The ZX calculus is a language for surface code lattice surgery

Niel de Beaudrap1 and Dominic Horsman2

1Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD
2Department of Physics, Durham University, South Road, Durham, DH1 1LE

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Abstract

A leading choice of error correction for scalable quantum computing is the surface code with lattice surgery. The basic lattice surgery operations, the merging and splitting of logical qubits, act non-unitarily on the logical states and are not easily captured by standard circuit notation. This raises the question of how best to design, verify, and optimise protocols that use lattice surgery, in particular in architectures with complex resource management issues. In this paper we demonstrate that the operations of the ZX calculus --- a form of quantum diagrammatic reasoning based on bialgebras --- match exactly the operations of lattice surgery. Red and green ``spider'' nodes match rough and smooth merges and splits, and follow the axioms of a dagger special associative Frobenius algebra. Some lattice surgery operations require non-trivial correction operations, which are captured natively in the use of the ZX calculus in the form of ensembles of diagrams. We give a first taste of the power of the calculus as a language for lattice surgery by considering two operations (T gates and producing a CNOT) and show how ZX diagram re-write rules give lattice surgery procedures for these operations that are novel, efficient, and highly configurable.

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Cited by

[1] John H. Selby, Carlo Maria Scandolo, and Bob Coecke, "Reconstructing quantum theory from diagrammatic postulates", arXiv:1802.00367.

[2] Stefano Gogioso and Subhayan Roy Moulik, "Purification and time-reversal deny entanglement in LOCC-distinguishable orthonormal bases", arXiv:1902.00316.

[3] Craig Gidney and Austin G. Fowler, "Efficient magic state factories with a catalyzed |CCZ> to 2|T> transformation", arXiv:1812.01238.

[4] Titouan Carette, Dominic Horsman, and Simon Perdrix, "SZX-calculus: Scalable Graphical Quantum Reasoning", arXiv:1905.00041.

[5] Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart, "Completeness of the ZX-Calculus", arXiv:1903.06035.

[6] John van de Wetering and Sal Wolffs, "Completeness of the Phase-free ZH-calculus", arXiv:1904.07545.

[7] Renaud Vilmart, "A Near-Optimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics", arXiv:1812.09114.

[8] Titouan Carette, Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart, "Completeness of Graphical Languages for Mixed States Quantum Mechanics", arXiv:1902.07143.

[9] Christophe Vuillot, Lingling Lao, Ben Criger, Carmen García Almudéver, Koen Bertels, and Barbara M. Terhal, "Code deformation and lattice surgery are gauge fixing", New Journal of Physics 21 3, 033028 (2019).

[10] Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart, "A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness", arXiv:1805.05296.

[11] Craig Gidney and Austin G. Fowler, "Flexible layout of surface code computations using AutoCCZ states", arXiv:1905.08916.

[12] Renaud Vilmart, "A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond", arXiv:1804.03084.

[13] Stach Kuijpers, John van de Wetering, and Aleks Kissinger, "Graphical Fourier Theory and the Cost of Quantum Addition", arXiv:1904.07551.

[14] Bob Coecke, Stefano Gogioso, and John H. Selby, "The time-reverse of any causal theory is eternal noise", arXiv:1711.05511.

[15] Niel de Beaudrap, Ross Duncan, Dominic Horsman, and Simon Perdrix, "Pauli Fusion: a computational model to realise quantum transformations from ZX terms", arXiv:1904.12817.

[16] Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart, "Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics", arXiv:1801.10142.

[17] Bob Coecke and Quanlong Wang, "ZX-Rules for 2-qubit Clifford+T Quantum Circuits", arXiv:1804.05356.

[18] Michael Hanks, Marta P. Estarellas, William J. Munro, and Kae Nemoto, "Efficient compression of quantum braided circuits", arXiv:1912.11503.

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