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Surface code is one of types of Quantum error correction code. Also capability of error correction of surface code depends on the code distance. As I know, if surface code's distance is 5, then it can detect up to 'd-1' (4, in this case) and correct up to '(d-1)/2' (2, in this case) errors. Is this error meaning logical-level error (error cause by logical operation applied on surface code) or physical qubit error occurred by physical error chain to apply logical operation?

Also I want to know what is the exact condition for occurring logical operation error on surface code. What I know is, logical operation can be applied to surface code by applying gates to the same number of physical qubits as code distance. What is the condition for occurring error on this logical operator?

If you have any reference (Research paper) please let me know.

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Is this error meaning logical-level error (error cause by logical operation applied on surface code) or physical qubit error occurred by physical error chain to apply logical operation?

A logical error consists of physical qubit errors. For the distance 5 surface code you mention, a logical error consists of 5 physical qubit errors (single qubit Pauli gates). A logical error can't be detected or corrected.

What is the condition for occurring error on this logical operator?

A weight 3 error (error consisting of 3 single qubit Paulis) may cause a logical error because a decoder may decide to try and correct the error by applying a weight 2 correction.

I recommend reading upto section VI of the following paper: https://journals.aps.org/pra/pdf/10.1103/PhysRevA.86.032324

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  • $\begingroup$ What is "weight" do you mean ? $\endgroup$
    – 김동민
    Jan 18, 2023 at 4:25
  • $\begingroup$ The weight of an error is the number of Paulis that are not dentity. For example, the d=3 surface code has 9 data qubits and $X_1 \otimes X_2 \otimes I_3 \otimes I_4 \otimes I_5 \otimes I_6 \otimes I_7 \otimes I_8 \otimes I_9$ is a weight 2 error. Here the indexes indicate the data qubit the Pauli matrix acts on. $\endgroup$
    – Peter-Jan
    Jan 18, 2023 at 8:32
  • $\begingroup$ A logical error cannot be detected or corrected, then surface code cannot be used anymore and it would be defected ? $\endgroup$
    – 김동민
    Jan 18, 2023 at 12:43
  • $\begingroup$ One more thing, does surface code cannot detect correct logical error ? why ? $\endgroup$
    – 김동민
    Jan 19, 2023 at 0:28
  • $\begingroup$ A logical error commutes with all stabilizers, $\endgroup$
    – Peter-Jan
    Jan 19, 2023 at 9:59

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