Multidimensional Quantum Generative Modeling by Quantum Hartley Transform

TL;DR

This paper introduces a quantum Hartley transform-based approach for multidimensional generative modeling, leveraging real-valued quantum states and kernels to improve quantum AI capabilities.

Contribution

It develops a novel quantum Hartley transform circuit and a differentiable Hartley feature map for enhanced quantum generative modeling and solving stochastic differential equations.

Findings

Quantum Hartley models enable real-valued amplitude states.

The approach improves sampling from complex distributions.

Tools for multivariate quantum generative modeling are provided.

Abstract

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.