Representation theory of inhomogeneous Gaussian unitaries

TL;DR

This paper extends the parameterization of Gaussian unitaries to include inhomogeneous cases, providing a detailed group structure and factorization into squeezing and displacement transformations, which is fundamental for quantum optics and continuous-variable quantum computing.

Contribution

It introduces a new parameterization for inhomogeneous Gaussian unitaries and derives their group multiplication law, expanding upon previous work on homogeneous cases.

Findings

Extended the framework to inhomogeneous Gaussian unitaries with parameters (M,z,Ψ)

Derived the group multiplication law for these unitaries

Provided a factorization into squeezing and displacement transformations

Abstract

Gaussian unitaries, generated by quadratic Hamiltonians, are fundamental in quantum optics and continuous-variable computing. Their structures correspond to symplectic (bosons) and orthogonal (fermions) groups, but physical realizations give rise to their respective double covers, introducing phase and sign ambiguities. The homogeneous (quadratic-only) case has been resolved through a parameterization constructed in a recent work [arXiv:2409.11628]. We extend the previous framework to inhomogeneous Gaussian unitaries parameterized by $(M,z,\Psi)$. The Baker-Campbel-Hausdorff formula allows us then to factor any Gaussian unitary into a squeezing and a displacement transformation, from which we derive the group multiplication law.