© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1903.06035 · Logical Methods in Computer Science

Completeness of the ZX-Calculus

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart

Published Jun 4, 2020• 18 citations

Abstract

The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. It comes equipped with an equational presentation. We focus here on a very important property of the language: completeness, which roughly ensures the equational theory captures all of quantum mechanics. We first improve on the known-to-be-complete presentation for the so-called Clifford fragment of the language - a restriction that is not universal - by adding some axioms. Thanks to a system of back-and-forth translation between the ZX-Calculus and a third-party complete graphical language, we prove that the provided axiomatisation is complete for the first approximately universal fragment of the language, namely Clifford+T. We then prove that the expressive power of this presentation, though aimed at achieving completeness for the aforementioned restriction, extends beyond Clifford+T, to a class of diagrams that we call linear with Clifford+T constants. We use another version of the third-party language - and an adapted system of back-and-forth translation - to complete the language for the ZX-Calculus as a whole, that is, with no restriction. We briefly discuss the added axioms, and finally, we provide a complete axiomatisation for an altered version of the language which involves an additional generator, making the presentation simpler.

DOI: 10.23638/lmcs-16(2:11)2020

Related papers

Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart

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 Clifford+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 ZX-calculus for pure quantum mechanics than the one recently introduced by Ng&Wang.

arXiv:1801.10142•ACM/IEEE Symposium on Logic in Computer Science (LICS)•Jun 27, 2018•55 citations

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

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart

We introduce the first complete and approximately 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 -- also called the π/4-fragment of the ZX-Calculus -- was one of the main open questions in categorical quantum mechanics. We prove the completeness of this fragment 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.

•May 31, 2017•110 citations

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

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart

We introduce the first complete and approximately 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 -- also called the π/4-fragment of the ZX-Calculus -- was one of the main open questions in categorical quantum mechanics. We prove the completeness of this fragment 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.

arXiv:1705.11151•ACM/IEEE Symposium on Logic in Computer Science (LICS)•Jun 27, 2018•6 citations

ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart +1 more

The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language -- i.e. the ability to derive any true equation -- is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity -- called cyclotomic supplementarity -- which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning.We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.

arXiv:1702.01945•Leibniz international proceedings in informatics•Jan 1, 2017•12 citations

The ZX Calculus is incomplete for Clifford+T quantum mechanics

Simon Perdrix, Quanlong Wang

The ZX calculus is a diagrammatic language for quantum mechanics and quantum information processing. We prove that the ZX-calculus is not complete for Clifford+T quantum mechanics. The completeness for this fragment has been stated as one of the main current open problems in categorical quantum mechanics. The ZX calculus was known to be incomplete for quantum mechanics, on the other hand, it has been proved complete for Clifford quantum mechanics (a.k.a. stabilizer quantum mechanics), and for single-qubit Clifford+T quantum mechanics. The question of the completeness of the ZX calculus for Clifford+T quantum mechanics is a crucial step in the development of the ZX calculus because of its (approximate) universality for quantum mechanics (i.e. any unitary evolution can be approximated using Clifford and T gates only). We exhibit a property which is know to be true in Clifford+T quantum mechanics and prove that this equation cannot be derived in the ZX calculus by introducing a new sound interpretation of the ZX calculus in which this particular property does not hold. Finally, we propose to extend the language with a new axiom.

•Jun 8, 2015•2 citations

ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces

Boldizsár Poór, Razin Abdulrauf Shaikh, Quanlong Wang

The ZX-calculus is a graphical language for reasoning about quantum computing and quantum information theory.As a complete graphical language, it incorporates a set of axioms rich enough to derive any equation of the underlying formalism.While completeness of the ZX-calculus has been established for qubits and the Clifford fragment of prime-dimensional qudits, universal completeness beyond two-level systems has remained unproven until now.In this paper, we present a proof establishing the completeness of finite-dimensional ZX-calculus, incorporating only the mixeddimensional Z-spider and the qudit X-spider as generators.Our approach builds on the completeness of another graphical language, the finite-dimensional ZW-calculus, with direct translations between these two calculi.By proving its completeness, we lay a solid foundation for the ZX-calculus as a versatile tool not only for quantum computation but also for various fields within finite-dimensional quantum theory.

arXiv:2405.10896•Electronic Proceedings in Theoretical Computer Science•Aug 19, 2025•2 citations

The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. It comes equipped with an equational presentation. We focus here on a very important property of the language: completeness, which roughly ensures the equational theory captures all of quantum mechanics. We first improve on the known-to-be-complete presentation for the so-called Clifford fragment of the language - a restriction that is not universal - by adding some axioms. Thanks to a system of back-and-forth translation between the ZX-Calculus and a third-party complete graphical language, we prove that the provided axiomatisation is complete for the first approximately universal fragment of the language, namely Clifford+T. We then prove that the expressive power of this presentation, though aimed at achieving completeness for the aforementioned restriction, extends beyond Clifford+T, to a class of diagrams that we call linear with Clifford+T constants. We use another version of the third-party language - and an adapted system of back-and-forth translation - to complete the language for the ZX-Calculus as a whole, that is, with no restriction. We briefly discuss the added axioms, and finally, we provide a complete axiomatisation for an altered version of the language which involves an additional generator, making the presentation simpler.

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 Clifford+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 ZX-calculus for pure quantum mechanics than the one recently introduced by Ng&Wang.

We introduce the first complete and approximately 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 -- also called the π/4-fragment of the ZX-Calculus -- was one of the main open questions in categorical quantum mechanics. We prove the completeness of this fragment 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.

We introduce the first complete and approximately 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 -- also called the π/4-fragment of the ZX-Calculus -- was one of the main open questions in categorical quantum mechanics. We prove the completeness of this fragment 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.

The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language -- i.e. the ability to derive any true equation -- is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity -- called cyclotomic supplementarity -- which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning.We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.

The ZX calculus is a diagrammatic language for quantum mechanics and quantum information processing. We prove that the ZX-calculus is not complete for Clifford+T quantum mechanics. The completeness for this fragment has been stated as one of the main current open problems in categorical quantum mechanics. The ZX calculus was known to be incomplete for quantum mechanics, on the other hand, it has been proved complete for Clifford quantum mechanics (a.k.a. stabilizer quantum mechanics), and for single-qubit Clifford+T quantum mechanics. The question of the completeness of the ZX calculus for Clifford+T quantum mechanics is a crucial step in the development of the ZX calculus because of its (approximate) universality for quantum mechanics (i.e. any unitary evolution can be approximated using Clifford and T gates only). We exhibit a property which is know to be true in Clifford+T quantum mechanics and prove that this equation cannot be derived in the ZX calculus by introducing a new sound interpretation of the ZX calculus in which this particular property does not hold. Finally, we propose to extend the language with a new axiom.

The ZX-calculus is a graphical language for reasoning about quantum computing and quantum information theory.As a complete graphical language, it incorporates a set of axioms rich enough to derive any equation of the underlying formalism.While completeness of the ZX-calculus has been established for qubits and the Clifford fragment of prime-dimensional qudits, universal completeness beyond two-level systems has remained unproven until now.In this paper, we present a proof establishing the completeness of finite-dimensional ZX-calculus, incorporating only the mixeddimensional Z-spider and the qudit X-spider as generators.Our approach builds on the completeness of another graphical language, the finite-dimensional ZW-calculus, with direct translations between these two calculi.By proving its completeness, we lay a solid foundation for the ZX-calculus as a versatile tool not only for quantum computation but also for various fields within finite-dimensional quantum theory.