study, we present NISQ-TDA, a fully implemented end-to-end quantum machine
learning algorithm needing only a short circuit-depth, that is applicable to
high-dimensional classical data, and with provable asymptotic speedup for
certain classes of problems. The algorithm neither suffers from the
data-loading problem nor does it need to store the input data on the quantum
computer explicitly. The algorithm was successfully executed on quantum
computing devices, as well as on noisy quantum simulators, applied to small
datasets. Preliminary empirical results suggest that the algorithm is robust to
noise.
