2017/04/30 by Niel de Beaudrap, Dominic Horsman · 81 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebra over a field #Algorithm #Arithmetic #Associative property #Axiom #Calculus (dental) #Computer science #Diagrammatic reasoning #Geometry #Lattice (music) #Mathematics #Notation #Physics #Programming language #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum and electron transport phenomena #Quantum mechanics #Satisfiability #Scalability #cs.LO #quant-ph
paper · pdf · doi:10.22331/q-2020-01-09-218
published in Quantum 4, 218 (Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften) · 20 pages, many figures. Minor revisions. Accepted to Quantum Journal
openalex publication_date 2020/01/09 · arxiv created 2020/06/04 · arxiv updated 2020/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.