Past exam of the laser science and quantum information processing course of the University of Oxford / 2015 / 5 / d / Solution

Let $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$ be the Pauli X eigenstates. EncodeBoth have zero total $X$ eigenvalue, so $R_x(\varphi )\otimes R_x(\varphi )$ is the identity on their span. This is the decoherence-free subspace obtained from the previous code by applying a Hadamard gate to each physical qubit.

Since $Z|+⟩=|-⟩$ and $Z|-⟩=|+⟩$, a correlated $Z\otimes Z$ error exchanges the codewords. It acts as logical $X_L$, not as harmless identity. The Pauli Y gate obeys $Y|+⟩=-i|-⟩$ and $Y|-⟩=i|+⟩$, soThe phases cancel within each codeword. Thus either unknown correlated gate gives a logical bit error on a general encoded state, though a logical $X_L$ eigenstate is unchanged as a physical ray. The code prevents collective $x$ rotations, not every possible correlated error.