The ZX-calculus is complete for the single-qubit Clifford+T group
The ZX-calculus is a graphical calculus for reasoning about pure state qubit quantum mechanics. It is complete for pure qubit stabilizer quantum mechanics, meaning any equality involving only stabilizer operations that can be derived using matrices can also be derived pictorially. Stabilizer operati...
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| Format: | Article |
| Language: | English |
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Open Publishing Association
2014-12-01
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| Series: | Electronic Proceedings in Theoretical Computer Science |
| Online Access: | http://arxiv.org/pdf/1412.8553v1 |
| _version_ | 1818854276676452352 |
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| author | Miriam Backens |
| author_facet | Miriam Backens |
| author_sort | Miriam Backens |
| collection | DOAJ |
| description | The ZX-calculus is a graphical calculus for reasoning about pure state qubit quantum mechanics. It is complete for pure qubit stabilizer quantum mechanics, meaning any equality involving only stabilizer operations that can be derived using matrices can also be derived pictorially. Stabilizer operations include the unitary Clifford group, as well as preparation of qubits in the state |0>, and measurements in the computational basis. For general pure state qubit quantum mechanics, the ZX-calculus is incomplete: there exist equalities involving non-stabilizer unitary operations on single qubits which cannot be derived from the current rule set for the ZX-calculus. Here, we show that the ZX-calculus for single qubits remains complete upon adding the operator T to the single-qubit stabilizer operations. This is particularly interesting as the resulting single-qubit Clifford+T group is approximately universal, i.e. any unitary single-qubit operator can be approximated to arbitrary accuracy using only Clifford operators and T. |
| first_indexed | 2024-12-19T07:50:08Z |
| format | Article |
| id | doaj.art-f8ff38f956c24de7a774d993c7e0fce1 |
| institution | Directory Open Access Journal |
| issn | 2075-2180 |
| language | English |
| last_indexed | 2024-12-19T07:50:08Z |
| publishDate | 2014-12-01 |
| publisher | Open Publishing Association |
| record_format | Article |
| series | Electronic Proceedings in Theoretical Computer Science |
| spelling | doaj.art-f8ff38f956c24de7a774d993c7e0fce12022-12-21T20:30:12ZengOpen Publishing AssociationElectronic Proceedings in Theoretical Computer Science2075-21802014-12-01172Proc. QPL 201429330310.4204/EPTCS.172.21:9The ZX-calculus is complete for the single-qubit Clifford+T groupMiriam Backens0 University of Oxford The ZX-calculus is a graphical calculus for reasoning about pure state qubit quantum mechanics. It is complete for pure qubit stabilizer quantum mechanics, meaning any equality involving only stabilizer operations that can be derived using matrices can also be derived pictorially. Stabilizer operations include the unitary Clifford group, as well as preparation of qubits in the state |0>, and measurements in the computational basis. For general pure state qubit quantum mechanics, the ZX-calculus is incomplete: there exist equalities involving non-stabilizer unitary operations on single qubits which cannot be derived from the current rule set for the ZX-calculus. Here, we show that the ZX-calculus for single qubits remains complete upon adding the operator T to the single-qubit stabilizer operations. This is particularly interesting as the resulting single-qubit Clifford+T group is approximately universal, i.e. any unitary single-qubit operator can be approximated to arbitrary accuracy using only Clifford operators and T.http://arxiv.org/pdf/1412.8553v1 |
| spellingShingle | Miriam Backens The ZX-calculus is complete for the single-qubit Clifford+T group Electronic Proceedings in Theoretical Computer Science |
| title | The ZX-calculus is complete for the single-qubit Clifford+T group |
| title_full | The ZX-calculus is complete for the single-qubit Clifford+T group |
| title_fullStr | The ZX-calculus is complete for the single-qubit Clifford+T group |
| title_full_unstemmed | The ZX-calculus is complete for the single-qubit Clifford+T group |
| title_short | The ZX-calculus is complete for the single-qubit Clifford+T group |
| title_sort | zx calculus is complete for the single qubit clifford t group |
| url | http://arxiv.org/pdf/1412.8553v1 |
| work_keys_str_mv | AT miriambackens thezxcalculusiscompleteforthesinglequbitcliffordtgroup AT miriambackens zxcalculusiscompleteforthesinglequbitcliffordtgroup |