Fermionic Gaussian Testing and Non-Gaussian Measures via Convolution

Recent works in quantum computation have recognized that Gaussianity is key to the classical simulation of quantum systems. This is supported by two celebrated results:

1. The Gottesmann-Knill theorem: classical simulation of discrete bosonic Gaussian circuits. Stabilizer states can be viewed as discrete bosonic Gaussian states and Clifford unitaries Gaussian unitaries.
   • Discrete bosonic Gaussian computation corresponds to the mathematical structure of abelian (sub)groups.
2. The matchgate formalism proposed by Leslie Valiant (or fermionic linear optics), consisting of fermionic Gaussian states and unitaries.
   • Fermionic Gaussian computation fundamentally depends on the existence of a small poly-dimensional Lie subalgebra of Gaussian operators.

Gaussian forms have been extensively studied in classical theories; their defining properties include but are not limited to:

1. Exponential-quadratic form in both density and characteristic representations.
2. Maximum entropy subject to power constraint.
3. Extremality with respect to convolution:
   • Fixed points of convolution: central limit theorem.
   • Entropy-power inequality.

The shortlist above emphasizes the pivotal role of the characteristic function (i.e. Fourier transform) and convolution in characterizing Gaussian states. In particular, convolution provides a quantitative tool for analyzing Gaussian behavior. In the quantum case, fermionic Gaussian states have been defined and studied-but not convolution. In this work, we look for a convolution operation satisfying the classical analogues of the properties; the successful definition and proof of all the properties above constitute our work.

Key perspectives in addition to the technical results:

1. The fermionic landscape of convolution and Gaussianity neatly parallels the classical and bosonic ones, suggesting the existence of a more general mathematical theory.
   • The quantum perspective is a nontrivial generalization. For example, super-selection rules from particle statistics restrict the applicability of the convolution theory. The overcoming of this physical (but not computational) restriction motivates our subsequent work.
2. Characteristic functions in classical statistics manifest as physical phase space distributions.
   • The exchange statistics of the particle determine the structure of phase space: e.g. the characteristic functions of discrete bosonic states have integer coordinates, while that of fermionic states have Grassmann coordinates.
3. Quantum generalizations highlight the additional perspective of efficient-computability on Gaussian forms, an aspect missing from a classical understanding.
   • Gaussian forms lie at the intersection of physics, computing, and information.
   • This suggests the possibility of identifying novel frameworks for classical simulation by inspecting other particles (e.g. non-abelian anyons).

My advisor, Kaifeng Bu, presented this project within the larger context of convolution and universal Gaussian testing–including continuous and discrete bosonic systems–at Harvard’s mathematical picture language seminar.