The three-qubit bit-flip code has

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Open quantum systems, decoherence and error correction · Open quantum systems, decoherence and error correction

Quantum Information and Technology, Ch Quantum error correction

Problem

The three-qubit bit-flip code has ∣0L⟩=∣000⟩|0_L\rangle=|000\rangle, ∣1L⟩=∣111⟩|1_L\rangle=|111\rangle and P=∣0L⟩⟨0L∣+∣1L⟩⟨1L∣P=|0_L\rangle\langle0_L|+|1_L\rangle\langle1_L|. Test the error list {I,X1,X2X3}\{I,X_1,X_2X_3\} against the Knill-Laflamme conditions. ((a)) Compute PX1PPX_1P and PX2X3PPX_2X_3P, and say whether each is a multiple of PP. ((b)) Compute P(X1†X2X3)P=P(X1X2X3)PP(X_1^\dagger X_2X_3)P=P(X_1X_2X_3)P, and say whether it is a multiple of PP. ((c)) State whether {I,X1,X2X3}\{I,X_1,X_2X_3\} is a correctable list of errors for this code, and contrast the outcome with the list {I,X1,X2,X3}\{I,X_1,X_2,X_3\} already checked in the text just before this problem.

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