[論文レビュー] Representation theory of inhomogeneous Gaussian unitaries

Jingqi Sun, Joshua Combes+1 more|arXiv (Cornell University)|Feb 9, 2026

Matrix Theory and Algorithms被引用数 0

ひとこと要約

Extends the representation theory of Gaussian unitaries to inhomogeneous (quadratic plus linear) cases, deriving the full group multiplication law including the inhomogeneous cocycle for bosons and fermions.

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.

研究の動機と目的

• Motivate the study of Gaussian unitaries beyond purely quadratic Hamiltonians by including linear terms in the Hamiltonian.
• Develop a complete parametrization of inhomogeneous Gaussian unitaries in terms of (M, z, Psi).
• Derive the full group multiplication law including the inhomogeneous cocycle for phase consistency.
• Connect the inhomogeneous construction to the known homogeneous case and displacement group to enable phase-aware circuit analysis.

提案手法

• Review the structure of Gaussian states and their phase-space representations via quadrature operators.
• Use Baker-Campbell-Hausdorff (BCH) and Cartan decomposition to factor Gaussian unitaries into squeezing and displacement parts.
• Introduce a reference Gaussian state |J> and define a reference phase Phi_J to track unitary phases.
• Decompose general inhomogeneous unitaries as U(M, z, Psi) and derive the exact multiplication law with a cocycle zeta(M1, M2, z1, z2).
• Employ a double-cover construction to manage phase ambiguities, leading to the inhomogeneous Mp group extension.
• Provide explicit expressions for the phase gamma(M, z) and the inhomogeneous cocycle zeta that govern composition.
• Relate the bosonic and fermionic cases through unified notation and their respective symplectic/orthogonal structures.

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 $(

実験結果

リサーチクエスチョン

• RQ1 How can inhomogeneous Gaussian unitaries (including linear terms) be parametrized to capture their full phase information?
• RQ2 What is the exact group multiplication law for inhomogeneous Gaussian unitaries, including the inhomogeneous cocycle?
• RQ3 How does one extend the homogeneous parametrization to include displacements while preserving a true representation (via a double cover)?
• RQ4 How can the phase of a general Gaussian unitary be computed from the Hamiltonian and reference Gaussian structure?
• RQ5 How do the bosonic and fermionic cases differ in their group structures and phase behavior, and how are they unified?

主な発見

• A complete parametrization of inhomogeneous Gaussian unitaries is established as U(M, z, Psi) with a detailed multiplication law that includes a cocycle term zeta(M1, M2, z1, z2).
• The group structure is identified as the double cover of the inhomogeneous group, IMp(2N, R), with a central U(1) phase factored in.
• Explicit expressions are provided for the displacement-squeezing phase gamma(M, z) and for the cocycle zeta that encode phase changes under composition.
• A decomposition of a general unitary into displacement and squeezing parts is used to derive the exact phase relations and to ensure associativity of the product.
• A unified treatment for bosons and fermions is developed, including their respective symplectic and orthogonal structures and the corresponding double covers Mp(2N, R) and Pin(2N, R).
• The work shows how to deduce the phase Psi from the Hamiltonian H when the unitary is e^{-i H}, linking algebraic data to dynamical parameters.

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 [ 36 , Fig. 6] . We plot $\braket{J|e^{\widehat{K}+\widehat{f}}|J}$ for displace