ZX-calculus: Cyclotomic supplementarity and incompleteness for Clifford+T quantum mechanics
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 n...
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| Fformat: | Conference item |
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Schloss Dagstuhl
2017
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| _version_ | 1826265790778703872 |
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| author | Jeandel, E Perdrix, S Vilmart, R Wang, Q |
| author_facet | Jeandel, E Perdrix, S Vilmart, R Wang, Q |
| author_sort | Jeandel, E |
| collection | OXFORD |
| description | 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 ZXCalculus, 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. |
| first_indexed | 2024-03-06T20:29:10Z |
| format | Conference item |
| id | oxford-uuid:3077f5fa-bf9b-4419-9b5f-66ac5e2c2e0e |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:29:10Z |
| publishDate | 2017 |
| publisher | Schloss Dagstuhl |
| record_format | dspace |
| spelling | oxford-uuid:3077f5fa-bf9b-4419-9b5f-66ac5e2c2e0e2022-03-26T13:01:34ZZX-calculus: Cyclotomic supplementarity and incompleteness for Clifford+T quantum mechanicsConference itemhttp://purl.org/coar/resource_type/c_5794uuid:3077f5fa-bf9b-4419-9b5f-66ac5e2c2e0eSymplectic Elements at OxfordSchloss Dagstuhl2017Jeandel, EPerdrix, SVilmart, RWang, QThe 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 ZXCalculus, 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. |
| spellingShingle | Jeandel, E Perdrix, S Vilmart, R Wang, Q ZX-calculus: Cyclotomic supplementarity and incompleteness for Clifford+T quantum mechanics |
| title | ZX-calculus: Cyclotomic supplementarity and incompleteness for Clifford+T quantum mechanics |
| title_full | ZX-calculus: Cyclotomic supplementarity and incompleteness for Clifford+T quantum mechanics |
| title_fullStr | ZX-calculus: Cyclotomic supplementarity and incompleteness for Clifford+T quantum mechanics |
| title_full_unstemmed | ZX-calculus: Cyclotomic supplementarity and incompleteness for Clifford+T quantum mechanics |
| title_short | ZX-calculus: Cyclotomic supplementarity and incompleteness for Clifford+T quantum mechanics |
| title_sort | zx calculus cyclotomic supplementarity and incompleteness for clifford t quantum mechanics |
| work_keys_str_mv | AT jeandele zxcalculuscyclotomicsupplementarityandincompletenessforcliffordtquantummechanics AT perdrixs zxcalculuscyclotomicsupplementarityandincompletenessforcliffordtquantummechanics AT vilmartr zxcalculuscyclotomicsupplementarityandincompletenessforcliffordtquantummechanics AT wangq zxcalculuscyclotomicsupplementarityandincompletenessforcliffordtquantummechanics |