correcting notation

Author: mhjensen

doc/pub/week6/html/week6-bs.html

@@ -819,17 +819,17 @@ <h2 id="expectation-values" class="anchor">Expectation values </h2>
 for a Pauli-\( \boldsymbol{X} \) matrix
 </p>
 $$
-\boldsymbol{X}=R_{\sigma}\boldsymbol{Z}R_{\sigma} = HZH
+\boldsymbol{X}=\boldsymbol{R}_{\sigma}\boldsymbol{Z}\boldsymbol{R}_{\sigma} = \boldsymbol{HZH}
 $$
 
 <p>for a Pauli-\( \boldsymbol{Y} \) matrix</p>
 $$
-\boldsymbol{Y}=R_{\sigma}\boldsymbol{Z}R_{\sigma}=\boldsymbol{HS}^{\dagger}\boldsymbol{ZHS},
+\boldsymbol{Y}=\boldsymbol{R}_{\sigma}\boldsymbol{Z}\boldsymbol{R}_{\sigma}=\boldsymbol{HS}^{\dagger}\boldsymbol{ZHS},
 $$
 
 <p>and</p>
 $$
-\boldsymbol{Z}=R_{\sigma}ZR_{\sigma}=\boldsymbol{I}\boldsymbol{Z}\boldsymbol{I}=\boldsymbol{Z}.
+\boldsymbol{Z}=\boldsymbol{R}_{\sigma}\boldsymbol{Z}\boldsymbol{R}_{\sigma}=\boldsymbol{I}\boldsymbol{Z}\boldsymbol{I}=\boldsymbol{Z}.
 $$
 
 
@@ -888,7 +888,7 @@ <h2 id="reminder-on-rotations" class="anchor">Reminder on rotations </h2>
 
 <p>Note the following identity of the basis rotator</p>
 $$
-R^\dagger_\sigma \boldsymbol{Z} R_\sigma = \sigma,
+\boldsymbol{R}^\dagger_\sigma \boldsymbol{Z} \boldsymbol{R}_\sigma = \boldsymbol{\sigma,}
 $$
 
 <p>which follows from the fact that \( \boldsymbol{HZH}=\boldsymbol{X} \) and \( \boldsymbol{SXS}^\dagger=\boldsymbol{Y} \).</p>
@@ -1084,13 +1084,13 @@ <h2 id="non-interacting-solution" class="anchor">Non-interacting solution </h2>
 <h2 id="rewriting-with-pauli-matrices" class="anchor">Rewriting with Pauli matrices </h2>
 <p>We rewrite \( H \) (and \( H_0 \) and \( H_I \))  via Pauli matrices</p>
 $$
-\mathcal{H}_0 = \mathcal{E} I + \Omega \sigma_z, \quad \mathcal{E} = \frac{E_1
+\mathcal{H}_0 = \mathcal{E} I + \Omega \boldsymbol{Z}, \quad \mathcal{E} = \frac{E_1
   + E_2}{2}, \; \Omega = \frac{E_1-E_2}{2},
 $$
 
 <p>and</p>
 $$
-\mathcal{H}_I = c \boldsymbol{I} +\omega_z\sigma_z + \omega_x\sigma_x,
+\mathcal{H}_I = c \boldsymbol{I} +\omega_z\boldsymbol{Z} + \omega_x\boldsymbol{X},
 $$
 
 <p>with \( c = (V_{11}+V_{22})/2 \), \( \omega_z = (V_{11}-V_{22})/2 \) and \( \omega_x = V_{12}=V_{21} \).
@@ -1221,7 +1221,7 @@ <h2 id="measurements-and-computational-basis" class="anchor">Measurements and co
 and the identity matrix \( \boldsymbol{I} \). Let us make this Hamiltonian that involves only one qubit somewhat more general
 </p>
 $$
-\left\langle \psi \right| \mathcal{H} \left| \psi \right\rangle = a \cdot \left\langle \psi \right| I \left| \psi \right\rangle + b \cdot \left\langle \psi \right| Z \left| \psi \right\rangle + c \cdot \left\langle \psi \right| X \left| \psi \right\rangle + d \cdot \left\langle \psi \right| Y \left| \psi \right\rangle.
+\left\langle \psi \right| \mathcal{H} \left| \psi \right\rangle = a \cdot \left\langle \psi \right| \boldsymbol{I} \left| \psi \right\rangle + b \cdot \left\langle \psi \right| \boldsymbol{Z} \left| \psi \right\rangle + c \cdot \left\langle \psi \right| \boldsymbol{X} \left| \psi \right\rangle + d \cdot \left\langle \psi \right| \boldsymbol{Y} \left| \psi \right\rangle.
 $$
 
