Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics

Abstract

The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. An axiomatisation has recently been proven to be complete for an approximatively universal fragment of quantum mechanics, the so-called Clif-ford+T fragment. We focus here on the expressive power of this axiomatisation beyond Clifford+T Quantum mechanics. We consider the full pure qubit quantum mechanics, and mainly prove two results: (i) First, the axiomatisation for Clifford+T quantum mechanics is also complete for all equations involving some kind of linear diagrams. The linearity of the diagrams reflects the phase group structure, an essential feature of the ZX-calculus. In particular all the axioms of the ZX-calculus are involving linear diagrams. (ii) We also show that the axiomatisation for Clifford+T is not complete in general but can be completed by adding a single (non linear) axiom, providing a simpler axiomatisation of the ZXcalculus for pure quantum mechanics than the one recently introduced by Ng&Wang.

A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond

“…We want to know what we can prove without the limitation to α, β ∈ {0, π} in the set of rules. To do so, we are going to adapt a result that was developed in [19], and see what is missing to prove equalities that are valid for linear diagrams with constants in {0, π}. The following of the section will give a sketch of the proof but will not dive into its details.…”

“…The following of the section will give a sketch of the proof but will not dive into its details. For more, see [19]. We call a linear diagram, a diagram where some angles are treated as variables, and where the parameters of the green and red dots are affine combinations of the variables, where coefficients are in Z:…”

“…A first step was made when completeness was discovered for the 1-qubit operations of the restriction [3]. A complete axiomatisation for the many-qubits Clifford+T fragment was finally discovered in [18], and lead to a complete axiomatisation in the general case, first in [23,16] where two new generators were added, and then in [19] with the original syntax.…”

“…We prove it is universal, sound and complete in Section 3. In Section 4, we try and adapt a theorem from [19] which gives a completeness result on a broader restriction of the language. We finally give in Section 5 a simple axiomatisation for the Clifford+T fragment of the ZX-Calculus with triangles (that corresponds to the (Toffoli, H, R Z (π/4)) gate set in circuitry), and prove it is complete.…”

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

“…-Firstly, we wish to understand the utility of the ZX-rules. In the case of the full completion [24,21] these were added using a purely theoretical methodology which consisted of translating Hilbert space structure into diagrams, passing via another graphical calculus [17,18]. However, a natural question concerns the actual practical use of each of these rules, as well as of other rules derived from them.…”