Graphical Calculus for Fermionic Tensors

TL;DR

This paper develops a graphical calculus for fermionic tensors that enables diagrammatic computations in fermionic many-body physics, extending existing qubit-based formalisms like the ZX calculus.

Contribution

It introduces a new graphical calculus for fermionic tensors, incorporating fermionic modes, qubits, and odd-parity states, expanding the tools available for fermionic quantum systems.

Findings

Represented fermionic Gaussian states diagrammatically

Applied calculus to fermionic partial trace and purification protocols

Constructed fermionic codes using the graphical formalism

Abstract

We introduce a graphical calculus, consisting of a set of fermionic tensors with tensor-network equations, which can be used to perform various computations in fermionic many-body physics purely diagrammatically. The indices of our tensors primarily correspond to fermionic modes, but also include qubits and fixed odd-parity states. Our graphical calculus extends the ZX calculus for systems involving qubits. We apply the calculus in order to represent various objects, operations, and computations in physics, including fermionic Gaussian states, the partial trace of Majorana modes, purification protocols, fermionization and bosonization maps, and the construction of fermionic codes.