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:

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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?


  • $\begingroup$ Can you elaborate what you mean by "how exactly to decompose into them tell me that I will get equivalent Pauli errors?" $\endgroup$ Commented Feb 23, 2022 at 15:14
  • $\begingroup$ In the example I showed, what was the recipe to find these channels? it is from the article $\endgroup$
    – Ron Cohen
    Commented Feb 23, 2022 at 15:18
  • $\begingroup$ 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. $\endgroup$ Commented Feb 28, 2022 at 10:29

1 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".
  • $\begingroup$ about CPTP- Notice that it is collapsing bigger space into its subpace after measure ancillas. and for the operation to be mathematically decompose correctly, so you make sure they are CPTP. $\endgroup$
    – Ron Cohen
    Commented Feb 23, 2022 at 15:20
  • $\begingroup$ So I understand this decomposing is not trivial at all, and has a big run time? What is the complexity of this decompostion? $\endgroup$
    – Ron Cohen
    Commented Feb 23, 2022 at 15:22
  • $\begingroup$ And, if it so difficult, how did they do it in the article for such big surface code? It seems like the docmposition was done only for 1 qubit? $\endgroup$
    – Ron Cohen
    Commented Feb 23, 2022 at 15:50
  • $\begingroup$ 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 $c_k$ $\endgroup$
    – Ron Cohen
    Commented Feb 23, 2022 at 15:52
  • $\begingroup$ @RonCohen sure if you use quantum channels, you have to make sure that they are CPTP ... that's the pleonasm I mentioned. Anyway, not important for the whole story. $\endgroup$ Commented Feb 28, 2022 at 10:31

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