Consider the five-qubit code generator

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

Consider the five-qubit code generator g4=ZXIXZg_4 = ZXIXZ, that is ZZ on qubit 1, XX on qubit 2, identity on qubit 3, XX on qubit 4, and ZZ on qubit 5. Its binary vector is g4=(01010∣10001)g_4 = (01010|10001). ((a)) Describe the one-ancilla circuit that measures g4g_4: state which single-qubit controlled gate acts on each qubit in its support, and count the two-qubit gates used. ((b)) An error Y4Y_4 (a YY error on qubit 4, binary vector (00010∣00010)(00010|00010)) occurs on a codeword for which ⟨g4⟩=+1\langle g_4\rangle = +1 before the error. Using the symplectic product of Y4Y_4 with g4g_4's binary vector, find ⟨g4⟩\langle g_4\rangle after the error and hence the probability of each outcome of the circuit in part (a).

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