Quadratic tensors as a unification of Clifford, Gaussian, and free-fermion physics

TL;DR

This paper introduces a unifying algebraic framework using quadratic functions over abelian groups and Hopf algebras to describe and efficiently analyze various quantum models, including Clifford circuits, fermionic systems, and stabilizer codes.

Contribution

It presents a novel formalism based on quadratic tensors that unifies multiple quantum models and enables efficient tensor network contractions, extending to models with mixed degrees of freedom.

Findings

Quadratic tensors fully specify certain quantum models with O(n^2) parameters.

Tensor networks of quadratic tensors can be contracted efficiently using a Schur complement-like operation.

The formalism includes models with mixed degrees of freedom and generalizes to higher-order tensors.

Abstract

Certain families of quantum mechanical models can be described and solved efficiently on a classical computer, including qubit or qudit Clifford circuits and stabilizer codes, free-boson or free-fermion models, and certain rotor and GKP codes. We show that all of these families can be described as instances of the same algebraic structure, namely quadratic functions over abelian groups, or more generally over (super) Hopf algebras. Different kinds of degrees of freedom correspond to different "elementary" abelian groups or Hopf algebras: $\mathbb{Z}_2$ for qubits, $\mathbb{Z}_d$ for qudits, $\mathbb{R}$ for continuous variables, both $\mathbb{Z}$ and $\mathbb{R}/\mathbb{Z}$ for rotors, and a super Hopf algebra $\mathcal F$ for fermionic modes. Objects such as states, operators, superoperators, or projection-operator valued measures, etc, are tensors. For the solvable models above, these tensors are quadratic tensors based on quadratic functions. Quadratic tensors with $n$ degrees of freedom are fully specified by only $O(n^2)$ coefficients. Tensor networks of quadratic tensors can be contracted efficiently on the level of these coefficients, using an operation reminiscent of the Schur complement. Our formalism naturally includes models with mixed degrees of freedom, such as qudits of different dimensions. We also use quadratic functions to define generalized stabilizer codes and Clifford gates for arbitrary abelian groups. Finally, we give a generalization from quadratic (or 2nd order) to $i$th order tensors, which are specified by $O(n^i)$ coefficients but cannot be contracted efficiently in general.