More improvements to knowledge base

1 files changed, 43 insertions, 42 deletions

diff --git a/source/know/concept/quantum-gate/index.md b/source/know/concept/quantum-gate/index.md
index 9704e53..dd198f2 100644
--- a/source/know/concept/quantum-gate/index.md
+++ b/source/know/concept/quantum-gate/index.md
@@ -17,15 +17,15 @@ so we only consider the most important examples here.
 
 ## One-qubit gates
 
-As an example, consider the following must general single-qubit state $$\Ket{\psi}$$:
+As an example, consider the following most general single-qubit state $$\ket{\psi}$$:
 
 $$\begin{aligned}
-    \Ket{\psi}
-    = \alpha \Ket{0} + \beta \Ket{1}
+    \ket{\psi}
+    = \alpha \ket{0} + \beta \ket{1}
     = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}
 \end{aligned}$$
 
-Arguably the most famous and/or most fundamental quantum gates are the **Pauli matrices**:
+Arguably the most famous and most fundamental quantum gates are the **Pauli matrices**:
 
 $$\begin{aligned}
     \boxed{
@@ -53,19 +53,19 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-They have the following effect on $$\Ket{\psi}$$.
+They have the following effect on $$\ket{\psi}$$.
 Note that $$X$$ is equivalent to the classical $$\mathrm{NOT}$$ gate
 (and is often given that name),
 and $$Z$$ is sometimes called the **phase-flip gate**:
 
 $$\begin{aligned}
-    X \Ket{\psi}
+    X \ket{\psi}
     = \begin{bmatrix} \beta \\ \alpha \end{bmatrix}
     \qquad
-    Y \Ket{\psi}
+    Y \ket{\psi}
     = \begin{bmatrix} -i \beta \\ i \alpha \end{bmatrix}
     \qquad
-    Z \Ket{\psi}
+    Z \ket{\psi}
     = \begin{bmatrix} \alpha \\ -\beta \end{bmatrix}
 \end{aligned}$$
 
@@ -87,7 +87,7 @@ For $$\phi = \pi$$, we recover the Pauli-$$Z$$ gate.
 In general, the action of $$R_\phi$$ is as follows:
 
 $$\begin{aligned}
-    R_\phi \Ket{\psi}
+    R_\phi \ket{\psi}
     = \begin{bmatrix} \alpha \\ e^{i \phi} \beta \end{bmatrix}
 \end{aligned}$$
 
@@ -128,10 +128,11 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Its action consists of rotating the qubit
-by $$\pi$$ around the axis $$(X + Z) / \sqrt{2}$$ of the Bloch sphere:
+by $$\pi$$ around the axis $$(X + Z) / \sqrt{2}$$ of
+the [Bloch sphere](/know/concept/bloch-sphere/):
 
 $$\begin{aligned}
-    H \Ket{\psi}
+    H \ket{\psi}
     = \frac{1}{\sqrt{2}} \begin{bmatrix} \alpha + \beta \\ \alpha - \beta \end{bmatrix}
 \end{aligned}$$
 
@@ -139,20 +140,20 @@ Notably, it maps the eigenstates of $$X$$ and $$Z$$ to each other,
 and is its own inverse (i.e. unitary):
 
 $$\begin{aligned}
-    H \Ket{0} = \Ket{+}
+    H \ket{0} = \ket{+}
     \qquad
-    H \Ket{1} = \Ket{-}
+    H \ket{1} = \ket{-}
     \qquad
-    H \Ket{+} = \Ket{0}
+    H \ket{+} = \ket{0}
     \qquad
-    H \Ket{-} = \Ket{1}
+    H \ket{-} = \ket{1}
 \end{aligned}$$
 
 The **Clifford gates** are a set including $$X$$, $$Y$$, $$Z$$, $$H$$ and $$S$$,
 or more generally any gates that rotate
 by multiples of $$\pi/2$$ around the Bloch sphere.
-This set is **not universal**, meaning that if we start from $$\Ket{0}$$,
-we can only reach $$\Ket{0}$$, $$\Ket{1}$$, $$\Ket{+}$$, $$\Ket{-}$$, $$\Ket{+i}$$ $$\Ket{-i}$$ using these gates.
+This set is **not universal**, meaning that if we start from $$\ket{0}$$,
+we can only reach $$\ket{0}$$, $$\ket{1}$$, $$\ket{+}$$, $$\ket{-}$$, $$\ket{+i}$$ $$\ket{-i}$$ using these gates.
 
