The three-qubit GHZ state is stabilized by , ,

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 The stabilizer formalism

Problem

The three-qubit GHZ state is stabilized by g1=XXXg_1=XXX, g2=ZZIg_2=ZZI, g3=IZZg_3=IZZ. Consider measuring g3=Z2Z3g_3 = Z_2Z_3 by the one-ancilla recipe: an ancilla starts in ∣0⟩|0\rangle, is Hadamarded to ∣+⟩|+\rangle, controlled-g3g_3 is applied from the ancilla onto the data qubits, the ancilla is Hadamarded again, and then measured in the computational basis. ((a)) Because g3g_3 is a product of ZZ's only, show that this reduces to the two-CNOT parity-check circuit and state how many two-qubit gates it uses. ((b)) Suppose an error has left the data qubits in a state with ⟨g3⟩=−1\langle g_3\rangle = -1. Give the probability of each of the two ancilla outcomes.

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