Trichromatic Open Digraphs for Understanding Qubits

TL;DR

This paper introduces a new trichromatic graphical calculus for quantum computing that symmetrically represents three observables, enabling derivation of new qubit equations beyond existing Z/X-calculus.

Contribution

It develops a novel trichromatic calculus that extends the Z/X-calculus, capturing symmetries of the Bloch sphere and deriving new qubit relationships.

Findings

Derived Euler angle decomposition of the Hadamard gate.

Established supplementary relationships valid for qubits.

Extended the Z/X-calculus with a third observable.

Abstract

We introduce a trichromatic graphical calculus for quantum computing. The generators represent three complementary observables that are treated on equal footing, hence reflecting the symmetries of the Bloch sphere. We derive the Euler angle decomposition of the Hadamard gate within it as well as the so-called supplementary relationships, which are valid equations for qubits that were not derivable within Z/X-calculus of Coecke and Duncan. More specifically, we have: dichromatic Z/X-calculus + Euler angle decomposition of the Hadamard gate = trichromatic calculus.