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Pivoting makes the ZX-calculus complete for real stabilizers

Duncan, Ross and Perdrix, Simon (2014) Pivoting makes the ZX-calculus complete for real stabilizers. Electronic Proceedings in Theoretical Computer Science, 171. pp. 50-62. ISSN 2075-2180

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Abstract

We show that pivoting property of graph states cannot be derived from the axioms of the ZX-calculus, and that pivoting does not imply local complementation of graph states. Therefore the ZX-calculus augmented with pivoting is strictly weaker than the calculus augmented with the Euler decomposition of the Hadamard gate. We derive an angle-free version of the ZX-calculus and show that it is complete for real stabilizer quantum mechanics.