The ZX-calculus is complete for the single-qubit Clifford+T group

Miriam Backens*

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

13 Citations (Scopus)


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 j0i, 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 = ( 1 0 0 eip/4) 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.

Original languageEnglish
Pages (from-to)293-303
JournalElectronic Proceedings in Theoretical Computer Science, EPTCS
Publication statusPublished - 28 Dec 2014
Event11th Workshop on Quantum Physics and Logic, QPL 2014 - Kyoto, Japan
Duration: 4 Jun 20146 Jun 2014

ASJC Scopus subject areas

  • Software


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