In error correction, like the bit flip, you perform a measurement which corresponds to different projections so that the outcomes can teach you about the error. What does it mean? How do you actually do it? Do you have to do the measurement in different bases?

Same with arbitrary errors which are composed of $I, X, Y, Z$ matrices and you want to perform a measurement which will collapse the state to either one of them. How is it performed? Do you measure in different bases?

What's the difference between measurement in different bases to measurement with different projections?


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Let's get some terminology correct. A particular measurement basis corresponds to a set of projectors, $\{P_i\}$ (satisfying $\sum_iP_i=\mathbb{I}$), where each $P_i$ corresponds to a measurement outcome. Sometimes, measurements are specified by giving a Hermitian operator $H$ which is not a set of projectors. Most often, we'd be talking about something with eigenvalues $\pm 1$, so you'd have $$ H=P_1-P_2, $$ so that defines the corresponding projectors.

From a theorist's perspective you can 'just' measure in that particular basis. From a practical perspective, what you often want to do is convert the measurement into a measurement on the computational basis. You can find a unitary $U$ such that $$ \tilde P_i=UP_iU^\dagger $$ is diagonal. So, instead, you can apply unitary $U$ and then measure in the computational basis (although if the projectors are not rank 1, there would be some grouping of terms).

With regards to error correction, there is a particularly natural way to measure the $\pm 1$ eigenvalue of a stabilizer. enter image description here You measure the top qubit in the computational basis, where the $\sigma_1\otimes\ldots\otimes\sigma_n$ is the stabilizer operator that you want to measure.


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