Fermionic Gaussian Testing and Non-Gaussian Measures via Convolution

TL;DR

Using fermionic convolution, an efficient protocol is proposed that tests the fermionic Gaussianity of pure states using three copies of the input state and non-Gaussian entropy is introduced, an experimentally accessible resource measure that quantifies fermionic non-Gaussianity.

Abstract

We define fermionic convolution and demonstrate its utility in characterizing fermionic non-Gaussian components, which are essential to the computational advantage of fermionic systems. Using fermionic convolution, we propose an efficient protocol that tests the fermionic Gaussianity of pure states using three copies of the input state. We also introduce "Non-Gaussian Entropy," an experimentally accessible resource measure that quantifies fermionic non-Gaussianity. These results provide new insights into the study of fermionic quantum computation.

Figures (9)

• Figure 1: The Gaussification $\mathcal{G}(\rho)$ is the closest state to the states $\rho$ and $\boxtimes^k\rho$.
• Figure 2: non-Gaussian entropy $NG^{(k)}(\psi_\phi)$ of the $4$-qubit state $\psi_\phi$ in \ref{['def:4QubitMagic']} for $k=1, 2, 3, \infty$.
• Figure 3: Non-Gaussian $K_M(\rho)$(blue) and Gaussian weight $K_G(\rho)$(red) of the $4$-qubit family \ref{['eq:4qubitFamily']}.
• Figure 4: Total cumulant weight $K(\rho)$ (Definition \ref{['def:cumulantWeights']}) of the $4$-qubit family \ref{['eq:4qubitFamily']}.
• Figure 5: Elementary decomposition of the $1$-qubit convolution unitary $W_\theta^{(1)}$. Here $R_z(\theta) = \exp(-i\theta Z/2)$.

Theorems & Definitions (93)

• Definition 1: Convolution
• Proposition 2
• Proposition 3
• Proposition 4
• Theorem 5
• Theorem 6
• Theorem 7
• Definition A.1: Clifford algebra
• Definition A.2: Grassmann algebra
• Definition A.3: Anti-commuting tensor product