# What is a code in the context of "stabilizer code" and a "toric code" in quantum computation?

I've been reading on topological quantum computation, and something that is repeatedly brought up is the notion of a code. Is this just a series of operations?

For example, it is used in this paper without much context

https://arxiv.org/pdf/1311.0277.pdf

Also what is a stabilizer generator? How can an element be both a stabilizer and a generator?

• Here they mean coding theory code, as in a function $f : C \to E$ from some message space $C$ to some code space $E$ that "from $f(x)$ allows you to recover $x$ (or most of $x$) even in the presence of errors". The exact definition of $f$ depends on what kind of $f$ you allow, and what kind of recovery you want. Does that help? Jun 21, 2017 at 23:14
• For an introduction to quantum codes, this thesis seems pretty comprehensive: thesis.library.caltech.edu/2900/2/THESIS.pdf. This might be a prerequisite before passing to topological quantum codes. Jun 21, 2017 at 23:23

Given $n$ qubits that reside in an Euclidean space $\mathcal E$ of dimension $2^n$, a quantum code is a subspace of $\mathcal E$.

Note then that a quantum code is just a subspace of a larger space, and we have not mentioned anything about the definitions of its properties pertaining to error-correction.

Here, a stabilizer $\mathcal S$ refers really to an abelian subgroup of $\mathbb F_4^n$. In this sense, the words, element' andgenerator' just inherit the definitions from group theory.

The connection between operator theory and this discrete set $\mathcal S$ is well known, and can be found in Daniel Gottesman's thesis on the arxiv.

If you'd like a formal operator theory introduction to quantum codes, there are many papers you can find, but you can for example find a very terse introduction in my paper: https://arxiv.org/abs/1604.07925

Do note that not all quantum codes are `stabilizer codes'.

The toric code can be understood to be a special stabilizer code, where the stabilizer is generated by elements of $\mathbb F_4^n$ of weight 4.