Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  What is "code" in "toric code"?

+ 7 like - 0 dislike
1459 views

When I first heard people talking about using Kitaev's toric code to do topological quantum computation, I was thinking how many lines does the toric code have. Then I was told that the "code" really represents quantum states. Later, I understood that the "code" is the degenerate ground subspace of a gapped Hamiltonian. But then I was corrected by people from quantum information who pointed out that code is a subspace of the Hilbert space of a quantum system, but code is also more than a subspace.

So now I am wondering what is "code"?

This post imported from StackExchange Physics at 2014-04-05 17:30 (UCT), posted by SE-user Xiao-Gang Wen
asked Jun 1, 2012 in Theoretical Physics by Xiao-Gang Wen (3,485 points) [ no revision ]
There is a link now provided by the editing. paragraph 2.2.1 seems to be using the definition of "code" as "The toric code is the simplest topological lattice model that supports Abelian anyons." I guess the word "code" is used as in "encode", a code encodes a specific model?

This post imported from StackExchange Physics at 2014-04-05 17:30 (UCT), posted by SE-user anna v
Here we have yet another difinition of "code": "code" is a quantum qubit model. This is very different from "code" is a subspace.

This post imported from StackExchange Physics at 2014-04-05 17:30 (UCT), posted by SE-user Xiao-Gang Wen
@anna - In my opinion this question is trying to imply subspace is logically similar to a qbit model and therefore mathematically predictable.

This post imported from StackExchange Physics at 2014-04-05 17:30 (UCT), posted by SE-user Argus

2 Answers

+ 8 like - 0 dislike

Let me try to give you the answer in just the right amount of generality. A quantum code is just a short way to say a quantum error-correcting code. It is a special embedding of one vector space into another larger one that satisfies some additional properties. If we start with a Hilbert space $H$, then a code is a decomposition into $H = (A \otimes B) \oplus C$. The quantum information is encoded into system $A$. In the event that $B$ is trivial, then indeed this is just a subspace of $H$. When $B$ is nontrivial, we say call it a subsystem code. Let's specialize to the case of $n$ qubits, so that $H = (\mathbb{C}^2)^{\otimes n}$, and it is easiest to imagine that the dimensions of $A$, $B$, and $C$ are all powers of 2, though of course this discussion could be generalized.

Let $P$ be the orthogonal projector onto $A\otimes B$, and let $\mathcal{E}$ be an arbitrary quantum channel, i.e. a completely positive trace preserving linear map. We say that $\mathcal{E}$ is recoverable if there exists another quantum channel $\mathcal{R}$ such that for all states $\rho_A \otimes \rho_B$, we have $$\mathcal{R}\circ\mathcal{E}(\rho_A \otimes \rho_B) = \rho_A \otimes \rho'_B,$$ where $\rho'_B$ is arbitrary. This says that for any state which is supported on $A\otimes B$ and is initially separable, we can reverse the effects of $\mathcal{E}$ up to a change on system $B$.

Fortunately, there are simpler equivalent conditions that one can check instead. For example, an equivalent condition can be stated in terms of the Kraus operators $E_j$ for the channel $\mathcal{E}$. The subsystem $A$ is correctable for $\mathcal{E}(\rho) = \sum_j E_j \rho E_j^\dagger$ if an only if for all $i,j$, there exists a $g^{ij}_B$ on subsystem $B$ such that $$ P E_i^\dagger E_j P = 1\hspace{-3pt}\mathrm{l}_A \otimes g^{ij}_B.$$ You can interpret this condition as saying that no error process associated to the channel $\mathcal{E}$ can gain any information about subsystem $A$.

Consider error channels which consist of Kraus operators that, when expanded in the Pauli basis, only have support on at most $d$ of the $n$ qubits in our Hilbert space. If every such channel is correctable for subsystem $A$, then we say our code has distance $d$. The largest such $d$ is called the distance of the code. For the toric code, this is the linear size of the lattice.

This post imported from StackExchange Physics at 2014-04-05 17:30 (UCT), posted by SE-user Steve Flammia
answered Jun 4, 2012 by Steve Flammia (110 points) [ no revision ]

Thanks for the detailed description of "code". According your description, "code" is a subspace, but "code" also has something more. It is still not very clear what is "code". If "code" is subspace + something, then what is this "something". You explained why code is important for error correction in quantum computation, which is very nice. But do we have a straight forward definition of "code"?

This post imported from StackExchange Physics at 2014-04-05 17:30 (UCT), posted by SE-user Xiao-Gang Wen

A code is simply the decomposition $H = (A \otimes B)\oplus C$. By itself, this is too general to be interesting. Therefore, people only emphasize that something is a code when it has additional properties. For example, people call it the "toric code" because it can correct quantum errors. The simple 2D ferromagnetic Ising model is also a code... but it only encodes a single classical bit, and corrects no quantum errors. So you will never hear people talk about the "Ising code". Sorry if the answer is disappointing, but it is really just that decomposition!

This post imported from StackExchange Physics at 2014-04-05 17:30 (UCT), posted by SE-user Steve Flammia

Now I see: "code" is like a piece of a drawing, it may be an art, or it may be not. (BTW I do like your answer very much.)

+ 1 like - 0 dislike

In general a "code" in quantum information is a collection of "codewords". One takes a (relatively large) number of physical qubits and considers only a limited number of states (i.e. a low-dimensional subspace) out of the full Hilbert space (formally, the code is the subspace). For stabilizer codes, all of these states are ground states of some gapped hamiltonian. Because there is a good deal of redundancy, the system is more robust against errors and decoherence.

Perhaps a clearer example would be the repetition code. Say you have six qubits, but you restrict yourself to the subspace spanned by $$\{|000,000\rangle,|000,111\rangle,|111,000\rangle,|111,111\rangle\}.$$ You can then think of these long combinations as "codewords" for the logical states $$\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\},$$ but they can still be recognized if any given qubit suffers some error.

This post imported from StackExchange Physics at 2014-04-05 17:30 (UCT), posted by SE-user Emilio Pisanty
answered Jun 3, 2012 by Emilio Pisanty (520 points) [ no revision ]
Thanks for the intuitive and pictorial explain of code. But I still like to know what is "code" exactly. If "code" is subspace + something, then what is this "something". Or maybe the "code" is just a subspace with nothing more (?).

This post imported from StackExchange Physics at 2014-04-05 17:30 (UCT), posted by SE-user Xiao-Gang Wen

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$y$\varnothing$icsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...