Pivoting makes the ZX-calculus complete for real stabilizers

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.

[1]  Peter Selinger,et al.  Dagger Compact Closed Categories and Completely Positive Maps: (Extended Abstract) , 2007, QPL.

[2]  Ross Duncan,et al.  Types for quantum computing , 2006 .

[3]  Bart De Moor,et al.  Edge-local equivalence of graphs , 2005 .

[4]  Bart De Moor,et al.  Graphical description of the action of local Clifford transformations on graph states , 2003, quant-ph/0308151.

[5]  Simon Perdrix,et al.  Rewriting Measurement-Based Quantum Computations with Generalised Flow , 2010, ICALP.

[6]  Simon Perdrix,et al.  Graph States and the Necessity of Euler Decomposition , 2009, CiE.

[7]  Umesh V. Vazirani,et al.  Quantum complexity theory , 1993, STOC.

[8]  Miriam Backens,et al.  The ZX-calculus is complete for stabilizer quantum mechanics , 2013, 1307.7025.

[9]  Bob Coecke,et al.  Trichromatic Open Digraphs for Understanding Qubits , 2011, ArXiv.

[10]  Ross Duncan,et al.  A graphical approach to measurement-based quantum computing , 2012, Quantum Physics and Linguistics.

[11]  Bob Coecke,et al.  Interacting quantum observables: categorical algebra and diagrammatics , 2009, ArXiv.

[12]  Matthew G. Parker,et al.  Edge local complementation and equivalence of binary linear codes , 2008, Des. Codes Cryptogr..

[13]  Mehdi Mhalla,et al.  Graph States, Pivot Minor, and Universality of (X, Z)-Measurements , 2012, Int. J. Unconv. Comput..

[14]  A. Joyal,et al.  The geometry of tensor calculus, I , 1991 .

[15]  B. Coecke,et al.  Classical and quantum structuralism , 2009, 0904.1997.

[16]  李幼升,et al.  Ph , 1989 .