A diagrammatic calculus of fermionic quantum circuits

TL;DR

This paper introduces a diagrammatic calculus for fermionic quantum circuits, providing a complete axiomatization and demonstrating its application to fermionic systems and quantum computing models.

Contribution

It develops the fermionic ZW calculus, a new string-diagrammatic language with complete axioms for fermionic quantum computing, enabling diagrammatic reasoning and calculations.

Findings

Complete axiomatization of the fermionic ZW calculus

Diagrammatic representation of key fermionic quantum gates

Application to fermionic Mach-Zehnder interferometer and dual-rail encoding

Abstract

We introduce the fermionic ZW calculus, a string-diagrammatic language for fermionic quantum computing (FQC). After defining a fermionic circuit model, we present the basic components of the calculus, together with their interpretation, and show how the main physical gates of interest in FQC can be represented in our language. We then list our axioms, and derive some additional equations. We prove that the axioms provide a complete equational axiomatisation of the monoidal category whose objects are systems of finitely many local fermionic modes (LFMs), with maps that preserve or reverse the parity of states, and the tensor product as monoidal product. We achieve this through a procedure that rewrites any diagram in a normal form. As an example, we show how the statistics of a fermionic Mach-Zehnder interferometer can be calculated in the diagrammatic language. We conclude by giving a diagrammatic treatment of the dual-rail encoding, a standard method in optical quantum computing used to perform universal quantum computation.