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 stores α∣0⟩+β∣1⟩ as α∣000⟩+β∣111⟩. An X3 error strikes the third data qubit. Syndrome extraction then uses two ancilla qubits, prepared in ∣0⟩ and read out after CNOTs, to measure Z1Z2 and Z2Z3. The Z1Z2 ancilla is read out correctly. The Z2Z3 ancilla itself suffers a bit-flip error during its own readout, so the classical bit reported for Z2Z3 is the opposite of what the data actually gave.
((a)) Write the state of the three data qubits immediately after the X3 error (before any syndrome measurement).
((b)) Someone reasons: "The reported syndrome is (Z1Z2,Z2Z3)=(+1,+1), which is the syndrome for no error, so nothing needs correcting." State the syndrome that a fault-free measurement would actually have returned, and explain what is wrong with this reasoning.
((c)) Given that the faulty readout leads to applying no correction, write the state of the three data qubits after this (mistaken) decision, and say whether the logical information α,β has been damaged at this stage.