Multidimensional Quantum Generative Modeling by Quantum Hartley Transform

Abstract

An approach for building quantum models based on the exponentially growing orthonormal basis of Hartley kernel functions is developed. First, a differentiable Hartley feature map parameterized by real-valued argument that enables quantum models suitable for solving stochastic differential equations and regression problems is designed. 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, a quantum Hartley transform circuit is proposed as a map between computational and Hartley basis. The developed paradigm is applied 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, the capability of multivariate quantum generative modeling is demonstrated for both correlated and uncorrelated distributions. As a result, the developed quantum Hartley-based generative models (QHGMs) offer a distinct quantum approach to generative AI at increasing scale.