Algebraic quantum coding theory

Evans, David, Holt, Jeffrey J., Jones, Chris +4 more

Steiner, Lisa

This work has reached several results. The first is, that it was possible to find a new, algebraic frame in which we can formulate stabilizer codes and show, that the choice of generators of a stabilizer algebra corresponds to choosing a representation of finitely many Rademacher functions in a matrix algebra. The second part of this work was to develop a quantum coding theory as a quantum analogue of classical coding theory. We do this by using a systematical view of quantum probability theory that was introduced by Kümmerer [1]. We follow this way of algebraization and develop analogously a quantum coding theory. Our result differs in some points from what has been developed so far, mainly because we are working not only with pure but also arbitrary states as well as infinitely many coupled qubits. We were able to integrate the common examples of quantum codes into our theory. The third main result is that we were able to show that the most important quantum algorithms, including stabilizer codes and the Shor algorithm, are in some sense commutative and thus classical. This could be done as quantum algorithms fit into the notion of quantum measurements, and our calculations imply that they can be represented as a coupling to a classical Bernoulli shift. [1] B. Kümmerer, Markov Dilations on W*-Algebras, Journal Functional Analysis, 63:139-177, 1985.

Hao Chen

It is shown that quantum error correction can be achieved by the use of classical binary codes or additive codes over F/sub 4/. In this correspondence, with the help of some algebraic techniques the theory of algebraic-geometric codes is used to construct an asymptotically good family of quantum error-correcting codes and other classes of good quantum error-correcting codes. Our results are compared with the known best quantum codes.

Sankalpa Khadka

Hao Chen

Chiu Fan Lee

This is a Part III essay aiming to discuss the contruction of quantum error-correcting codes through the use of the theory of algebraic function fields, which produces codes with asymptotically good parameters. This exposition emphasises constructibility and several applications of the main theorem (theorem 6.2) are given. Prerequisites are the first 12 lectures of the Part III course Quantum Information Theory [16] or chapter 7 of Preskill’s Lecture Notes [15], and the first two chapters of Stichtenoth’s Algebraic Function Fields and Codes [19].