## 14.5 … and error families

The quotient group^{297}

The cosets of **error families** on the codespace **physical errors**.

Again, we can write ^{298}

These errors also let us understand how the structure of the codespace is mirrored across each of the cosets.
In other words, we picked ^{299}

- We can write
\mathcal{C}_1 as the stabiliser space of\mathcal{S} conjugated byX\mathbf{1}\mathbf{1} , i.e.\begin{aligned} (X\mathbf{1}\mathbf{1})\langle ZZ\mathbf{1}, \mathbf{1}ZZ \rangle(X\mathbf{1}\mathbf{1})^{-1} &= \langle (X\mathbf{1}\mathbf{1})(ZZ\mathbf{1})(X\mathbf{1}\mathbf{1})^{-1}, (X\mathbf{1}\mathbf{1})(\mathbf{1}ZZ)(X\mathbf{1}\mathbf{1})^{-1} \rangle \\&= \langle -ZZ\mathbf{1}, \mathbf{1}ZZ \rangle \end{aligned} and, indeed,|100\rangle and|011\rangle are both stabilised by this group. - The logical states of
\mathcal{C}_1 are, by definition as our chosen basis, the elements|100\rangle and|011\rangle , but note that these are exactly the images of the logical states of\mathcal{C} under the errorX\mathbf{1}\mathbf{1} , i.e.\begin{aligned} |0\rangle_{L,1} &\coloneqq |100\rangle = X\mathbf{1}\mathbf{1}|000\rangle \\|1\rangle_{L,1} &\coloneqq |011\rangle = X\mathbf{1}\mathbf{1}|111\rangle \end{aligned} - The logical operators on
\mathcal{C}_1 are the logical operators on\mathcal{C} conjugated byX\mathbf{1}\mathbf{1} , i.e\begin{aligned} X_{L,1} &\coloneqq (X\mathbf{1}\mathbf{1})(XXX)(X\mathbf{1}\mathbf{1})^{-1} \\&= XXX \\Z_{L,1} &\coloneqq (X\mathbf{1}\mathbf{1})(ZZZ)(X\mathbf{1}\mathbf{1})^{-1} \\&= -ZZZ \end{aligned} and, indeed,X_{L,1} andZ_{L,1} behave as expected on the new logical states, i.e.\begin{aligned} X_{L,1}\colon |0\rangle_{L,1} &\longmapsto |1\rangle_{L,1} \\|1\rangle_{L,1} &\longmapsto |0\rangle_{L,1} \\Z_{L,1}\colon |0\rangle_{L,1} &\longmapsto |0\rangle_{L,1} \\|1\rangle_{L,1} &\longmapsto -|1\rangle_{L,1} \end{aligned} as you can check by hand.

All in all, the chain of normal subgroups

But this stabiliser formalism introduces some new ambiguity.
In Section 14.3, we saw how measuring the three ancilla qubits in the Hamming *family* has occurred, not which specific *physical* error like they did before.
At first, this seems like a definite downgrade from our previous theory — the actual errors that affect our circuits are still the *physical* errors, but now we have no way of knowing which one occurred, only which family it lives in!
How are we to pick which coset representative to apply in order to correct the error?

As you might expect, the story is not yet over.
Depending on the specifics of the scenario, sometimes knowing the error family is enough to be able to correct not just one physical error, but *many*.
In order to give a more precise explanation, we need to take a step back and look at the scenarios that we’re actually trying to model — we do this in Section 14.7.

Recall that the elements of the quotient group

G/H are exactly the cosets ofH\triangleleft G .↩︎We sometimes denote the coset

P\cdot N(\mathcal{S}) simply by[P] , just to save space.↩︎Recall that conjugation expresses a change of basis: given an invertible

(n\times n) matrixB , we can turn a basis\{v_1,\ldots,v_n\} into a new basis\{Bv_1,\ldots,Bv_n\} , and to write any operatorA in this new basis we simply calculateBAB^{-1} (“undo the change of basis, applyA , then redo the change of basis”).↩︎