A Simplified Stabilizer ZX-calculus

Abstract

The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics. The language is sound and complete: a stabilizer ZX-diagram can be transformed into another one if and only if these two diagrams represent the same quantum evolution or quantum state. We show that the stabilizer ZX-calculus can be simplified, removing unnecessary equations while keeping only the essential axioms which potentially capture fundamental structures of quantum mechanics. We thus give a significantly smaller set of axioms and prove that meta-rules like 'colour symmetry' and 'upside-down symmetry', which were considered as axioms in previous versions of the language, can in fact be derived. In particular, we show that the additional symbol and one of the rules which had been recently introduced to keep track of scalars (diagrams with no inputs or outputs) are not necessary.

A Near-Minimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics

“…We conjecture that all the axioms in Figure 1 are necessary. Indeed, in [1], nearly all the rules for Clifford -i.e. all of the axioms in Figure 1 except (E) and (EU)-are proven to be necessary, and all arguments stand here: -(S): It is the only axiom that can transform a node of degree four or higher into a diagram containing lower-degree nodes -(I g ) or (I r ): These are the only two axioms that can transform a diagram with nodes connected to a boundary to a node-free diagram -(CP): It is the only axiom that can transform a diagram with two connected outputs into one with two disconnected outputs -(HD): The necessity of this axiom requires a non-trivial interpretation given in [15,17], and given again in the Appendix at page 15.…”

“…As we just said, a first and easy step to do is to show that we can recover the rules that are known to make the language complete for Clifford [1]. This will allow us to freely use in the following all the equations of the π 2 -fragment that are sound.…”

Optimising Clifford Circuits with Quantomatic

“…The ZX-calculus has a rich equational theory based on the theory of Frobenius-Hopf algebras [7,10]. Various axiomatisations have been proposed ( [12,13,5,23,16,22]) with various advantages and drawbacks. Here we adopt the scheme of Backens [3] which is clean, concise, and adequate for the treatment of the Clifford group.…”

ZX-Rules for 2-Qubit Clifford+T Quantum Circuits

“…We will do so too here, since this makes that the rules of the ZXcalculus appear much simpler (see e.g. [5] for a presentation of the ZX-calculus rules with explicit scalars that make equations hold on-the-nose). Due to the diagrammatic underpinning, in addition to the rules given below, there is one meta-rule that ZX-calculus obeys, namely:…”