ZX-Rules for 2-Qubit Clifford+T Quantum Circuits

ZX-calculus is a high-level graphical formalism for qubit computation. In this paper we give the ZX-rules that enable one to derive all equations between 2-qubit Clifford+T quantum circuits. Our rule set is only a small extension of the rules of stabilizer ZX-calculus, and substantially less than those needed for the recently achieved universal completeness. One of our rules is new, and we expect it to also have other utilities. These ZX-rules are much simpler than the complete of set Clifford+T circuit equations due to Selinger and Bian, which indicates that ZX-calculus provides a more convenient arena for quantum circuit rewriting than restricting oneself to circuit equations. The reason for this is that ZX-calculus is not constrained by a fixed unitary gate set for performing intermediate computations.

[1]  Aleks Kissinger,et al.  Strong Complementarity and Non-locality in Categorical Quantum Mechanics , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

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

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

[4]  B. Coecke.  Quantum picturalism , 2009, 0908.1787.

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

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

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

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

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

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

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

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

[13]  Samson Abramsky,et al.  A categorical semantics of quantum protocols , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[14]  Simon Perdrix,et al.  Towards a Minimal Stabilizer ZX-calculus , 2017, Log. Methods Comput. Sci..

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

[16]  Aleks Kissinger,et al.  Generalised compositional theories and diagrammatic reasoning , 2015, 1506.03632.

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

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

[19]  Bob Coecke,et al.  Tutorial: Graphical Calculus for Quantum Circuits , 2012, International Workshop on Reversible Computation.

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

[21]  Bob Coecke,et al.  Interacting Quantum Observables , 2008, ICALP.

[22]  Aleks Kissinger,et al.  Picturing Quantum Processes by Bob Coecke , 2017 .

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

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

[25]  Aleks Kissinger,et al.  Tensors, !-graphs, and non-commutative quantum structures. , 2014 .

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

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