The commutant of fermionic Gaussian unitaries

Abstract

In this work, we characterize the $t$-th order commutants of fermionic Gaussian unitaries and of their particle-preserving subgroup acting on $n$ fermionic modes. These commutants govern Haar averages over the corresponding groups and therefore play a central role in fermionic randomized protocols, invariant theory, and resource quantification. Using Howe dualities, we show that the particle-preserving commutant is generated by generalized copy-hopping operators, while that for general Gaussian commutant is generated by generalized quadratic Majorana bilinears together with parity. We then derive closed formulas for the dimensions of both commutants as functions of $t$ and $n$, and develop constructive Gelfand--Tsetlin procedures to obtain explicit orthonormal bases, with detailed low-$t$ examples. Our framework also clarifies the structure of replicated fermionic states and connects naturally to measures of fermionic correlations, generalized Plücker-type constraints, and the stabilizer entropy of fermionic Gaussian states. These results provide a unified algebraic description of higher-order invariants for fermionic Gaussian dynamics.

Figures (3)

• Figure 1: Dimensions of the commutants. We plot in a logarithmic scale the dimensions of $\mathcal{C}_{t,n}$ and $\mathcal{C}_{t,n}^{\rm PP}$ as a function of the number of fermionic modes $n$, for values of $t=2,3,4,5$.
• Figure 2: Comparison of numerical estimates and analytical predictions for $\mathbb{E}_{U\sim\mathbb{U}(n)}S_4(R_r(U)\ket{r})$. The solid dots correspond to the average over $10^4$ independent samples of particle-preserving fermionic gaussian unitaries, and the associated error bars show one standard deviation from the mean. The dashed lines instead are the closed formula shown in Eq. \ref{['eq:avg_pp_stab_sector_formula_1']}. The system sizes considered are $n=2,\dots,8$ and for each $n$ we show the results from the sectors with $r=1,\dots,\min(n,4)$ particles, corresponding to colors blue, orange, green and red, respectively.
• Figure 3: Comparison of numerical estimates and analytical predictions for $\mathbb{E}_{U\sim\mathbb{SPIN}(2n)}S_4(U\ket{0})$. For increasing system sizes $n=2,\dots,8$, the dots show the average of $S_4(U\ket{0})$ over $10^4$ independent samples of $U\sim\mathbb{SPIN}(2n)$ and the associated error bars show one standard deviation from the mean. The dashed line instead shows the analytical expression for $\mathbb{E}_{U\sim\mathbb{SPIN}(2n)}S_4(U\ket{0})$ given in Eq. \ref{['eq:avg_gauss_stab']}.

Theorems & Definitions (32)

• Lemma 1
• Lemma 2
• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4
• Theorem 3: Commutant generators
• Theorem 4: Commutant dimension
• Lemma 5
• Lemma 1