update week 6

Author: mhjensen

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

@@ -42,10 +42,21 @@
                None,
                'plans-for-the-week-of-february-24-28-solving-quantum-mechanical-problems'),
               ('Readings', 2, None, 'readings'),
-              ('Gates and measurements, reminder from previous two weeks',
+              ('States, gates and measurements, reminder from preview lectures',
                2,
                None,
-               'gates-and-measurements-reminder-from-previous-two-weeks'),
+               'states-gates-and-measurements-reminder-from-preview-lectures'),
+              ('Single qubit gates', 2, None, 'single-qubit-gates'),
+              ('Pauli-$X$ gate', 2, None, 'pauli-x-gate'),
+              ('Hadamard gate', 2, None, 'hadamard-gate'),
+              ('Phase Gates', 2, None, 'phase-gates'),
+              ('The  inverse of the $S$-gate',
+               2,
+               None,
+               'the-inverse-of-the-s-gate'),
+              ('Two-qubit gates', 2, None, 'two-qubit-gates'),
+              ('The SWAP gate', 2, None, 'the-swap-gate'),
+              ('Pauli Strings', 2, None, 'pauli-strings'),
               ('Variational Quantum Eigensolver',
                2,
                None,
@@ -160,7 +171,15 @@
         <ul class="dropdown-menu">
      <!-- navigation toc: --> <li><a href="#plans-for-the-week-of-february-24-28-solving-quantum-mechanical-problems" style="font-size: 80%;">Plans for the week of February 24-28, Solving quantum mechanical problems</a></li>
      <!-- navigation toc: --> <li><a href="#readings" style="font-size: 80%;">Readings</a></li>
-     <!-- navigation toc: --> <li><a href="#gates-and-measurements-reminder-from-previous-two-weeks" style="font-size: 80%;">Gates and measurements, reminder from previous two weeks</a></li>
+     <!-- navigation toc: --> <li><a href="#states-gates-and-measurements-reminder-from-preview-lectures" style="font-size: 80%;">States, gates and measurements, reminder from preview lectures</a></li>
+     <!-- navigation toc: --> <li><a href="#single-qubit-gates" style="font-size: 80%;">Single qubit gates</a></li>
+     <!-- navigation toc: --> <li><a href="#pauli-x-gate" style="font-size: 80%;">Pauli-\( X \) gate</a></li>
+     <!-- navigation toc: --> <li><a href="#hadamard-gate" style="font-size: 80%;">Hadamard gate</a></li>
+     <!-- navigation toc: --> <li><a href="#phase-gates" style="font-size: 80%;">Phase Gates</a></li>
+     <!-- navigation toc: --> <li><a href="#the-inverse-of-the-s-gate" style="font-size: 80%;">The  inverse of the \( S \)-gate</a></li>
+     <!-- navigation toc: --> <li><a href="#two-qubit-gates" style="font-size: 80%;">Two-qubit gates</a></li>
+     <!-- navigation toc: --> <li><a href="#the-swap-gate" style="font-size: 80%;">The SWAP gate</a></li>
+     <!-- navigation toc: --> <li><a href="#pauli-strings" style="font-size: 80%;">Pauli Strings</a></li>
      <!-- navigation toc: --> <li><a href="#variational-quantum-eigensolver" style="font-size: 80%;">Variational Quantum Eigensolver</a></li>
      <!-- navigation toc: --> <li><a href="#the-vqe" style="font-size: 80%;">The VQE</a></li>
      <!-- navigation toc: --> <li><a href="#expectation-value-of-hamiltonian" style="font-size: 80%;">Expectation value of Hamiltonian</a></li>
@@ -232,8 +251,7 @@ <h2 id="plans-for-the-week-of-february-24-28-solving-quantum-mechanical-problems
 <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
 <ol>
  <li> Repetition from last week on gates, measurements and one-qubit systems</li>
- <li> Introducing the Variational Quantum Eigensolver (VQE) and discussion of project 1</li>
- <li> <b>Note on project work:</b> feel free to collaborate on the projects
+ <li> Introducing the Variational Quantum Eigensolver (VQE) and discussion of project 1
 <!-- o <a href="https://youtu.be/" target="_self">Video of lecture to be added</a> -->
 <!-- o <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/HandWrittenNotes/2024/NotesFebruary21.pdf" target="_self">Whiteboard notes</a> --></li>
 </ol>
@@ -250,9 +268,200 @@ <h2 id="readings" class="anchor">Readings </h2>
 <li> <a href="https://www.sciencedirect.com/science/article/pii/S0370157322003118?via%3Dihub" target="_self">See also the VQE review article by Tilly et al.</a></li>
 </ol>
 <!-- !split -->
-<h2 id="gates-and-measurements-reminder-from-previous-two-weeks" class="anchor">Gates and measurements, reminder from previous two weeks </h2>
+<h2 id="states-gates-and-measurements-reminder-from-preview-lectures" class="anchor">States, gates and measurements, reminder from preview lectures </h2>
+
+<p>Mathematically, quantum gates are a series of unitary operators in the
+operator space $ \mathcal{H} \otimes \mathcal{H}^{*}$ which evolve the
+state. The unitary nature preserves the norm of the state vector,
+ensuring the probabilities sum to unity. Since not all gates
+correspond to an observable, they are not necessarily hermitian.
+</p>
+
+<!-- !split -->
+<h2 id="single-qubit-gates" class="anchor">Single qubit gates </h2>
+
+<p>The Pauli matrices (and gate operations following therefrom) are defined as</p>
+$$
+	X \equiv \sigma_x = \begin{pmatrix}
+		0 & 1 \\
+		1 & 0
+	\end{pmatrix}, \quad
+	Y \equiv \sigma_y = \begin{pmatrix}
+		0 & -i \\
+		i & 0
+	\end{pmatrix}, \quad
+	Z \equiv \sigma_z = \begin{pmatrix}
+		1 & 0 \\
+		0 & -1
+	\end{pmatrix}.
+$$
+
+
+<!-- !split -->
+<h2 id="pauli-x-gate" class="anchor">Pauli-\( X \) gate </h2>
+
+<p>The Pauli-\( X \) gate is also known as the <b>NOT</b> gate, which flips the state of the qubit.</p>
+$$
+\begin{align}
+	X\vert 0\rangle &= \vert 1\rangle, 
+\label{_auto1}\\
+	X\vert 1\rangle &= \vert 0\rangle.	
+\label{_auto2}
+\end{align}
+$$
+
+<p>The Pauli-\( Y \) gate flips the bit and multiplies the phase by $ i $. </p>
+$$
+\begin{align}
+	Y\vert 0\rangle &= i\vert 1\rangle, 
+\label{_auto3}\\
+	Y\vert 1\rangle &= -i\vert 0\rangle.
+\label{_auto4}
+\end{align}
+$$
+
+<p>The Pauli-\( Z \) gate multiplies only the phase of \( \vert 1\rangle \) by $ -1 \( .</p>
+$$
+\begin{align}
+	Z\vert 0\rangle &= \vert 0\rangle, 
+\label{_auto5}\\
+	Z\vert 1\rangle &= -\vert 1\rangle.
+\label{_auto6}
+\end{align}
+$$
+
+<!-- !split -->
+<h2 id="hadamard-gate" class="anchor">Hadamard gate </h2>
+
+<p>The Hadamard gate is defined as</p>
+$$
+	H = \frac{1}{\sqrt{2}} \begin{pmatrix}
+		1 & 1 \\
+		1 & -1
+	\end{pmatrix}.
+$$
+
+<p>It creates a superposition of the  \) \vert 0\rangle $ and $ \vert 1\rangle $ states.</p>
+$$
+\begin{align}
+	H\vert 0\rangle &= \frac{1}{\sqrt{2}} \left( \vert 0\rangle + \vert 1\rangle \right), 
+\label{_auto7}\\
+	H\vert 1\rangle &= \frac{1}{\sqrt{2}} \left( \vert 0\rangle - \vert 1\rangle \right).
+\label{_auto8}
+\end{align}
+$$
+
+
+<!-- !split -->
+<h2 id="phase-gates" class="anchor">Phase Gates </h2>
+<p>The phase gate is usually denoted as \( S \) and is defined as</p>
+$$
+\begin{equation}
+	S = \begin{pmatrix}
+		1 & 0 \\
+		0 & i
+	\end{pmatrix}.
+\label{_auto9}
+\end{equation}
+$$
+
+<p>It multiplies only the phase of the $ \vert 1\rangle $ state by $ i $.</p>
+$$
+\begin{align}
+	S\vert 0\rangle &= \vert 0\rangle, 
+\label{_auto10}\\
+	S\vert 1\rangle &= i\vert 1\rangle.
+\label{_auto11}
+\end{align}
+$$
+
+
+<!-- !split -->
+<h2 id="the-inverse-of-the-s-gate" class="anchor">The  inverse of the \( S \)-gate </h2>
+
+<p>The inverse</p>
+$$
+\begin{equation}
+	S^\dagger = \begin{pmatrix}
+		1 & 0 \\
+		0 & -i
+	\end{pmatrix}
+\label{_auto12}
+\end{equation}
+$$
 
