Efficient Representation of Gaussian Fermionic Pure States in Non-Computational Bases

This paper introduces an innovative approach for representing Gaussian fermionic states, pivotal in quantum spin systems and fermionic models, within a range of alternative quantum bases. We focus on transitioning these states from the conventional computational (\sigma^z) basis to more complex bases, such as ((\phi, \frac{\pi}{2}, \alpha)), which are essential for accurately calculating critical quantities like formation probabilities and Shannon entropy. We present a novel algorithm that not only simplifies the basis transformation but also reduces computational complexity, making it feasible to calculate amplitudes of large systems efficiently. Our key contribution is a technique that translates amplitude calculations into the Pfaffian computation of submatrices from an antisymmetric matrix, a process facilitated by understanding domain wall relationships across different bases. As an application, we will determine the formation probabilities for various bases and configurations within the critical transverse field Ising chain, considering both periodic and open boundary conditions. We aim to categorize the configurations and bases by examining the universal constant term that characterizes the scaling of the logarithm of the formation probability in the periodic system, as well as the coefficient of the logarithmic term in the case of open systems. In the open system scenario, this coefficient is influenced by the central charge and the conformal weight of the boundary condition-changing operator. This work is set to expand the toolkit available for researchers in quantum information theory and many-body physics, providing a more efficient and elegant solution for exploring Gaussian fermionic states in non-standard quantum bases.