  A hybrid classical-quantum approach for the solution of nonlinear ordinary
differential equations using Walsh-Hadamard basis functions is proposed.
Central to this hybrid approach is the computation of the Walsh-Hadamard
transform of arbitrary vectors, which is enabled in our framework using quantum
Hadamard gates along with state preparation, shifting, scaling, and measurement
operations. It is estimated that the proposed hybrid classical-quantum approach
for the Walsh-Hadamard transform of an input vector of size N results in a
considerably lower computational complexity (O(N) operations) compared to the
Fast Walsh-Hadamard transform (O(N log2(N)) operations). This benefit will also
be relevant in the context of the proposed hybrid classical-quantum approach
for the solution of nonlinear differential equations. Comparisons of results
corresponding to the proposed hybrid classical-quantum approach and a purely
classical approach for the solution of nonlinear differential equations (for
cases involving one and two dependent variables) were found to be satisfactory.
Some new perspectives relevant to the natural ordering of Walsh functions (in
the context of both classical and hybrid approaches for the solution of
nonlinear differential equations) and representation theory of finite groups
are also presented here.
