The ZX calculus is a language for surface code lattice surgery - Niel de Beaudrap

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.
arxiv.org/abs/...
Erratum from speaker:
There is an error from 51:24 onwards, in the form of an invalid rewrite involving a green merge (a Z spider) with two inputs and one output. What I intended to convey was that an X-eigenstate which had eigenvalues described by phases of 'aπ' and 'bπ' on the inputs, would yield a state with eigenvalue given by a phase '(a+b)π' on the output. However, this was not a correct representation of that fact. The simplest correct representation would instead involve projections on various wires, though the diagrams describing this would perhaps not have been very informative.
Recorded on July 20th 2020.
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