Matrix Elements of Fermionic Gaussian Operators in Arbitrary Pauli Bases: A Pfaffian Formula

TL;DR

This paper derives a general Pfaffian formula for calculating matrix elements of fermionic Gaussian operators in arbitrary Pauli bases, facilitating scalable quantum computations and revealing underlying algebraic structures.

Contribution

It introduces a Pfaffian formula and sign-encoding matrices for fermionic Gaussian operators in arbitrary bases, connecting to Lie and Clifford algebras.

Findings

Provides a scalable Pfaffian formula for matrix elements

Establishes algebraic structure linking to Lie and Clifford algebras

Enables applications in quantum tomography and fermionic simulations

Abstract

Fermionic Gaussian operators are foundational tools in quantum many-body theory, numerical simulation of fermionic dynamics, and fermionic linear optics. While their structure is fully determined by two-point correlations, evaluating their matrix elements in arbitrary local spin bases remains a nontrivial task, especially in applications involving quantum measurements, tomography, and basis-rotated simulations. In this work, we derive a fully explicit and general Pfaffian formula for the matrix elements of fermionic Gaussian operators between arbitrary Pauli product states. Our approach introduces a pair of sign-encoding matrices whose classification leads to a Lie algebra isomorphic to $\mathfrak{so}(2L)$. This algebraic structure not only guarantees consistency of the Pfaffian signs but also reveals deep connections to Clifford algebras. The resulting framework enables scalable computations across diverse fields -- from quantum tomography and entanglement dynamics to algebraic structure in fermionic circuits and matchgate computation. Beyond its practical utility, our construction sheds light on the internal symmetries of Gaussian operators and offers a new lens through which to explore their role in quantum information and computational models.