Graduate Knill-Laflamme condition Shors code

[steve1763]

TL;DR

How would one apply the Knill-Leflamme condition to Shors code?

The K-L condition has projection operators onto the codespace for the error correction code, as I understand it. My confusion I think comes primarily from what exactly these projections are? As in, how would one find these projections for say, the Shor 9-qubit code?

 

[azntoon]

The K-L condition is a mathematical theorem that states that a given error correction code must have certain properties in order for it to be valid and correct errors in quantum systems. To satisfy this condition, projection operators onto the codespace must be found. These projection operators represent the orthogonal basis of the codespace and they are used to measure the error syndrome, which gives information about the type of errors that have occurred. Finding these projection operators for the Shor 9-qubit code is a bit complicated since the codespace is composed of multiple subspaces. Specifically, the codespace is composed of two four-dimensional subspaces and one one-dimensional subspace. The first step is to find a set of nine commuting operators that span the nine-dimensional codespace. Then, one needs to solve for the eigenvalues and eigenvectors of these operators. The projection operators can then be constructed from the eigenvectors using the outer product.