# Is there a tool that shows me all $2^n$ stabilizers for a given graph state?

Is there a tool which takes the adjacency matrix of a graph as input and prints out a table with all stabilizer measurements?

## 1 Answer

To obtain the stabilisers of a graph state, from its adjacency matrix:

1. Change all 1s to Zs
2. Change all 0s to identity operators
3. Put X operators on the diagonal

Each row then represents a stabiliser of the graph state, and any nontrivial stabiliser is a product of one or more rows.

• Thanks for the answer. That one is clear. I'm looking for a tool because for more than 5 qubits calculating all stabilizers by hand to try to recognize patterns in them gets really exhausting over time. That's why I'm looking for an automated version. If I can teach sympy the Pauli multiplication table and feed it with the matrix or the generators that's fine as well but I have never used a symbolic library before. May 30, 2020 at 21:38
• If you're determined to try to look for patterns in exponentially many stabilisers, perhaps you should use a representation of stabilisers in terms of boolean vectors as in [arXiv:quant-ph/9605005]. This will allow you to represent the stabilisers as the image of all $n$ bit-vectors under a $n \times 2n$ matrix, though if you care about the signs of the operators (and you should think about whether or not you might), some extra work is required to make sure you get those right. May 31, 2020 at 9:13
• Alternatively you can use a representation in which these sign issues don't arise, e.g. [arXiv:1102.3354] (of which I'm the author). This uses 'vectors' of integers mod 4. You'd have to be comfortable with a redundant representation of Pauli operators, but then if you use a 'proper' tableau --- e.g., if you start from a tableau representing just X generators or Z generators, and apply H and CZ as needed to get a matrix representing the graph state generators --- then you can just rely on anything that does linear algebra over integers (and reduce them mod 4). May 31, 2020 at 9:22