Quantum Algorithms and Oracles with the Scalable ZX-calculus

Categorifying the zx-calculus

This paper presents a symmetric monoidal and compact closed bicategory that categorifies the zx-calculus developed by Coecke and Duncan. The 1-cells in this bicategory are certain graph morphisms that correspond to the string diagrams of the zx-calculus, while the 2-cells are rewrite rules.

The ZX Calculus is incomplete for Clifford+T quantum mechanics

The ZX calculus is a diagrammatic language for quantum mechanics and quantum information processing. We prove that the ZX-calculus is not complete for the Clifford+T quantum mechanics. The completeness for this fragment has been stated as one of the main current open problems in categorical quantum mechanics [8]. The ZX calculus was known to be incomplete for quantum mechanics [7], on the other...

The ZX calculus is incomplete for quantum mechanics

Backens recently proved that the ZX-calculus is complete for an important subset of quantum mechanics, namely stabilizer quantum mechanics, i.e. that for stabilizer quantum mechanics, any equation that can be shown to hold in the Dirac formalism can also be shown to hold within the ZX-calculus[2]. For her proof, she relied on operations on a special class of quantum states, namely graph states....

The ZX calculus is incomplete for non-stabilizer quantum mechanics

The ZX-calculus is complete for stabilizer quantum mechanics

The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics (QM), meaning any pure state, unitary operation and post-selected pure projective measurement can be expressed in the ZX-calculus. The calculus is also sound, i.e. any equality that can be derived graphically can also be derived using matr...