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Graphical structures for quantum error correction

Dominic Horsman ( Joint Quantum Centre, Durham University )

In this talk I introduce a new construction for quantum error correction (QEC) that admits the use of the ZX calculus as a high-level graphical language. QEC as standardly constructed uses the mathematical machinery of group theory and many body quantum theory in ways that can be impenetrable to non-specialists, and is unwieldy beyond small number of qubits. Some codes can use notions of topology as high-level descriptors, but an important class of codes, CSS codes, have previously lacked a user-friendly high-level language for describing and developing new codes. With the imminent development of first-generation quantum computers of 50-100 qubits, finding codes that make best use of limited qubit resources and that are tailored to error models of specific hardware devices has become a pressing practical issue.

I will go through the recently-introduced CPC construction for CSS codes (arXiv:1611.08012). Rather than dealing directly with stabilizer subspaces etc., the use of a basic coherent (rather than decoherent) parity check between qubits allows codes to be modelled simply in terms of constraints between qubits. I will demonstrate how to build error detecting and correcting codes from the basic coherent check, and how known classical codes can act as ‘seeds’ of quantum codes in a way that differs from the usual CSS construction. I will show how the construction maps to the ZX calculus of observables, from a basic coherent check up through more complex codes. I give three new CSS codes for small scale devices, and will detail how the use of the calculus, as well as the automated tool Quantomatic, aided both the discovery and verification (in terms of stabilizers and error tables) of these codes. This application of the ZX calculus as a user-friendly language for CSS codes opens the door to exciting possibilities for Quantomatic-based design and verification tools for QEC, and I will finish by discussing applications to specific quantum devices that are being developed.




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