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+---
+title: "Repetition code"
+date: 2021-05-07
+categories:
+- Quantum information
+layout: "concept"
+---
+
+A **repetition code** is a simple approach to error correction:
+to protect a bit $x$, make two copies:
+
+$$\begin{aligned}
+    0 \to 000
+    \qquad \quad
+    1 \to 111
+\end{aligned}$$
+
+If a single-bit error occurs, e.g. $000 \to 100$,
+a majority vote resets the minority bit.
+Clearly, this does not protect against multi-bit errors,
+but that is usually not necessary.
+
+In quantum computing, where error correction is much more important,
+repetition codes can also be used,
+albeit with some complications,
+as discussed below.
+
+
+## Bit flip code
+
+Suppose that we want to detect errors in
+the following arbitrary qubit state $\Ket{\psi}$:
+
+$$\begin{aligned}
+    \Ket{\psi}
+    = \alpha \Ket{0} + \beta \Ket{1}
+\end{aligned}$$
+
+For now, let us limit ourselves to detecting **bit flips**,
+where $\alpha$ and $\beta$ get switched:
+
+$$\begin{aligned}
+    \alpha \Ket{0} + \beta \Ket{1}
+    \quad \to \boxed{\mathrm{Error}} \to \quad
+    \beta \Ket{0} \!+\! \alpha \Ket{1}
+\end{aligned}$$
+
+One way to defend against this is
+the quantum version of a classical repetition code:
+
+$$\begin{aligned}
+    \Ket{\psi}
+    \quad \to \boxed{\mathrm{Encoder}} \to \quad
+    \ket{\overline{\psi}}
+    = \alpha \Ket{000} + \beta \Ket{111}
+\end{aligned}$$
+
+In other words, a *logical* $\Ket{0}$ (written $\ket{\overline{0}}$)
+is represented by 3 *physical* qubits, and vice versa:
+
+$$\begin{aligned}
+    \boxed{
+        \Ket{0}
+        \to
+        \ket{\overline{0}}
+        = \Ket{000}
+        \qquad \quad
+        \Ket{1}
+        \to
+        \ket{\overline{1}}
+        = \Ket{111}
+    }
+\end{aligned}$$
+
+Such a transformation is easy to achieve with the following sequence
+of [quantum gates](/know/concept/quantum-gate/):
+
+<a href="bit-flip-encode.png">
+<img src="bit-flip-encode.png" style="width:32%">
+</a>
+
+So, a little while after encoding the state $\Ket{\psi}$ like that,
+a bit flip occurs on the 2nd qubit:
+
+$$\begin{aligned}
+    \ket{\overline{\psi}}
+    \quad \to \boxed{\mathrm{Error}} \to \quad
+    \alpha \Ket{010} + \beta \Ket{101}
+\end{aligned}$$
+
+But now there is a problem: how do we detect this error?
+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 $ZZI = Z_1 \otimes Z_2 \otimes I_3$,
+where $I$ is identity and $Z$ is the Pauli-$Z$ gate.
+The 3-qubit basis states are its eigenvectors:
+
+$$\begin{alignedat}{2}
+    ZZI \Ket{000}
+    &= + \Ket{000}
+    \qquad
+    ZZI \Ket{001}
+    &&= + \Ket{001}
+    \\
+    ZZI \Ket{010}
+    &= - \Ket{010}
+    \qquad
+    ZZI \Ket{011}
+    &&= - \Ket{011}
+    \\
+    ZZI \Ket{100}
+    &= - \Ket{100}
+    \qquad
+    ZZI \Ket{101}
+    &&= - \Ket{101}
+    \\
+    ZZI \Ket{110}
+    &= + \Ket{110}
+    \qquad
+    ZZI \Ket{111}
+    &&= + \Ket{111}
+\end{alignedat}$$
+
+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 $+1$,
+there is *maybe* no error ($\Ket{001}$ and $\Ket{110}$ are false negatives).
+
+These false negatives are fixed by including another stabilizer $IZZ$,
+with these eigenvectors:
+
+$$\begin{alignedat}{2}
+    IZZ \Ket{000}
+    &= + \Ket{000}
+    \qquad
+    IZZ \Ket{001}
+    &&= - \Ket{001}
+    \\
+    IZZ \Ket{010}
+    &= - \Ket{010}
+    \qquad
+    IZZ \Ket{011}
+    &&= + \Ket{011}
+    \\
+    IZZ \Ket{100}
+    &= + \Ket{100}
+    \qquad
+    IZZ \Ket{101}
+    &&= - \Ket{101}
+    \\
+    IZZ \Ket{110}
+    &= - \Ket{110}
+    \qquad
+    IZZ \Ket{111}
+    &&= + \Ket{111}
+\end{alignedat}$$
+
+In which case $\Ket{100}$ and $\Ket{011}$ are false negatives.
+In other words, $IZZ$ cannot detect if the 1st qubit was flipped,
+while $ZZI$ cannot protect the 3rd qubit.
+But by using both, we know exactly which qubit was flipped
+thanks to the eigenvalues:
+
+<table style="width:30%;margin:auto;text-align:center;">
+    <tr>
+        <th>Error</th>
+        <th>$ZZI$</th>
+        <th>$IZZ$</th>
+    </tr>
+    <tr>
+        <td>$I$</td>
+        <td>$+1$</td>
+        <td>$+1$</td>
+    </tr>
+    <tr>
+        <td>$X_1$</td>
+        <td>$-1$</td>
+        <td>$+1$</td>
+    </tr>
+    <tr>
+        <td>$X_2$</td>
+        <td>$-1$</td>
+        <td>$-1$</td>
+    </tr>
+    <tr>
+        <td>$X_1$</td>
+        <td>$+1$</td>
+        <td>$-1$</td>
+    </tr>
+</table>
+
+Where e.g. $X_3$ denotes that the 3rd qubit was flipped.