-<p>Material to come here</p>
+<p>is known as the $ S^\dagger$ gate which applies an \( \imath \) phase shift to \( \vert 1\rangle \).</p>
+$$
+\begin{align}
+	S^\dagger\vert 0\rangle &= \vert 0\rangle, 
+\label{_auto13}\\
+	S^\dagger\vert 1\rangle &= -i\vert 1\rangle.
+\label{_auto14}
+\end{align}
+$$
+
+<!-- !split -->
+<h2 id="two-qubit-gates" class="anchor">Two-qubit gates </h2>
+
+<p>The CNOT gate is a two-qubit gate which acts on two qubits, a control qubit and a target qubit. The CNOT gate is defined as</p>
+$$
+\begin{equation}
+	\text{CNOT} = \begin{pmatrix}
+		1 & 0 & 0 & 0 \\
+		0 & 1 & 0 & 0 \\
+		0 & 0 & 0 & 1 \\
+		0 & 0 & 1 & 0
+	\end{pmatrix}.
+\label{_auto15}
+\end{equation}
+$$
+
+<p>It is often used to perform linear entanglement on qubits.</p>
+$$
+\begin{align*}
+	\text{CNOT} \vert 00\rangle &= \vert 00\rangle, \\
+	\text{CNOT} \vert 01\rangle &= \vert 01\rangle, \\
+	\text{CNOT} \vert 10\rangle &= \vert 11\rangle, \\
+	\text{CNOT} \vert 11\rangle &= \vert 10\rangle.
+\end{align*}
+$$
+
+
+<!-- !split -->
+<h2 id="the-swap-gate" class="anchor">The SWAP gate </h2>
+<p>The SWAP gate is a two-qubit gate which swaps the state of two qubits. It is defined as</p>
+$$
+\begin{equation}
+	\text{SWAP} = \begin{pmatrix}
+		1 & 0 & 0 & 0 \\
+		0 & 0 & 1 & 0 \\
+		0 & 1 & 0 & 0 \\
+		0 & 0 & 0 & 1
+	\end{pmatrix}.
+\label{_auto16}
+\end{equation}
+$$
+
+$$
+\begin{align*}
+	\text{SWAP}\vert 00\rangle &= \vert 00\rangle, \\
+	\text{SWAP} \vert 01\rangle &= \vert 10\rangle, \\
+	\text{SWAP} \vert 10\rangle &= \vert 01\rangle, \\
+	\text{SWAP} \vert 11\rangle &= \vert 11 \rangle.
+\end{align*}
+$$
+
+
+<!-- !split -->
+<h2 id="pauli-strings" class="anchor">Pauli Strings </h2>
+<p>A Pauli string, such as $ XIYZ $ is a tensor product of Pauli matrices acting on different qubits.
+The Pauli string $ XIYZ $ is defined as (from qubit one to qubit four, from left to right)
+</p>
+$$
+	XIYZ \equiv X_0 \otimes I_1 \otimes Y_2 \otimes Z_3.
+$$
+
+<p>Hamiltonians are often rewritten or decomposed in terms of Pauli string as they can be easily implemented on quantum computers. </p>
 