 If we add *any* non-Clifford gate, for example $$T$$,
 then we can reach any point on the Bloch sphere,
@@ -170,15 +171,15 @@ any state can be approximated.
 ## Two-qubit gates
 
 As an example, let us consider
-the following two pure one-qubit states $$\Ket{\psi_1}$$ and $$\Ket{\psi_2}$$:
+the following two pure one-qubit states $$\ket{\psi_1}$$ and $$\ket{\psi_2}$$:
 
 $$\begin{aligned}
-    \Ket{\psi_1}
-    = \alpha_1 \Ket{0} + \beta_1 \Ket{1}
+    \ket{\psi_1}
+    = \alpha_1 \ket{0} + \beta_1 \ket{1}
     = \begin{bmatrix} \alpha_1 \\ \beta_1 \end{bmatrix}
     \qquad \quad
-    \Ket{\psi_2}
-    = \alpha_2 \Ket{0} + \beta_2 \Ket{1}
+    \ket{\psi_2}
+    = \alpha_2 \ket{0} + \beta_2 \ket{1}
     = \begin{bmatrix} \alpha_2 \\ \beta_2 \end{bmatrix}
 \end{aligned}$$
 
@@ -186,23 +187,22 @@ The composite state of both qubits, assuming they are pure,
 is then their tensor product $$\otimes$$:
 
 $$\begin{aligned}
-    \Ket{\psi_1 \psi_2}
-    = \Ket{\psi_1} \otimes \Ket{\psi_2}
-    &= \alpha_1 \alpha_2 \Ket{00} + \alpha_1 \beta_2 \Ket{01} + \beta_1 \alpha_2 \Ket{10} + \beta_1 \beta_2 \Ket{11}
+    \ket{\psi_1 \psi_2}
+    = \ket{\psi_1} \otimes \ket{\psi_2}
+    &= \alpha_1 \alpha_2 \ket{00} + \alpha_1 \beta_2 \ket{01} + \beta_1 \alpha_2 \ket{10} + \beta_1 \beta_2 \ket{11}
     \\
-    &= c_{00} \Ket{00} + c_{01} \Ket{01} + c_{10} \Ket{10} + c_{11} \Ket{11}
+    &= c_{00} \ket{00} + c_{01} \ket{01} + c_{10} \ket{10} + c_{11} \ket{11}
 \end{aligned}$$
 
 Note that a two-qubit system may be [entangled](/know/concept/quantum-entanglement/),
 in which case the coefficients $$c_{00}$$ etc. cannot be written as products,
-i.e. $$\Ket{\psi_2}$$ cannot be expressed separately from $$\Ket{\psi_1}$$, and vice versa.
+i.e. $$\ket{\psi_2}$$ cannot be expressed separately from $$\ket{\psi_1}$$, and vice versa.
+In other words, the action of a two-qubit gate
+can be expressed in the basis of $$\ket{00}$$, $$\ket{01}$$, $$\ket{10}$$ and $$\ket{11}$$,
+but not always in the basis of $$\ket{0}_1$$, $$\ket{1}_1$$, $$\ket{0}_2$$ and $$\ket{1}_2$$.
 
