Towards a Minimal Stabilizer ZX-calculus
The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics. The language is sound and complete: one can transform a stabilizer ZX-diagram into another one using the graphical rewrite rules if and only if these two diagrams represent the same quantum evolution o...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Logical Methods in Computer Science e.V.
2020-12-01
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| Series: | Logical Methods in Computer Science |
| Subjects: | |
| Online Access: | https://lmcs.episciences.org/3961/pdf |
| _version_ | 1797268598739173376 |
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| author | Miriam Backens Simon Perdrix Quanlong Wang |
| author_facet | Miriam Backens Simon Perdrix Quanlong Wang |
| author_sort | Miriam Backens |
| collection | DOAJ |
| description | The stabilizer ZX-calculus is a rigorous graphical language for reasoning
about quantum mechanics. The language is sound and complete: one can transform
a stabilizer ZX-diagram into another one using the graphical rewrite rules if
and only if these two diagrams represent the same quantum evolution or quantum
state. We previously showed that the stabilizer ZX-calculus can be simplified
by reducing the number of rewrite rules, without losing the property of
completeness [Backens, Perdrix & Wang, EPTCS 236:1--20, 2017]. Here, we show
that most of the remaining rules of the language are indeed necessary. We do
however leave as an open question the necessity of two rules. These include,
surprisingly, the bialgebra rule, which is an axiomatisation of
complementarity, the cornerstone of the ZX-calculus. Furthermore, we show that
a weaker ambient category -- a braided autonomous category instead of the usual
compact closed category -- is sufficient to recover the meta rule 'only
connectivity matters', even without assuming any symmetries of the generators. |
| first_indexed | 2024-04-25T01:35:02Z |
| format | Article |
| id | doaj.art-eeef2137aee6430a8967e3125c74e58c |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:35:02Z |
| publishDate | 2020-12-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-eeef2137aee6430a8967e3125c74e58c2024-03-08T10:32:06ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742020-12-01Volume 16, Issue 410.23638/LMCS-16(4:19)20203961Towards a Minimal Stabilizer ZX-calculusMiriam BackensSimon PerdrixQuanlong WangThe stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics. The language is sound and complete: one can transform a stabilizer ZX-diagram into another one using the graphical rewrite rules if and only if these two diagrams represent the same quantum evolution or quantum state. We previously showed that the stabilizer ZX-calculus can be simplified by reducing the number of rewrite rules, without losing the property of completeness [Backens, Perdrix & Wang, EPTCS 236:1--20, 2017]. Here, we show that most of the remaining rules of the language are indeed necessary. We do however leave as an open question the necessity of two rules. These include, surprisingly, the bialgebra rule, which is an axiomatisation of complementarity, the cornerstone of the ZX-calculus. Furthermore, we show that a weaker ambient category -- a braided autonomous category instead of the usual compact closed category -- is sufficient to recover the meta rule 'only connectivity matters', even without assuming any symmetries of the generators.https://lmcs.episciences.org/3961/pdfquantum physicscomputer science - logic in computer sciencef.1.1f.3.2 |
| spellingShingle | Miriam Backens Simon Perdrix Quanlong Wang Towards a Minimal Stabilizer ZX-calculus Logical Methods in Computer Science quantum physics computer science - logic in computer science f.1.1 f.3.2 |
| title | Towards a Minimal Stabilizer ZX-calculus |
| title_full | Towards a Minimal Stabilizer ZX-calculus |
| title_fullStr | Towards a Minimal Stabilizer ZX-calculus |
| title_full_unstemmed | Towards a Minimal Stabilizer ZX-calculus |
| title_short | Towards a Minimal Stabilizer ZX-calculus |
| title_sort | towards a minimal stabilizer zx calculus |
| topic | quantum physics computer science - logic in computer science f.1.1 f.3.2 |
| url | https://lmcs.episciences.org/3961/pdf |
| work_keys_str_mv | AT miriambackens towardsaminimalstabilizerzxcalculus AT simonperdrix towardsaminimalstabilizerzxcalculus AT quanlongwang towardsaminimalstabilizerzxcalculus |