The five-qubit code has generators , , , .

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 five-qubit code has generators g1=XZZXIg_1=XZZXI, g2=IXZZXg_2=IXZZX, g3=XIXZZg_3=XIXZZ, g4=ZXIXZg_4=ZXIXZ. ((a)) Using the qubit-by-qubit anticommutation rule already used to test whether two Pauli strings commute, compute the syndrome of the two-qubit error X1X2X_1X_2 against all four generators. ((b)) Find which one of the fifteen single-qubit errors Xj,Yj,ZjX_j,Y_j,Z_j (j=1,…,5j=1,\ldots,5) produces exactly this same syndrome. ((c)) Say what the decoder does on seeing this syndrome, and give the net operator left on the five qubits once its correction has been applied. Is the code's guarantee against a single-qubit error violated?

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