+The measurement outcomes on the last three rows are called **error syndromes**,
+and are obtained by a **syndrome measurement**.
+
+Fortunately, we can measure $ZZI$ and $IZZ$
+without affecting $\ket{\overline{\psi}}$ itself,
+by applying $\mathrm{CNOT}$s to some ancillary qubits
+and then measuring those:
+
+<a href="bit-flip-detect.png">
+<img src="bit-flip-detect.png" style="width:62%">
+</a>
+
+The two measurements, respectively representing $ZZI$ and $IZZ$,
+yield $\Ket{1}$ if a bit flip definitely occurred,
+and $\Ket{0}$ otherwise.
+There is no entanglement,
+so the input is untouched.
+
+
+## Phase flip code
+
+The above system protects us against all single-qubit bit flips.
+Unfortunately, that is not enough:
+qubits can also experience a **phase flip**:
+
+$$\begin{aligned}
+    \alpha \Ket{0} + \beta \Ket{1}
+    \quad \to \boxed{\mathrm{Error}} \to \quad
+    \alpha \Ket{0} - \beta \Ket{1}
+\end{aligned}$$
+
+How to detect that?
+If we want to protect against phase flips *instead of* bit flips,
+we can simply do the same as before,
+but along the $X$-axis intead of the $Z$-axis:
+
+$$\begin{aligned}
+    \boxed{
+        \Ket{0}
+        \to
+        \ket{\overline{0}}
+        = \Ket{+\!+\!+}
+        \qquad \quad
+        \Ket{1}
+        \to
+        \ket{\overline{1}}
+        = \Ket{-\!-\!-}
+    }
+\end{aligned}$$
+
+Such that an arbitrary state $\Ket{\psi}$ is encoded as follows,
+by the circuit shown below:
+
+$$\begin{aligned}
+    \Ket{\psi}
+    \quad \to \boxed{\mathrm{Encoder}} \to \quad
+    \ket{\overline{\psi}}
+    = \alpha \Ket{+\!+\!+} + \beta \Ket{-\!-\!-}
+\end{aligned}$$
+
+<a href="phase-flip-encode.png">
+<img src="phase-flip-encode.png" style="width:40%">
+</a>
+
+A phase flip along the $Z$-axis
+corresponds to a bit flip along the $X$-axis $\Ket{+} \to \Ket{-}$.
+In this case, the stabilizers are $XXI$ and $IXX$,
+and the error detection circuit is as follows:
+
+<a href="phase-flip-detect.png">
+<img src="phase-flip-detect.png" style="width:70%">
+</a>
+
+This system protects us against all single-qubit phase flips,
+but not against bit flips.
+
+
+## Shor code
+
+What kind of repetition code would we need
+if we want to detect both bit flips *and* phase flips?
+The most straightforward option is the **Shor code**.
+Starting from a phase flip encoding:
+
+$$\begin{aligned}
+    \Ket{0} \to
+    \ket{\overline{0}}
+    &= \Ket{+\!+\!+}
+    = \bigg( \frac{\Ket{0} + \Ket{1}}{\sqrt{2}} \bigg)^{\otimes 3}
+    \\
+    \Ket{1} \to
+    \ket{\overline{1}}
+    &= \Ket{-\!-\!-}
+    = \bigg( \frac{\Ket{0} - \Ket{1}}{\sqrt{2}} \bigg)^{\otimes 3}
+\end{aligned}$$
+
+We add protection against bit flips
+by using a repetition code for each physical qubit:
+
+$$\begin{aligned}
+    \boxed{
+        \ket{\overline{0}}
+        = \bigg( \frac{\Ket{000} + \Ket{111}}{\sqrt{2}} \bigg)^{\otimes 3}
+        \qquad \quad
+        \ket{\overline{1}}
+        = \bigg( \frac{\Ket{000} - \Ket{111}}{\sqrt{2}} \bigg)^{\otimes 3}
+    }
+\end{aligned}$$
+
+This encoding is achieved by the following quantum circuit,
+which simply consists of the phase flip encoder,
+followed by 3 copies of the bit flip encoder:
+
+<a href="shor-code-encode.png">
+<img src="shor-code-encode.png" style="width:55%">
+</a>
+
+We thus use 9 physical qubits to store 1 logical qubit.
+Fortunately, more efficient schemes exist.
+
+The bit flip stabilizers $ZZI$ and $IZZ$
+are applied on a per-block basis, like so:
+
+$$\begin{aligned}
+    ZZI \: III \: III \qquad\quad III \: ZZI \: III \qquad\quad III \: III \: ZZI
+    \\
+    IZZ \: III \: III \qquad\quad III \: IZZ \: III \qquad\quad III \: III \: IZZ
+\end{aligned}$$
+
+Whereas the phase flip stabilizers $XXI$ and $IXX$
+are applied to entire blocks at once:
+
+$$\begin{aligned}
+    XXX \: XXX \: III
+    \qquad \quad
+    III \: XXX \: XXX
+\end{aligned}$$
+
+
+
+## References
+1.  J.S. Neergaard-Nielsen,
+    *Quantum information: lectures notes*,
+    2021, unpublished.
+2.  S. Aaronson,
+    *Introduction to quantum information science: lecture notes*,
+    2018, unpublished.
+