bibkey: negari2026gaussianapproximation authors: Amir-Reza Negari; Farzin Salek; Zoltán Zimborás; Aram Harrow; Patrick Hayden; Jens Eisert year: 2026 title: “Approximation theorems for fermionic Gaussian states” doi: 10.48550/arXiv.2610.01860 url: https://arxiv.org/abs/2610.01860v1 claim: “Section VI asks whether the constants in the high-degree energy and free-energy estimates are optimal.” strata_touched:

• D5/S3/Quantum/Fermionic/FockMajoranaCarrier
• D5/S3/Quantum/Fermionic/ConferenceMatrices
• D5/S3/Quantum/Fermionic/CompleteCartesianGraph
• D5/S3/Quantum/Fermionic/CoordinateCliffordSpectrum
• D5/S3/Quantum/Fermionic/CoordinateEdgeHamiltonian
• D5/S3/Quantum/Fermionic/CoordinateAverage
• D5/S3/Quantum/Fermionic/JointSignProjectors
• D5/S3/Quantum/Fermionic/FlatCliffordGround
• D5/S3/Quantum/Fermionic/CliffordGroundDensity
• D5/S3/Quantum/Fermionic/PhysicalSiteProducts
• D5/S3/Quantum/Fermionic/GibbsProductGap
• D5/S3/Quantum/Fermionic/GaussianHighDegreeSharpness license: citation-only triage: anchor

Approximation theorems for fermionic Gaussian states

Section VI, printed page 26, states:

It remains open, however, whether the constants in the resulting high-degree energy and free-energy estimates are optimal, since their derivation also uses edge averaging, Cauchy–Schwarz, and only the operator-norm normalization of the interactions.

The estimates are Theorem III.1, equation (44), and Theorem III.5, equation (79). Both have bound $\sqrt{2m/D}$ for a finite simple $D$-regular graph with $m$ fermionic modes per vertex and $D\ge 1$. The Hamiltonian is the edge average $H=|E{|}^{-1}{\sum }_{e\in E}{H}_e$, with real quadratic inter-site Majorana coefficients and operator norm $\Vert H_e\Vert \le 1$.

The Majorana convention is ${\gamma }_{2j-1}=c_j+c_j^{†}$ and ${\gamma }_{2j}=i(c_j-c_j^{†})$. Physical density matrices satisfy $[\rho ,P]=0$, where $P=(-1{)}^{\hat{N}}$ is number parity.

The energy comparison minimizes $\mathrm{tr}(H\rho )$ over physical density matrices and over physical product density matrices. The free-energy comparison uses the actual Gibbs density ${\rho }_{\beta }=e^{-\beta H}/\mathrm{tr}(e^{-\beta H})$ and the tensor product ${\sigma }_{\beta }$ of its one-site partial traces, with $F_{\beta }(\omega )=\mathrm{tr}(H\omega )-S(\omega )/\beta$ and $S(\omega )=-\mathrm{tr}(\omega \log \omega )$ in nats.

Uniform optimality of the prefactor does not require equality at every degree or at any finite inverse temperature. These are distinct questions.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2610.01860
• Source: https://arxiv.org/abs/2610.01860v1

Pages 3–4 fix the carrier and operators:

The number operator and parity operator are given by

$\hat{N}={\sum }_{j=1}^m{c}_j^{†}{c}_j$, $P=(-1{)}^{\hat{N}}$.

From the creation and annihilation operators we define Hermitian Majorana operators by

${\gamma }_{2j-1}=c_j+c_j^{†}$, ${\gamma }_{2j}=i(c_j-c_j^{†})$, $j=1,\dots ,m$.

They satisfy the Clifford relations

$\{{\gamma }_p,{\gamma }_q\}=2{\delta }_{p,q}1$, ${\gamma }_p^{†}={\gamma }_p$.

Physical fermionic states obey the parity superselection rule

$[\rho ,P]=0$. The Lean reading of the operator power is fixed by preregistration #15127.