 
@@ -1254,7 +1254,7 @@ <h2 id="in-more-detail" class="anchor">In more detail </h2>
 <p>We have</p>
 $$
 \begin{align*}
-&\text{Z eigenvectors} \qquad
+&\text{\boldsymbol{Z}-eigenvectors} \qquad
 \left| 0 \right\rangle = \begin{bmatrix}
 1\\
 0
@@ -1272,7 +1272,7 @@ <h2 id="for-the-other-two-matrices" class="anchor">For the other two matrices </
 
 $$
 \begin{align*}
-&\text{X eigenvectors} \qquad
+&\text{\boldsymbol{X}-eigenvectors} \qquad
 \left| + \right\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix}
 1\\
 1
@@ -1282,7 +1282,7 @@ <h2 id="for-the-other-two-matrices" class="anchor">For the other two matrices </
 -1
 \end{bmatrix},
 \\
-&\text{Y eigenvectors} \qquad
+&\text{\boldsymbol{Y}-eigenvectors} \qquad
 \left| +i \right\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix}
 1\\
 i
@@ -1309,11 +1309,11 @@ <h2 id="explicit-eigenvalues" class="anchor">Explicit eigenvalues </h2>
 
 $$
 \begin{align*}
-\left\langle \psi \right| Z \left| \psi \right\rangle &= \left( {c_1^z}^* \cdot \left\langle 0 \right| + {c_2^z}^* \cdot \left\langle 1 \right| \right) Z \left( c_1^z \cdot \left| 0 \right\rangle + c_2^z \cdot \left| 1 \right\rangle \right) = {\left| c_1^z \right|}^2 - {\left| c_2^z \right|}^2,
+\left\langle \psi \right| \boldsymbol{Z} \left| \psi \right\rangle &= \left( {c_1^z}^* \cdot \left\langle 0 \right| + {c_2^z}^* \cdot \left\langle 1 \right| \right) Z \left( c_1^z \cdot \left| 0 \right\rangle + c_2^z \cdot \left| 1 \right\rangle \right) = {\left| c_1^z \right|}^2 - {\left| c_2^z \right|}^2,
 \\
-\left\langle \psi \right| X \left| \psi \right\rangle &= \left( {c_1^x}^* \cdot \left\langle + \right| + {c_2^x}^* \cdot \left\langle - \right| \right) X \left( c_1^x \cdot \left| + \right\rangle + c_2^x \cdot \left| - \right\rangle \right) = {\left| c_1^x \right|}^2 - {\left| c_2^x \right|}^2,
+\left\langle \psi \right| \boldsymbol{X} \left| \psi \right\rangle &= \left( {c_1^x}^* \cdot \left\langle + \right| + {c_2^x}^* \cdot \left\langle - \right| \right) X \left( c_1^x \cdot \left| + \right\rangle + c_2^x \cdot \left| - \right\rangle \right) = {\left| c_1^x \right|}^2 - {\left| c_2^x \right|}^2,
 \\
-\left\langle \psi \right| Y \left| \psi \right\rangle &= \left( {c_1^y}^* \cdot \left\langle +i \right| + {c_2^y}^* \cdot \left\langle -i \right| \right) Y \left( c_1^y \cdot \left| +i \right\rangle + c_2^y \cdot \left| -i \right\rangle \right) = {\left| c_1^y \right|}^2 - {\left| c_2^y \right|}^2.
+\left\langle \psi \right| \boldsymbol{Y} \left| \psi \right\rangle &= \left( {c_1^y}^* \cdot \left\langle +i \right| + {c_2^y}^* \cdot \left\langle -i \right| \right) Y \left( c_1^y \cdot \left| +i \right\rangle + c_2^y \cdot \left| -i \right\rangle \right) = {\left| c_1^y \right|}^2 - {\left| c_2^y \right|}^2.
 \end{align*}
 $$
 
@@ -1334,18 +1334,18 @@ <h2 id="unitary-transformation-of-boldsymbol-x" class="anchor">Unitary transform
 
 <p>If we use the Hadamard gate</p>
 $$
-H = \frac{1}{\sqrt{2}}\begin{bmatrix}
+\boldsymbol{H} = \frac{1}{\sqrt{2}}\begin{bmatrix}
 1 & 1\\
 1 & -1
 \end{bmatrix},
 $$
 
