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.
Highlights
With the development of devices, quantum computing is in the midst of moving from concept to mature technology [31, 26]
By considering other topologically equivalent presentations of the coloured graph of Eqn (5), and how those graphs describe operations which may be decomposed as ZX diagrams, we may obtain a further set of procedures, all of which implement the CNOT in the positive branch (in the Appendix we show how we may realise CNOT operations in the negative branch for each of these procedures, as well as for that given in Eqn (19)):
We have demonstrated how the ZX calculus acts as a precise and fundamental description of the splitting and merging operations of lattice surgery on the planar surface code
Summary
With the development of (small-scale, noisy) devices, quantum computing is in the midst of moving from concept to mature technology [31, 26]. The ZX calculus is a calculus: a formal language, with meaningpreserving rules for how those diagrams may be transformed, without the need to transcribe those operations as exponentially large matrices (even for full pure-state QM, that cannot be efficiently simulated using e.g. stabilizer notation [27]) By such transformations of these diagrams — corresponding to different sets of lattice surgery procedures that implement the same operation — new protocols may be discovered. The use of the ZX calculus for surface codes with lattice surgery makes the manipulation of error corrected operations visually intuitive, and capable of verification at large scales As these operations will likely form the basic operations that a fault-tolerant device will use at the logical level, ZX becomes the natural language and logic for programming large-scale quantum computing technologies
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