  A potential advantage of quantum machine learning stems from the ability of
encoding classical data into high dimensional complex Hilbert space using
quantum circuits. Recent studies exhibit that not all encoding methods are the
same when representing classical data since certain parameterized circuit
structures are more expressive than the others. In this study, we show the
difference in encoding techniques can be visualized by investigating the
topology of the data embedded in complex Hilbert space. The technique for
visualization is a hybrid quantum based topological analysis which uses simple
diagonalization of the boundary operators to compute the persistent Betti
numbers and the persistent homology graph. To augment the computation of Betti
numbers within a NISQ framework, we suggest a simple hybrid algorithm. Through
a illuminating example of a synthetic data set and the methods of angle
encoding, amplitude encoding, and IQP encoding, we reveal topological
differences with the encoding methods, as well as the original data.
Consequently, our results suggest the encoding method needs to be considered
carefully within different quantum machine learning models since it can
strongly affect downstream analysis like clustering or classification.
