can the package be extended to work with subsystem codes?

[qodesign]

I've been using the package with (subspace) stabilizer codes and its speed and accuracy have so far been impressive.

How difficult would it be to add support for subsystem codes?

For example, the BBS code described here https://arxiv.org/abs/2002.06257 is a $[[21,4,3]]$ code.
Running QDistRnd with the stabilizers alone ($H_X,H_Z$) gives a distance of 2.

[LeonidPryadko]

Glad you like the package. To work with a CSS subsystem code, where gauge generator matrices $G_X$ and $G_Z$ do not have orthogonal rows (and you are trying to find a minimum-weight $Z$-codeword), you need to construct the corresponding stabilizer generator matrix $H_X$, and use the matrices $(H_X,G_Z)$ to construct the $Z$ codewords as you would for a conventional stabilizer CSS code.

To this end, suppose the matrix $R=G_XG_Z^T$ has rank $p>0$. Do usual row transformations on $R$ to construct a row echelon form; denote the corresponding invertible row transformation matrix $U$ (so that $UR$ has row echelon form with first $p$ rows non-zero). Denote $U_p$ the matrix $U$ with the first $p$ rows removed. Then the check matrix is $H_X = U_p G_X$. For verification, $H_X$ should have rows orthogonal to those of $G_Z$, and the rank of $H_X$ should equal $\mathop{\rm rank}G_Z-p$.

Please let me know if this works for you.

[qodesign]

I think I'm looking at this from a different angle. I can calculate $H_X,H_Z$ (stabilizers) and $L_X,L_Z$ (logicals I care about..."dressed logicals" to use bad lingo). So the distance would be the minimum weight of an error in $&lt;H_x,H_z,L_x,L_z&gt;$ not in $<H_x,H_z>$; so a possible modification to the package would be to allow optional $L_x,L_z$ inputs.

[LeonidPryadko]

I see. Yes, I agree it would be a useful modification, and not difficult to do.

For now, if you can also calculate the gauge generator matrices $G_X$ and $G_Z$ (e.g., rows of $G_Z$ are orthogonal to combined rows of $H_X$ and $L_X$), you can just specify $H_X$ and $G_Z$ on the input to calculate the distance $d_Z$.

Alternatively, assuming you only need to deal with binary (qubit) CSS codes, you can use my package dist_m4ri. It can read MMX matrices (including logical generator matrices), implements the same random information set algorithm as the QDistRnd package, and also a deterministic distance calculation algorithm suitable for quantum LDPC codes with low weights of stabilizer generator rows -- the complexity scales like $n(w-1)^{d-1}$, where $n$ is the code length, $d$ is the distance in question, and $w$ is the maximum row weight of the check matrix (say, $H_X$ to calculate $d_Z$). If distance is too large, it can also guarantee the absence of codewords up to some weight.

[qodesign]

Working with $(H_x,G_z)$ and $(H_z,G_x)$ (so two calls to DistRandCSS) is working fine...thanks for the suggestion.

I'll take a look at the dist_m4ri package a bit later but so far staying within GAP is more convenient since I define most of the codes there and QDistRnd is handling these well...I only checked codes of a few hundred qubits so far.

[perlinm]

Any ideas for what to do for non-CSS subsystem codes?