Fernando Souza

Topological Codes I

Abstract:

This couple of talks is a tutorial on topological codes. We will first review what quantum error correction is about, particularly stabilizer codes, as needed. Then, we will introduce surface codes, explaining the necessary topological ideas. Finally, we will report on the computational aspects of these codes, and comment on the related topics as much as time and interest allow.

Surface codes are quantum error-correcting codes in which qubits are arranged in an array on a surface, and the encoded quantum operations are associated with topological features called non-trivial homological 1-cycles. They were discovered by Alexei Kitaev (1997), and are partially inspired by Kitaev's proposal for the realization of quantum computers via anyons (anyonic computation, a.k.a. topological computation, 1997.) Actually, this stream of ideas culminated with a deep correspondence between the quantum circuit model and some of the topological/categorical objects called unitary topological modular functors (2000.) Further developments on surface codes, and theoretical advantadges of both surface codes and topological computation have been pointed out by a handful of people. A particular interesting feature is the intrinsic fault tolerance resulting of the topological nature of the model.

References:

Introductions to quantum error-correcting codes (QECC) in general texts:

D. Bouwmeester, A. Ekert, and A. Zeilinger (Eds.); "The Physics of Quantum Information: Cryptography, Communication, and Computation", Springer-Verlag, Berlin (1998);

J. Gruska: "Quantum Computing", McGraw-Hill (1999);

A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi: "Classical and Quantum Computation" (GSM vol. 47), Amer. Math. Soc. (2002);

The articles by J. Preskill (quant-ph/9712048) and A. Steane in H. K. Lo, S. Popescu, and T. Spiller: "Introduction to Quantum Computation and Information", World Scientific (1998);

M.A. Nielsen and I.L. Chuang, "Quantum Computation and Quantum Information", Cambridge University Press (2000);

The notes by Z. Meglicki: http://beige.ucs.indiana.edu/M743/

A. Pittenger: "An Introduction to Quantum Computing and Algorithms", 2nd printing, Birkhauser (2001);

The notes by J Preskill: http://www.theory.caltech.edu/~preskill/ph229/

Texts on QECC:

quant-ph/0004072 and quant-ph/9705052 by Gottesman;

The lecture notes at http://www.c3.lanl.gov/~knill/cv/cv/index.html#cv:lectures, in particular those at http://www.fields.utoronto.ca/programs/scientific/00-01/quantum_computing/abstracts.htm l#Knill by Knill;

Kitaev, A. Yu: Quantum computations: algorithms and error correction. (Russian) Uspekhi Mat. Nauk 52 (1997), no. 6(318), 53--112; translation in Russian Math. Surveys 52 (1997), no. 6, 1191--1249.