Decompose into completely stabilizer preserving channel in surface codes

Question

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

Answer 1

First, the formulation is a bit unclear at that point. I don't know what they want to say with "and completely positive trace-reserving (CPTP) channels". Any quantum channel is by definition CPTP. I think what the authors want to do is to decompose the noise channel as an affine combination of CSP channels (which is always possible).

To answer your questions:

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".