  We develop an approach for building quantum models based on the exponentially
growing orthonormal basis of Hartley kernel functions. First, we design a
differentiable Hartley feature map parametrized by real-valued argument that
enables quantum models suitable for solving stochastic differential equations
and regression problems. Unlike the naturally complex Fourier encoding, the
proposed Hartley feature map circuit leads to quantum states with real-valued
amplitudes, introducing an inductive bias and natural regularization. Next, we
propose a quantum Hartley transform circuit as a map between computational and
Hartley basis. We apply the developed paradigm to generative modeling from
solutions of stochastic differential equations, and utilize the quantum Hartley
transform for fine sampling from parameterized distributions through an
extended register. Finally, we present tools for implementing multivariate
quantum generative modeling for both correlated and uncorrelated distributions.
As a result, the developed quantum Hartley models offer a distinct quantum
approach to generative AI at increasing scale.