 <p>we can rewrite</p>
 $$
-X=HZH.
+\boldsymbol{X}=\boldsymbol{HZH}.
 $$
 
-<p>The Hadamard gate/matrix is a unitary matrix with the property that \( H^2=\boldsymbol{I} \).</p>
+<p>The Hadamard gate/matrix is a unitary matrix with the property that \( \boldsymbol{H}^2=\boldsymbol{I} \).</p>
 
 <!-- !split -->
 <h2 id="generalizing" class="anchor">Generalizing </h2>
@@ -1393,7 +1393,7 @@ <h2 id="multiple-ansatzes" class="anchor">Multiple ansatzes </h2>
 </p>
 
 $$
-\langle \psi \vert (c+\mathcal{E})\boldsymbol{I} + (\Omega+\omega_z)\boldsymbol{\sigma}_z + \omega_x\boldsymbol{\sigma}_x\vert \psi \rangle.
+\langle \psi \vert (c+\mathcal{E})\boldsymbol{I} + (\Omega+\omega_z)\boldsymbol{Z} + \omega_x\boldsymbol{X}\vert \psi \rangle.
 $$
 
 
@@ -1410,13 +1410,13 @@ <h2 id="rotations-again" class="anchor">Rotations again </h2>
 </p>
 
 $$
-R_x(\theta)=\cos{\frac{\theta}{2}}\boldsymbol{I}-\imath \sin{\frac{\theta}{2}}\boldsymbol{\sigma}_x,
+R_x(\theta)=\cos{\frac{\theta}{2}}\boldsymbol{I}-\imath \sin{\frac{\theta}{2}}\boldsymbol{X},
 $$
 
 <p>and</p>
 
 $$
-R_y(\phi)=\cos{\frac{\phi}{2}}\boldsymbol{I}-\imath \sin{\frac{\phi}{2}}\boldsymbol{\sigma}_y.
+R_y(\phi)=\cos{\frac{\phi}{2}}\boldsymbol{I}-\imath \sin{\frac{\phi}{2}}\boldsymbol{Y}.
 $$
 
 

doc/pub/week6/html/week6-reveal.html

@@ -734,21 +734,21 @@ <h2 id="expectation-values">Expectation values </h2>
 </p>
 <p>&nbsp;<br>
 $$
-\boldsymbol{X}=R_{\sigma}\boldsymbol{Z}R_{\sigma} = HZH
+\boldsymbol{X}=\boldsymbol{R}_{\sigma}\boldsymbol{Z}\boldsymbol{R}_{\sigma} = \boldsymbol{HZH}
 $$
 <p>&nbsp;<br>
 
 <p>for a Pauli-\( \boldsymbol{Y} \) matrix</p>
 <p>&nbsp;<br>
 $$
-\boldsymbol{Y}=R_{\sigma}\boldsymbol{Z}R_{\sigma}=\boldsymbol{HS}^{\dagger}\boldsymbol{ZHS},
+\boldsymbol{Y}=\boldsymbol{R}_{\sigma}\boldsymbol{Z}\boldsymbol{R}_{\sigma}=\boldsymbol{HS}^{\dagger}\boldsymbol{ZHS},
 $$
 <p>&nbsp;<br>
 
 <p>and</p>
 <p>&nbsp;<br>
 $$
-\boldsymbol{Z}=R_{\sigma}ZR_{\sigma}=\boldsymbol{I}\boldsymbol{Z}\boldsymbol{I}=\boldsymbol{Z}.
+\boldsymbol{Z}=\boldsymbol{R}_{\sigma}\boldsymbol{Z}\boldsymbol{R}_{\sigma}=\boldsymbol{I}\boldsymbol{Z}\boldsymbol{I}=\boldsymbol{Z}.
 $$
 <p>&nbsp;<br>
 </section>
@@ -822,7 +822,7 @@ <h2 id="reminder-on-rotations">Reminder on rotations </h2>
 <p>Note the following identity of the basis rotator</p>
 <p>&nbsp;<br>
 $$
-R^\dagger_\sigma \boldsymbol{Z} R_\sigma = \sigma,
+\boldsymbol{R}^\dagger_\sigma \boldsymbol{Z} \boldsymbol{R}_\sigma = \boldsymbol{\sigma,}
 $$
 <p>&nbsp;<br>
 
