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Diffstat (limited to 'source/know/concept/repetition-code')
| -rw-r--r-- | source/know/concept/repetition-code/index.md | 9 |
1 files changed, 6 insertions, 3 deletions
diff --git a/source/know/concept/repetition-code/index.md b/source/know/concept/repetition-code/index.md index fa039a3..ba83c1a 100644 --- a/source/know/concept/repetition-code/index.md +++ b/source/know/concept/repetition-code/index.md @@ -94,7 +94,7 @@ We could measure the state, but that would make it collapse, which is probably not what we want. The trick is to use operators called **stabilizers**, -in this case for example $$ZZI = Z_1 \otimes Z_2 \otimes I_3$$, +in this case $$ZZI = Z_1 \otimes Z_2 \otimes I_3$$, where $$I$$ is identity and $$Z$$ is the Pauli-$$Z$$ gate. The 3-qubit basis states are its eigenvectors: @@ -127,7 +127,7 @@ $$\begin{alignedat}{2} We could measure $$ZZI$$ for $$\ket{\overline{\psi}}$$, and if the eigenvalue is $$-1$$, we know that a bit flip has occurred, -whereas if the eigenvalue is $$+1$$, +but if the eigenvalue is $$+1$$, there is *maybe* no error ($$\Ket{001}$$ and $$\Ket{110}$$ are false negatives). These false negatives are fixed by including another stabilizer $$IZZ$$, @@ -170,7 +170,7 @@ thanks to the eigenvalues: | $$I$$ | $$+1$$ | $$+1$$ | | $$X_1$$ | $$-1$$ | $$+1$$ | | $$X_2$$ | $$-1$$ | $$-1$$ | -| $$X_1$$ | $$+1$$ | $$-1$$ | +| $$X_3$$ | $$+1$$ | $$-1$$ | Where e.g. $$X_3$$ denotes that the 3rd qubit was flipped. The measurement outcomes on the last three rows are called **error syndromes**, @@ -309,6 +309,9 @@ $$\begin{aligned} III \: XXX \: XXX \end{aligned}$$ +In this way, we are protected against all single-qubit errors, +but at a significant physical cost. + ## References |