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2016

Weight Reduction for Quantum Codes

Hastings, M. B.

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We present an algorithm that takes a CSS stabilizer code as input, and outputs another CSS stabilizer code such that the stabilizer generators all have weights $O(1)$ and such that $O(1)$ generators act on any given qubit.

The number of logical qubits is unchanged by the procedure, while we give bounds on the increase in number of physical qubits and in the effect on distance and other code parameters, such as soundness (as a locally testable code) and "cosoundness" (defined later).
Applications are discussed, including to codes from high-dimensional manifolds which have logarithmic weight stabilizers.
Assuming a conjecture in geometry\cite{hdm}, this allows the construction of CSS stabilizer codes with generator weight $O(1)$ and almost linear distance.

Weight Reduction for Quantum Codes · Around

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A. Hatcher, “Algebraic Topology”, available at https://www.math.cornell.edu/ hatcher/AT/AT.pdf
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L. Eldar and D. Aharonov, “Quantum Locally Testable Codes”, arXiv:1310.5664
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A. Leverrier, J.-P. Tillich, and G. Zémor, “Quantum Expander Codes”, 015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pp. 810-824
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Then

J.-P. Tillich and G. Zemor, “Quantum LDPC codes with positive rate and minimum distance proportional to n 1 / 2 n^{1/2} ”, IEEE Trans. Inform. Theory. 60
2014
Later among the works it cites.
M.H. Freedman and M.B. Hastings, “Quantum systems on non-k-hyperfinite complexes: A generalization of classical statistical mechanics on expander graphs”, arXiv:1301.1363, QIC 14
Original
2014
Later among the works it cites.
Sergey Bravyi, Matthew B. Hastings, “Homological Product Codes”, Proc. of the 46th ACM Symposium on Theory of Computing (STOC 2014), pp. 273-282 (2014)
2014
Later among the works it cites.
D. Bacon, S. T. Flammia, A. W. Harrow, and J. Shi, “Sparse Quantum Codes from Quantum Circuits”, Proc. of STOC ’15, pp. 327-334, (2015)
2015
Later among the works it cites.

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