-In other words, the general action of a two-qubit quantum gate
-can be expressed in the basis of $$\Ket{00}$$, $$\Ket{01}$$, $$\Ket{10}$$ and $$\Ket{11}$$,
-but not always in the basis of $$\Ket{0}_1$$, $$\Ket{1}_1$$, $$\Ket{0}_2$$ and $$\Ket{1}_2$$.
-
-With that said, the first two-qubit gate is $$\mathrm{SWAP}$$,
-which simply swaps $$\Ket{\psi_1}$$ and $$\Ket{\psi_2}$$:
+With this noted, the first two-qubit gate is $$\mathrm{SWAP}$$,
+which simply swaps $$\ket{\psi_1}$$ and $$\ket{\psi_2}$$:
 
 {% include image.html file="swap.png" width="22%"
     alt="SWAP gate diagram" %}
@@ -219,18 +219,18 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-This matrix is given in the basis of $$\Ket{00}$$, $$\Ket{01}$$, $$\Ket{10}$$ and $$\Ket{11}$$.
+This matrix is given in the basis of $$\ket{00}$$, $$\ket{01}$$, $$\ket{10}$$ and $$\ket{11}$$.
 Note that $$\mathrm{SWAP}$$ cannot generate entanglement,
 so if its input is separable, its output is too.
 In any case, its effect is clear:
 
 $$\begin{aligned}
-    \mathrm{SWAP} \Ket{\psi_1 \psi_2}
-    &= c_{00} \Ket{00} + c_{10} \Ket{01} + c_{01} \Ket{10} + c_{11} \Ket{11}
+    \mathrm{SWAP} \ket{\psi_1 \psi_2}
+    &= c_{00} \ket{00} + c_{10} \ket{01} + c_{01} \ket{10} + c_{11} \ket{11}
 \end{aligned}$$
 
 Next, there is the **controlled NOT gate** $$\mathrm{CNOT}$$,
-which "flips" (applies $$X$$ to) $$\Ket{\psi_2}$$ if $$\Ket{\psi_1}$$ is true:
+which "flips" (applies $$X$$ to) $$\ket{\psi_2}$$ if $$\ket{\psi_1}$$ is true:
 
 {% include image.html file="cnot.png" width="22%"
     alt="CNOT gate diagram" %}
@@ -250,13 +250,13 @@ $$\begin{aligned}
 That is, it swaps the last two coefficients $$c_{10}$$ and $$c_{11}$$ in the composite state vector:
 
 $$\begin{aligned}
-    \mathrm{CNOT} \Ket{\psi_1 \psi_2}
-    &= c_{00} \Ket{00} + c_{01} \Ket{01} + c_{11} \Ket{10} + c_{10} \Ket{11}
+    \mathrm{CNOT} \ket{\psi_1 \psi_2}
+    &= c_{00} \ket{00} + c_{01} \ket{01} + c_{11} \ket{10} + c_{10} \ket{11}
 \end{aligned}$$
 
 More generally, from every one-qubit gate $$U$$,
 we can define a two-qubit **controlled U gate** $$\mathrm{CU}$$,
-which applies $$U$$ to $$\Ket{\psi_2}$$ if $$\Ket{\psi_1}$$ is true:
+which applies $$U$$ to $$\ket{\psi_2}$$ if $$\ket{\psi_1}$$ is true:
 
 {% include image.html file="cu.png" width="22%"
     alt="CU gate diagram" %}
@@ -277,8 +277,8 @@ Where the lower-right 2x2 block is simply $$U$$.
 The general action of this gate is given by:
 
 $$\begin{aligned}
-    \mathrm{CU} \Ket{\psi_1 \psi_2}
-    &= c_{00} \Ket{00} + c_{01} \Ket{01} + (c_{10} u_{00} + c_{11} u_{01}) \Ket{10} + (c_{10} u_{10} + c_{11} u_{11}) \Ket{11}
+    \mathrm{CU} \ket{\psi_1 \psi_2}
+    &= c_{00} \ket{00} + c_{01} \ket{01} + (c_{10} u_{00} + c_{11} u_{01}) \ket{10} + (c_{10} u_{10} + c_{11} u_{11}) \ket{11}
 \end{aligned}$$
 
 A set of gates is **universal** if all possible mappings
@@ -287,6 +287,7 @@ A minimal universal set is $$\{\mathrm{CNOT}, T, S\}$$,
 and there exist many others.
 
 
+
 ## References
 1.  J.S. Neergaard-Nielsen,
     *Quantum information: lectures notes*,