# Decompose into completely stabilizer preserving channel in surface codes

In the article "Sampling-based quasiprobability simulation for fault-tolerant quantum error correction on the surface codes under coherent noise" they are talking about decomposing (possibly non-Clifford) noise channels into the sum of completely stabilizer preserving (CSP) channels and completely positive trace-preserving (CPTP) channels $$S_{k}^{(i)}$$ in formula (5) page 2.

There is also an example of it on page 3, where they decompose:

into:

## My questions:

1. What is the meaning of the definition of CSP and CPTP and how exactly to decompose into them tell me that I will get equivalent Pauli errors?
2. Given any error channel, how am I finding those channels?

Thanks

• Can you elaborate what you mean by "how exactly to decompose into them tell me that I will get equivalent Pauli errors?" Commented Feb 23, 2022 at 15:14
• In the example I showed, what was the recipe to find these channels? it is from the article Commented Feb 23, 2022 at 15:18
• The recipe would be as sketched in my answer below. But I suspect that the authors made an educated guess for this single-qubit example. It might not even be the optimal decomposition. Commented Feb 28, 2022 at 10:29

1. CSP means "completely stabilizer-preserving". A quantum channel $$\mathcal{E}$$ is CSP if $$\mathcal{E}\otimes\mathrm{id}_n (|\psi\rangle\langle\psi|)$$ is a convex combination of stabilizer states, for any $$2n$$-qubit stabilizer state $$\psi$$. Equivalently, the Choi state of $$\mathcal{E}$$ is a convex combination of $$2n$$-qubit stabilizer states. (see Seddon and Campbell "Quantifying magic for multi-qubit operations")
2. The decomposition of $$\mathcal E$$ into CSP channels is not unique. You want the decomposition $$\mathcal E = \sum_k c_k S_k$$ for which $$\sum_k |c_k|$$ is minimal. This is called the channel robustness $$R_*(\mathcal E)$$. In principal, you can compute the optimal decomposition using a linear program (in practice, you can only solve that for up to 2-3 qubits). For more details, see again the above paper and also Howard and Campbell "Application of a Resource Theory for Magic States to Fault-Tolerant Quantum Computing" and Heinrich and Gross "Robustness of Magic and Symmetries of the Stabiliser Polytope".
• and , you also told me just how to find $c_k$ I am interested in how to find the $S_k$ of the channel, before I find $c_k$ Commented Feb 23, 2022 at 15:52