@@ -1051,15 +1051,15 @@ <h2 id="rewriting-with-pauli-matrices">Rewriting with Pauli matrices </h2>
 <p>We rewrite \( H \) (and \( H_0 \) and \( H_I \))  via Pauli matrices</p>
 <p>&nbsp;<br>
 $$
-\mathcal{H}_0 = \mathcal{E} I + \Omega \sigma_z, \quad \mathcal{E} = \frac{E_1
+\mathcal{H}_0 = \mathcal{E} I + \Omega \boldsymbol{Z}, \quad \mathcal{E} = \frac{E_1
   + E_2}{2}, \; \Omega = \frac{E_1-E_2}{2},
 $$
 <p>&nbsp;<br>
 
 <p>and</p>
 <p>&nbsp;<br>
 $$
-\mathcal{H}_I = c \boldsymbol{I} +\omega_z\sigma_z + \omega_x\sigma_x,
+\mathcal{H}_I = c \boldsymbol{I} +\omega_z\boldsymbol{Z} + \omega_x\boldsymbol{X},
 $$
 <p>&nbsp;<br>
 
@@ -1196,7 +1196,7 @@ <h2 id="measurements-and-computational-basis">Measurements and computational bas
 </p>
 <p>&nbsp;<br>
 $$
-\left\langle \psi \right| \mathcal{H} \left| \psi \right\rangle = a \cdot \left\langle \psi \right| I \left| \psi \right\rangle + b \cdot \left\langle \psi \right| Z \left| \psi \right\rangle + c \cdot \left\langle \psi \right| X \left| \psi \right\rangle + d \cdot \left\langle \psi \right| Y \left| \psi \right\rangle.
+\left\langle \psi \right| \mathcal{H} \left| \psi \right\rangle = a \cdot \left\langle \psi \right| \boldsymbol{I} \left| \psi \right\rangle + b \cdot \left\langle \psi \right| \boldsymbol{Z} \left| \psi \right\rangle + c \cdot \left\langle \psi \right| \boldsymbol{X} \left| \psi \right\rangle + d \cdot \left\langle \psi \right| \boldsymbol{Y} \left| \psi \right\rangle.
 $$
 <p>&nbsp;<br>
 </section>
@@ -1236,7 +1236,7 @@ <h2 id="in-more-detail">In more detail </h2>
 <p>&nbsp;<br>
 $$
 \begin{align*}
-&\text{Z eigenvectors} \qquad
+&\text{\boldsymbol{Z}-eigenvectors} \qquad
 \left| 0 \right\rangle = \begin{bmatrix}
 1\\
 0
@@ -1256,7 +1256,7 @@ <h2 id="for-the-other-two-matrices">For the other two matrices </h2>
 <p>&nbsp;<br>
 $$
 \begin{align*}
-&\text{X eigenvectors} \qquad
+&\text{\boldsymbol{X}-eigenvectors} \qquad
 \left| + \right\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix}
 1\\
 1
@@ -1266,7 +1266,7 @@ <h2 id="for-the-other-two-matrices">For the other two matrices </h2>
 -1
 \end{bmatrix},
 \\
-&\text{Y eigenvectors} \qquad
+&\text{\boldsymbol{Y}-eigenvectors} \qquad
 \left| +i \right\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix}
 1\\
 i
@@ -1296,11 +1296,11 @@ <h2 id="explicit-eigenvalues">Explicit eigenvalues </h2>
 <p>&nbsp;<br>
 $$
 \begin{align*}
-\left\langle \psi \right| Z \left| \psi \right\rangle &= \left( {c_1^z}^* \cdot \left\langle 0 \right| + {c_2^z}^* \cdot \left\langle 1 \right| \right) Z \left( c_1^z \cdot \left| 0 \right\rangle + c_2^z \cdot \left| 1 \right\rangle \right) = {\left| c_1^z \right|}^2 - {\left| c_2^z \right|}^2,
+\left\langle \psi \right| \boldsymbol{Z} \left| \psi \right\rangle &= \left( {c_1^z}^* \cdot \left\langle 0 \right| + {c_2^z}^* \cdot \left\langle 1 \right| \right) Z \left( c_1^z \cdot \left| 0 \right\rangle + c_2^z \cdot \left| 1 \right\rangle \right) = {\left| c_1^z \right|}^2 - {\left| c_2^z \right|}^2,
 \\
-\left\langle \psi \right| X \left| \psi \right\rangle &= \left( {c_1^x}^* \cdot \left\langle + \right| + {c_2^x}^* \cdot \left\langle - \right| \right) X \left( c_1^x \cdot \left| + \right\rangle + c_2^x \cdot \left| - \right\rangle \right) = {\left| c_1^x \right|}^2 - {\left| c_2^x \right|}^2,
+\left\langle \psi \right| \boldsymbol{X} \left| \psi \right\rangle &= \left( {c_1^x}^* \cdot \left\langle + \right| + {c_2^x}^* \cdot \left\langle - \right| \right) X \left( c_1^x \cdot \left| + \right\rangle + c_2^x \cdot \left| - \right\rangle \right) = {\left| c_1^x \right|}^2 - {\left| c_2^x \right|}^2,
 \\
-\left\langle \psi \right| Y \left| \psi \right\rangle &= \left( {c_1^y}^* \cdot \left\langle +i \right| + {c_2^y}^* \cdot \left\langle -i \right| \right) Y \left( c_1^y \cdot \left| +i \right\rangle + c_2^y \cdot \left| -i \right\rangle \right) = {\left| c_1^y \right|}^2 - {\left| c_2^y \right|}^2.
+\left\langle \psi \right| \boldsymbol{Y} \left| \psi \right\rangle &= \left( {c_1^y}^* \cdot \left\langle +i \right| + {c_2^y}^* \cdot \left\langle -i \right| \right) Y \left( c_1^y \cdot \left| +i \right\rangle + c_2^y \cdot \left| -i \right\rangle \right) = {\left| c_1^y \right|}^2 - {\left| c_2^y \right|}^2.
 \end{align*}
 $$
 <p>&nbsp;<br>
@@ -1324,7 +1324,7 @@ <h2 id="unitary-transformation-of-boldsymbol-x">Unitary transformation of \( \bo
 <p>If we use the Hadamard gate</p>
 <p>&nbsp;<br>
 $$
-H = \frac{1}{\sqrt{2}}\begin{bmatrix}
+\boldsymbol{H} = \frac{1}{\sqrt{2}}\begin{bmatrix}
 1 & 1\\
 1 & -1
 \end{bmatrix},
@@ -1334,11 +1334,11 @@ <h2 id="unitary-transformation-of-boldsymbol-x">Unitary transformation of \( \bo
 <p>we can rewrite</p>
 <p>&nbsp;<br>
 $$
-X=HZH.
+\boldsymbol{X}=\boldsymbol{HZH}.
 $$
 <p>&nbsp;<br>
 
