A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics

Abstract

We introduce the first complete and approximatively universal diagrammatic language for quantum mechanics. We make the ZX-Calculus, a diagrammatic language introduced by Coecke and Duncan, complete for the so-called Clifford+T quantum mechanics by adding two new axioms to the language. The completeness of the ZX-Calculus for Clifford+T quantum mechanics was one of the main open questions in categorical quantum mechanics. We prove the completeness of the π 4 -fragment of the ZX-Calculus using the recently studied ZW-Calculus, a calculus dealing with integer matrices. We also prove that the π 4 -fragment of the ZX-Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.Part 4 (Corollary 1) Let D 1 , D 2 be a diagram of the ZX-Calculus. If ZX π/4 ⊢ D 1 XW W X = D 2 XW W X then ZX π/4 ⊢ D 1 = D 2 . Our main theorem is now obvious:Proof (Proof of Theorem 1). Let D 1 , D 2 be two diagrams of the ZX π/4 -Calculus s.t. D 1 = D 2 . By Part 2, D 1 XW = D 2 XW . By Part 1, the ZW 1/2 -calculus is complete and therefore ZW 1/2 ⊢ D 1 XW = D 2 XW . By Part 3, ZX π/4 ⊢ D 1 XW W X = D 2 XW W X . By Part 4 this implies ZX π/4 ⊢ D 1 = D 2 .⊓ ⊔ This approach gives a completion procedure. It gives a set of equalities between ZX π/4 -diagrams whose derivability proves the completeness of the language. Hence, the new rules of the ZX π/4 -Calculus we introduced have obviously been chosen for Parts 4 and 3 to hold. However they have been greatly simplified from what one can obtain using the approach naively.

Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices

“…In Appendix E, we use the ZX-calculus [54,55] to explicitly compute the total non-unitary evolution operator, which is equivalent to the circuit shown in Fig. 18.…”

“…Contracting the upper and lower indices of these tensors around a ring gives a quantum circuit that contains an acausal component. To exorcise the acausality we appeal to the ZX-calculus [54,55], a diagramatic language for simplifying and rewriting quantum operations. The basic rules of this formalism needed for this calculation are summarized in Fig.…”

Universal Properties in Quantum Theory

“…In Appendix B we show that L(Isometry 2 ) is a full subcategory of CPTP, by using a variation of the proof of Theorem 5. We understand that there is ongoing work to add discarding to the ZX-calculus [14], which may be along similar lines.…”

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.…”

“…Backens [3] has shown that the ZX-calculus is sound and complete for stabilizer quantum theory -that is the fragment of quantum mechanics containing only the Clifford operations and states which can be produced from them. While recent extensions to the ZX-calculus have been proposed which are complete for the Clifford+T fragment [16] and for the full qubit theory [22], both these extensions are significantly larger and more complex than the stabilizer subtheory, and both axiomatisations are undergoing rapid development. Clifford circuits therefore provide a stable platform to develop techniques which could later be extended to a universal language.…”