The goal of the project is to develop a working group around "new quantum information theory" from viewpoint of "categorification," motivated by the application of Jones polynomial on topological quantum information.

量子誤り訂正理論と結び目ホモロジー

Invited Speaker: Benjamin Audoux (I2M, AMU)

Abstract: The goal of these lectures is to show how topological tools can be used to deal with quantum error correction. The first lecture will be an introduction to classical error correction, that is how computers encode messages so that it can detect and correct a reasonable number of errors occuring during a transmission. The second lecture will present a similar process in the realm of quantum computers, and a focus will be given in the case of CSS codes and the relationship between CSS codes and chain complexes. In the light of the previous lecture, the last one will describe several topology-flavoured error correcting quantum codes.

References: 

On classical codes :

- F.J. MacWilliams and N.J.A. Sloane, The Theory of Error Correcting Codes, North-Holland Mathematical Library, Vol. 16. Amsterdam etc.: North- Holland (Elsevier)

On quantum codes :

- A.R. Calderbank and P.W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54 (1996), Article Id. 1098

- A. Steane, Multiple particle interference and quantum error correction, Proc. Roy. Soc. Lond. A 452 (1996), 2551–2577

- J. Preskill, Quantum information and computation, www.theory.caltech.edu/~prekill/ph229/

On applications of topology to quantum codes :

- A.Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Physics 303 (2003), 2–30

- M.H. Freedman, D.A. Meyer, and F. Luo, Z2-systolic freedom and quantum codes, in Ch. Goong and R. K. Brylinski (eds.) Mathematics of quantum computation. Compu- tational Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2002, 287–320

- B. Audoux, An application of Khovanov homology to quantum codes, Ann. Inst. Henri Poincaré D 1 (2014), no. 2, 185–223

- S. Bravyi and M.B. Hastings, Homological product codes,  inT OC’14—Proceedings of the 2014 ACM Symposium on Theory of Computing. Held in New York, May 31–June 3, 2014. ACM Press, New York, 2014, 273–282

- B. Audoux and A. Couvreur, On tensor product of CSS codes, Ann. Inst. Henri Poincaré D 6 (2019), no. 2, 239–287

- M.B. Hastings, J. Haah and R. O'Donnell, Fiber Bundle Codes: Breaking the N1/2polylog(N) Barrier for Quantum LDPC Codes, arXiv:2009.03921

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