The ZX calculus is a language for surface code lattice surgery
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 nonunitarily 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 nontrivial 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 rewrite rules give lattice surgery procedures for these operations that are novel, efficient, and highly configurable.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 arXiv:
 arXiv:1704.08670
 Bibcode:
 2017arXiv170408670D
 Keywords:

 Quantum Physics;
 Computer Science  Logic in Computer Science
 EPrint:
 20 pages, many figures. Minor revisions. Accepted to Quantum Journal