-<p>The Hadamard gate/matrix is a unitary matrix with the property that \( H^2=\boldsymbol{I} \).</p>
+<p>The Hadamard gate/matrix is a unitary matrix with the property that \( \boldsymbol{H}^2=\boldsymbol{I} \).</p>
 </section>
 
 <section>
@@ -1393,7 +1393,7 @@ <h2 id="multiple-ansatzes">Multiple ansatzes </h2>
 
 <p>&nbsp;<br>
 $$
-\langle \psi \vert (c+\mathcal{E})\boldsymbol{I} + (\Omega+\omega_z)\boldsymbol{\sigma}_z + \omega_x\boldsymbol{\sigma}_x\vert \psi \rangle.
+\langle \psi \vert (c+\mathcal{E})\boldsymbol{I} + (\Omega+\omega_z)\boldsymbol{Z} + \omega_x\boldsymbol{X}\vert \psi \rangle.
 $$
 <p>&nbsp;<br>
 </section>
@@ -1412,15 +1412,15 @@ <h2 id="rotations-again">Rotations again </h2>
 
 <p>&nbsp;<br>
 $$
-R_x(\theta)=\cos{\frac{\theta}{2}}\boldsymbol{I}-\imath \sin{\frac{\theta}{2}}\boldsymbol{\sigma}_x,
+R_x(\theta)=\cos{\frac{\theta}{2}}\boldsymbol{I}-\imath \sin{\frac{\theta}{2}}\boldsymbol{X},
 $$
 <p>&nbsp;<br>
 
 <p>and</p>
 
 <p>&nbsp;<br>
 $$
-R_y(\phi)=\cos{\frac{\phi}{2}}\boldsymbol{I}-\imath \sin{\frac{\phi}{2}}\boldsymbol{\sigma}_y.
+R_y(\phi)=\cos{\frac{\phi}{2}}\boldsymbol{I}-\imath \sin{\frac{\phi}{2}}\boldsymbol{Y}.
 $$
 <p>&nbsp;<br>
 </section>