Topological color codes without translation symmetry and surface - - PowerPoint PPT Presentation

Topological color codes without translation symmetry and surface codes

Arjun Bhagoji1 Pradeep Sarvepalli1

1Department of Electrical Engineering

Indian Institute of Technology, Madras

SP Coding and Information School 2015 Unicamp, Brazil

Bhagoji, Sarvepalli Topological color codes without translation symmetry and surface codes SP Coding School 1 / 5

What are quantum codes?

Quantum error correction is used in quantum computing to protect information from quantum noise Pauli operators on a single quantum bit (qubit) are {I, X, Y , Z} Errors are usually defined in terms of elements of the Pauli group on n qubits, Pn Stabilizer formalism allows us to study quantum codes conveniently and leads to ideas of fault-tolerant quantum computation

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Stabilizer and topological codes

Stabilizer group S is a subgroup of Pn which has elements which all commute with each other and which does not contain the element I. Now, instead of specifying the states of the subspace making up the codespace we can just specify the generators of the stabilizer group Define the codespace S as all states |ψ which satisfy S |ψ = |ψ for all stabilizer elements S ∈ S Topological codes are types of local stabilizer codes, where stabilizer generators act only on a few nearby qubits

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Color codes and surface codes

Surface codes are topological codes with qubits attached to the edges

• f a lattice embedded in a given closed surface

Stabilizers are the faces and set of edges meeting at a vertex Color codes are topological codes which are very promising from the point of view of fault tolerant quantum computing and alternative decoders are of interest We have come up with an algorithm to generalize the map from color codes to surface codes for which efficient decoding algorithms are known

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Bibliography

Dave Bacon’s lecture notes for CSE 599d-Quantum Computing at the University of Washington Bomb´ ın, H. ”An Introduction to Topological Quantum Codes.” arXiv preprint arXiv:1311.0277 (2013). Bomb´ ın, H., Martin-Delgado, M.A. ”Topological quantum distillation.” Physical Review Letters 97.18 (2006): 180501. Bomb´ ın, H., Duclos-Cianci, G., Poulin, D. (2012). Universal topological phase of two-dimensional stabilizer codes. New Journal of Physics, 14(7), 073048.

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