This makes no difference to what we're after.
Note that, throughout, I've been lazy about imaginary numbers appearing, and have instead written $XZ=Y$. However, it will not be able to correct the product error Through the field survey, it was found that although the fractures on the exposed surface were adequately developed, the actual information of the aperture opening was lost as a result of weathering (Fig. Next, let's introduce a unitary $U$ such thatĭefining a new code $C'$ as $UCU^\dagger$ (please allow the abuse of notation I hope it's clear what I mean), it must be able to correct errors $X_1$ and $Z_1$. So, the code will not be able to correct this error (or, at least, there surely exist codes that do not correct this error). On the other hand, the product of the two errors is $Y_1Z_2Z_3Z_4$, which contains 4 $Z$ errors. This code can correct an error $X_1Z_2Z_3$ or an error $Z_1Z_4$ because CSS codes are independently distance 5 on the two X/Z types. If (a, b, c) is the starting normed vector and (x, y, z) is the end normed vector you can write b sin(f) + c cos(f) z based on rotation matrix around X-axis, where f is rotation angle around X-axis. In that context, the following construction may be of assistance:Ĭonsider a standard distance 5 CSS code, $C$. If you rotate first around X-axis and then around Z-axis your last rotation do not change z projection. I've understood from the comments that the OP is willing to consider a more general error set, rather than focussing specifically on the more standard case of distance being a measure of the number of single-qubit errors that can be tolerated.
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