Floquet engineering on non-commutative manifolds for nonlinear dimensionality reduction and quantum simulations of gauge fields on inflationary spacetimes

Inspired by the techniques used in classical simulations of large-N Yang-Mills matrix models, we investigate how to encode non-commutative geometries in quantum circuits and develop a new framework for quantum simulations of gauge theories. Within quantum theory, it is possible to associate geometric structures with sets of Hermitian operators, by constructing quasi-coherent states from these operators, and treating their state overlap as a metric within a non-commutative manifold. We develop a procedure for generating Floquet sequences that measure the curvature on these non-commutative manifolds for a given set of high-dimensional quantum gates. With the ability to measure such curvature, we study bundle geometries in which the Floquet sequence encodes the field Lagrangian of a non-commutative Yang-Mills theory. These bundles can exist on any background manifold, and so these methods enable analog and digital quantum simulations of quantum field theories on curved and dynamical spacetime backgrounds. We comment on the unitarity of these fields in an inflationary spacetime, and find that the simulation remains unitary throughout the expansion. Additionally, the matrix Laplacian operator generated by these Floquet sequences is equivalent to the eigenmaps used in manifold learning, a technique from classical machine learning used for nonlinear dimensionality reduction. We provide early considerations on how these Floquet sequences would be able to replicate a manifold learning algorithm, potentially opening up a new method for high-dimensional data analysis that is native to quantum platforms.