Supplementarity is Necessary for Quantum Diagram Reasoning

The ZX-calculus is a powerful diagrammatic language for quantum mechanics and quantum information processing. We prove that its π 4-fragment is not complete, in other words the ZX-calculus is not complete for the so called "Clifford+T quantum mechanics". The completeness of this fragment was one of the main open problems in categorical quantum mechanics, a programme initiated by Abramsky and Coecke. The ZX-calculus was known to be incomplete for quantum mechanics. On the other hand, its π

[1]  Miriam Backens,et al.  Making the stabilizer ZX-calculus complete for scalars , 2015, 1507.03854.

[2]  Dusko Pavlovic,et al.  A new description of orthogonal bases , 2008, Mathematical Structures in Computer Science.

[3]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[4]  P. Oscar Boykin,et al.  On universal and fault-tolerant quantum computing: a novel basis and a new constructive proof of universality for Shor's basis , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[5]  Peter Selinger.  Finite Dimensional Hilbert Spaces are Complete for Dagger Compact Closed Categories (Extended Abstract) , 2011, Electron. Notes Theor. Comput. Sci..

[6]  Simon Perdrix,et al.  Pivoting makes the ZX-calculus complete for real stabilizers , 2013, QPL.

[7]  Miriam Backens,et al.  The ZX-calculus is complete for the single-qubit Clifford+T group , 2014, QPL.

[8]  Ross Duncan,et al.  Interacting Frobenius Algebras are Hopf , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

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

[10]  Simon Perdrix,et al.  A Simplified Stabilizer ZX-calculus , 2016, QPL.

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

[12]  Miriam Backens,et al.  A Complete Graphical Calculus for Spekkens’ Toy Bit Theory , 2014, 1411.1618.

[13]  Bill Edwards,et al.  Three qubit entanglement within graphical Z/X-calculus , 2011, HPC.

[14]  Vladimir Zamdzhiev,et al.  The ZX calculus is incomplete for quantum mechanics , 2014, QPL.

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

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