Speaker: James D. Whitfield, Dartmouth College

Date: Nov 4, 2025

Abstract: This seminar will explore foundational aspects of free-fermion systems and their mathematical structures, focusing on the role of symplectic transformations in diagonalizing fermionic operators. We will introduce the algebraic framework of fermionic operators, Gaussian fermionic states, and their covariance matrices. Central to the discussion is the Bloch-Messiah theorem, which underpins the symplectic diagonalization of skew-symmetric matrices arising in these systems. The interplay between orthogonal and symplectic unitaries, as well as the construction of symplectic diagonalizing transformations, will be examined with some proofs, some examples, and some numerical implementations. Applications to quantum dynamics, particularly in the context of efficiently simulating free-fermion evolutions, will be highlighted, along with connections to combinatorial matrix functions such as determinants and pfaffians. This talk aims to provide a rigorous yet accessible bridge between fermionic antisymmetry and quantum information science.