Representation theory of inhomogeneous Gaussian unitaries

TL;DR

This work resolves the global-phase ambiguities in composing Gaussian unitaries by lifting the affine symplectic parameters $(M,z)$ to $(M,z,\Psi)$ and introducing an inhomogeneous cocycle $\zeta$. Building on the homogeneous parametrization and circle function, it derives a complete multiplication law and explicit phase formulas for unitaries generated by general quadratic Hamiltonians, including the displacement component. The paper also extends the framework to fermions and provides concrete mappings from general GQH data to the inhomogeneous unitary parameters $(M,z,\Psi)$, with case studies for a single bosonic mode validating the cocycle and phase formulas. The resulting phase-aware structure enables unambiguous, scalable simulation and variational optimization of Gaussian circuits, and it clarifies when non-quadratic (parity-induced) evolutions arise, informing potential non-Gaussian extensions. Together, these contributions bridge abstract representation theory with practical, phase-coherent Gaussian quantum computation and simulation.

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 $(M,z,Ψ)$. 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.

Figures (3)

• Figure 1: Time evolution of the product phase $\zeta(t)$ and $\braket{\hat{N}_J(t)}$. The complex phase $\zeta \in [-\pi,\pi]$ and the expectation value of the total number operator $\braket{\hat{N}_J(t)}$ are plotted as functions of time $t$ for randomly generated Hamiltonians with parameters $(h_\ell,f_\ell)$. Solid lines denote the analytical results given by \ref{['eq:def:zeta']} and \ref{['eq:def:NJ']}, while dashed lines represent truncated numerical approximations with different cutoff values $n_{\mathrm{max}}$, computed using \ref{['eq:num:zeta']} and \ref{['eq:NJ-time']}. All panels share a common legend shown in the lower-left panel. The unstable case has $h_1=\left(0.4-0.6-0.60.5\right)$ and $h_2=\left(0.50.40.4-0.4\right)$ with real eigenvalues $\pm 0.4$ and $\pm 0.6$, respectively. Whereas the stable case has $h_1=\left(0.40.20.20.5\right)$ and $h_2=\left(0.8-0.2-0.20.5\right)$ with purely imaginary eigenvalues $\pm 0.4\mathrm{i}$ and $\pm 0.6\mathrm{i}$, respectively. In all cases, the displacement parameters are fixed to $f_1=f_2=(0.5,0.5)$.
• Figure 2: Bosonic $\braket{J|e^{\widehat{K}+\widehat{f}}|J}$ for $\widehat{K}=a\widehat{X}+c\widehat{Z}$ and $f=\rho(\cos\tau,\sin\tau)$. These panels present the inhomogeneous case in exactly the same configuration as Hackl_2024. We plot $\braket{J|e^{\widehat{K}+\widehat{f}}|J}$ for displacement parameters $\tau=15^\circ,45^\circ$ and $\rho=0.1,1,5,15$. The complex phase \ref{['eq:arg-ekf']} is encoded by color, while the modulus \ref{['eq:mod-ekf']} is represented by brightness. White contour lines indicate respective values of the modulus, as specified by the brightness legend. In all panels, the horizontal axis corresponds to $a$ and the vertical axis to $c$.
• Figure 3: Planar section of bosonic fiber bundle for $\mathrm{Sp}(2,\mathbb{R})$. A general symplectic group element $M$ is parametrized by $(\rho,\theta,z)$, where $\rho$ and $\theta$ are polar coordinates of a horizontal plane and $z$ is a $2\pi$-periodic vertical coordinate, as $M= \cosh{\rho}\left(\cos{z}\sin{z}-\sin{z}\cos{z}\right)-\sinh{\rho}\left(-\sin{\theta}\cos{\theta}\cos{\theta}\sin{\theta}\right)$. We demonstrate a section of $\theta=0$. We also recognize the Lie algebra $\mathfrak{sp}(2,\mathbb{R})=\mathrm{span}(X,Y,Z)$ as tangent space at the identity. We further illustrate the different regions.

Theorems & Definitions (14)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• proof
• Lemma 4
• proof
• proof