A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond

We consider a ZX-calculus augmented with triangle nodes which is well-suited to reason on the so-called Toffoli-Hadamard fragment of quantum mechanics. We precisely show the form of the matrices it represents, and we provide an axiomatisation which makes the language complete for the Toffoli-Hadamard quantum mechanics. We extend the language with arbitrary angles and show that any true equation involving linear diagrams which constant angles are multiple of Pi are derivable. We show that a single axiom is then necessary and sufficient to make the language equivalent to the ZX-calculus which is known to be complete for Clifford+T quantum mechanics. As a by-product, it leads to a new and simple complete axiomatisation for Clifford+T quantum mechanics.

[1]  Dominic Horsman,et al.  Quantum picturalism for topological cluster-state computing , 2011, 1101.4722.

[2]  Dominic Horsman,et al.  The ZX calculus is a language for surface code lattice surgery , 2017, Quantum.

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

[4]  Aleks Kissinger,et al.  Quantomatic: A proof assistant for diagrammatic reasoning , 2015, CADE.

[5]  Amar Hadzihasanovic,et al.  A Diagrammatic Axiomatisation for Qubit Entanglement , 2015, 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science.

[6]  Yaoyun Shi.  Both Toffoli and controlled-NOT need little help to do universal quantum computing , 2003, Quantum Inf. Comput..

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

[8]  Amar Hadzihasanovic,et al.  The algebra of entanglement and the geometry of composition , 2017, ArXiv.

[9]  Ross Duncan,et al.  Verifying the Steane code with Quantomatic , 2013, QPL.

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

[11]  Aleks Kissinger,et al.  Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning , 2017 .

[12]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

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

[14]  Quanlong Wang,et al.  Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics , 2018, 1801.07993.

[15]  Quanlong Wang,et al.  A universal completion of the ZX-calculus , 2017, ArXiv.

[16]  Quanlong Wang,et al.  Two complete axiomatisations of pure-state qubit quantum computing , 2018, LICS.

[17]  J. Robin B. Cockett,et al.  The category TOF , 2018, QPL.

[18]  Isaac L. Chuang,et al.  Quantum Computation and Quantum Information (10th Anniversary edition) , 2011 .

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

[20]  D. Aharonov.  A Simple Proof that Toffoli and Hadamard are Quantum Universal , 2003, quant-ph/0301040.

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

[22]  Simon Perdrix,et al.  ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics , 2017, MFCS.

[23]  Peter Selinger,et al.  Quantum circuits of T-depth one , 2012, ArXiv.

[24]  Aleks Kissinger,et al.  The Compositional Structure of Multipartite Quantum Entanglement , 2010, ICALP.

[25]  Simon Perdrix,et al.  Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics , 2018, LICS.

[26]  Simon Perdrix,et al.  A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics , 2017, LICS.

[27]  Stefan Zohren,et al.  Graphical structures for design and verification of quantum error correction , 2016, Quantum Science and Technology.