Suppose a code has generators{Z⊗I⊗X,X⊗I⊗Z}{𝑍⊗𝐼⊗𝑋,𝑋⊗𝐼⊗𝑍} .Which of the following errors can be detected by the code?Group of answer choicesX1𝑋1 X2𝑋2 X0𝑋0 Z0𝑍0 Z2𝑍2 Z1𝑍1
Question
Suppose a code has generators{Z⊗I⊗X,X⊗I⊗Z}{𝑍⊗𝐼⊗𝑋,𝑋⊗𝐼⊗𝑍} .Which of the following errors can be detected by the code?Group of answer choicesX1𝑋1 X2𝑋2 X0𝑋0 Z0𝑍0 Z2𝑍2 Z1𝑍1
Solution
The given code generators are {Z⊗I⊗X, X⊗I⊗Z}.
Let's analyze each error:
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X1: This error cannot be detected by the code because it does not commute with the second generator X⊗I⊗Z.
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X2: This error cannot be detected by the code because it does not commute with the first generator Z⊗I⊗X.
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X0: This error can be detected by the code because it commutes with both generators.
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Z0: This error cannot be detected by the code because it does not commute with the second generator X⊗I⊗Z.
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Z2: This error cannot be detected by the code because it does not commute with the first generator Z⊗I⊗X.
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Z1: This error can be detected by the code because it commutes with both generators.
So, the errors that can be detected by the code are X0 and Z1.
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