From bcae81336764eb6c4cdf0f91e2fe632b625dd8b2 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 23 Oct 2022 22:18:11 +0200
Subject: Optimize and improve naming of all images in knowledge base
---
source/know/concept/repetition-code/index.md | 103 ++++++++++++++++-----------
1 file changed, 63 insertions(+), 40 deletions(-)
(limited to 'source/know/concept/repetition-code/index.md')
diff --git a/source/know/concept/repetition-code/index.md b/source/know/concept/repetition-code/index.md
index 99ac630..678211e 100644
--- a/source/know/concept/repetition-code/index.md
+++ b/source/know/concept/repetition-code/index.md
@@ -27,6 +27,7 @@ albeit with some complications,
as discussed below.
+
## Bit flip code
Suppose that we want to detect errors in
@@ -76,9 +77,7 @@ $$\begin{aligned}
Such a transformation is easy to achieve with the following sequence
of [quantum gates](/know/concept/quantum-gate/):
-
-
-
+{% include image.html file="bit-flip-encode.png" width="32%" alt="Bit flip code encoder" %}
So, a little while after encoding the state $$\Ket{\psi}$$ like that,
a bit flip occurs on the 2nd qubit:
@@ -166,31 +165,61 @@ But by using both, we know exactly which qubit was flipped
thanks to the eigenvalues:
-
- | Error |
- $$ZZI$$ |
- $$IZZ$$ |
-
-
- | $$I$$ |
- $$+1$$ |
- $$+1$$ |
-
-
- | $$X_1$$ |
- $$-1$$ |
- $$+1$$ |
-
-
- | $$X_2$$ |
- $$-1$$ |
- $$-1$$ |
-
-
- | $$X_1$$ |
- $$+1$$ |
- $$-1$$ |
-
+
+ |
+ Error
+ |
+
+ $$ZZI$$
+ |
+
+ $$IZZ$$
+ |
+
+
+ |
+ $$I$$
+ |
+
+ $$+1$$
+ |
+
+ $$+1$$
+ |
+
+
+ |
+ $$X_1$$
+ |
+
+ $$-1$$
+ |
+
+ $$+1$$
+ |
+
+
+ |
+ $$X_2$$
+ |
+
+ $$-1$$
+ |
+
+ $$-1$$
+ |
+
+
+ |
+ $$X_1$$
+ |
+
+ $$+1$$
+ |
+
+ $$-1$$
+ |
+
Where e.g. $$X_3$$ denotes that the 3rd qubit was flipped.
@@ -202,9 +231,7 @@ without affecting $$\ket{\overline{\psi}}$$ itself,
by applying $$\mathrm{CNOT}$$s to some ancillary qubits
and then measuring those:
-
-
-
+{% include image.html file="bit-flip-detect.png" width="62%" alt="Bit flip code decoder" %}
The two measurements, respectively representing $$ZZI$$ and $$IZZ$$,
yield $$\Ket{1}$$ if a bit flip definitely occurred,
@@ -213,6 +240,7 @@ There is no entanglement,
so the input is untouched.
+
## Phase flip code
The above system protects us against all single-qubit bit flips.
@@ -254,23 +282,20 @@ $$\begin{aligned}
= \alpha \Ket{+\!+\!+} + \beta \Ket{-\!-\!-}
\end{aligned}$$
-
-
-
+{% include image.html file="phase-flip-encode.png" width="40%" alt="Phase flip code encoder" %}
A phase flip along the $$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:
-
-
-
+{% include image.html file="phase-flip-detect.png" width="70%" alt="Phase flip code decoder" %}
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
@@ -307,9 +332,7 @@ 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:
-
-
-
+{% include image.html file="shor-code-encode.png" width="55%" alt="Shor code encoder" %}
We thus use 9 physical qubits to store 1 logical qubit.
Fortunately, more efficient schemes exist.
--
cgit v1.2.3