The Shor code

We see from the above that the starting idea of error correction is to encode a logical

qubit in multiple physical qubits. Such encoding is usually referred to as code; we could

call the first circuit a bit code and the second circuit a phase code. These are simple

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codes that only correct single discrete error on any of the qubits. The bit code only

corrects bit errors and the phase code only corrects phase errors.

There exists a code introduced by Shor, that can correct any single qubit error. This can

be seen as a combination of the bit code and the phase code, and it requires encoding

the logical qubits in nine physical qubits. This code is given by the following circuit:

rigure s: the snor code

What are the logical qubits $|0\_($L$)$:$)$ and $|1$:$)\_($L$)$ in the Shor code? $(0\mathrm{.}5$p$)$

It is pretty amazing that the Shor code can detect and correct any single qubit error,

since a general error is not going to be discrete but rather one of continuous set of error

$($for example one could instead of the phase flip have an error that multiples by a any

phase$).$ We won't show that this is a fact here $($though if you want, you can try to

convince yourself that it works$),$ but instead just look at a single discrete error in the

Shor code, namely a phase flip on any of the nine physical qubits.

Show that you can detect a phase flip error by measuring the two observables

X$\_(1)$X$\_(2)$X$\_(3)$X$\_(4)$X$\_(5)$X$\_(6)$ and X$\_(4)$X$\_(5)$X$\_(6)$X$\_(7)$X$\_(8)$X$\_(9).(0\mathrm{.}5$ p $)$

Show that you can recover from a phase flip on any of the first three qubits with

Z$\_(1)$Z$\_(2)$Z$\_(3).(0\mathrm{.}5$p$)$

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