 <!-- !split -->
 <h2 id="variational-quantum-eigensolver" class="anchor">Variational Quantum Eigensolver </h2>
@@ -366,15 +575,15 @@ <h2 id="expectation-values" class="anchor">Expectation values </h2>
 $$
 \begin{align}
 HS^{\dagger}, & \text{if} \ \sigma = Y,
-\label{_auto1}
+\label{_auto17}
 \end{align}
 $$
 
 <p>and</p>
 $$
 \begin{align}
 I, & \text{if} \ \sigma = Z.
-\label{_auto2}
+\label{_auto18}
 \end{align}
 $$
 
@@ -452,7 +661,7 @@ <h2 id="arbitrary-pauli-gate" class="anchor">Arbitrary Pauli gate </h2>
 &=\langle\phi\vert\left(\sum_{x\in\{0,1\}}(-1)^x\vert x\rangle\langle x\vert\right)\vert\phi\rangle \nonumber \\
 &=\sum_{x\in\{0,1\}}(-1)^x\vert\langle x\vert \phi\rangle\vert^2\nonumber \\
 &=\sum_{x\in\{0,1\}}(-1)^xP(\vert \phi\rangle\to\vert x\rangle),
-\label{_auto3}
+\label{_auto19}
 \end{align}
 $$
 
@@ -518,7 +727,7 @@ <h2 id="which-gives-us" class="anchor">Which gives us </h2>
 \\
 &=
 \sum_{x\in\{0,1\}^n}(-1)^{\sum_{p\in Q}x_p}P(\vert \phi\rangle\to\vert x\rangle),
-\label{_auto4}
+\label{_auto20}
 \end{align}
 $$
 
@@ -608,15 +817,15 @@ <h2 id="rewriting-the-hamiltonian" class="anchor">Rewriting the Hamiltonian </h2
 $$
 \begin{equation}
        H_0\vert 0 \rangle =E_1\vert 0 \rangle,
-\label{_auto5}
+\label{_auto21}
 \end{equation}
 $$
 
 <p>and</p>
 $$
 \begin{equation}
        H_0\vert 1\rangle =E_2\vert 1\rangle,
-\label{_auto6}
+\label{_auto22}
 \end{equation}
 $$