update

Author: mhjensen

doc/pub/week12/html/week12-bs.html

@@ -9,8 +9,8 @@
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Quantum Computing, Quantum Machine Learning and Quantum Information Theories">
-<title>Quantum Computing, Quantum Machine Learning and Quantum Information Theories</title>
+<meta name="description" content="Quantum Computing and Quantum Machine Learning">
+<title>Quantum Computing and Quantum Machine Learning</title>
 
 <!-- reveal.js: https://lab.hakim.se/reveal-js/ -->
 
@@ -168,7 +168,7 @@
 <section>
 <!-- ------------------- main content ---------------------- -->
 <center>
-<h1 style="text-align: center;">Quantum Computing, Quantum Machine Learning and Quantum Information Theories</h1>
+<h1 style="text-align: center;">Quantum Computing and Quantum Machine Learning</h1>
 </center>  <!-- document title -->
 
 <!-- author(s): Morten Hjorth-Jensen -->
@@ -225,14 +225,14 @@ <h2 id="classical-discrete-fourier-transform">Classical Discrete Fourier Transfo
 <p>Given a vector of \( N \) complex numbers \( \{x_0, x_1, \dots, x_{N-1}\} \), the DFT is defined as:</p>
 <p>&nbsp;<br>
 $$
-X_k = \sum_{n=0}^{N-1} x_n \exp{-2\pi i kn/N}, \quad k = 0, 1, \dots, N-1
+X_k = \sum_{n=0}^{N-1} x_n \exp{-(2\pi i kn/N)}, \quad k = 0, 1, \dots, N-1
 $$
 <p>&nbsp;<br>
 
 <p>The inverse transform is:</p>
 <p>&nbsp;<br>
 $$
-x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \exp{2\pi i kn/N}
+x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \exp{(2\pi i kn/N)}
 $$
 <p>&nbsp;<br>
 </section>
@@ -243,7 +243,7 @@ <h2 id="in-matrix-form">In matrix form </h2>
 <p>This can be expressed in matrix form:</p>
 <p>&nbsp;<br>
 $$
-\vec{X} = F_N \vec{x}, \quad \text{where} \quad (F_N)_{jk} = \frac{1}{\sqrt{N}} \exp{2\pi i jk/N}
+\boldsymbol{X} = F_N \boldsymbol{x}, \quad \text{where} \quad (F_N)_{jk} = \frac{1}{\sqrt{N}} \exp{(2\pi i jk/N)}
 $$
 <p>&nbsp;<br>
 </section>
@@ -254,14 +254,14 @@ <h2 id="quantum-fourier-transform">Quantum Fourier Transform </h2>
 <p>Let \( N = 2^n \). The QFT acts on the state:</p>
 <p>&nbsp;<br>
 $$
-\vert x\rangle = \vert x_1 x_2 \dots x_n\rangle
+\vert x\rangle = \vert x_0 x_1 \dots x_{n-1}\rangle
 $$
 <p>&nbsp;<br>
 
 <p>and transforms it into:</p>
 <p>&nbsp;<br>
 $$
-\mathrm{QFT} \vert x\rangle  = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} \exp{2\pi i xk/N} \vert k\rangle
+\mathrm{QFT} \vert x\rangle  = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} \exp{(2\pi i xk/N)} \vert k\rangle
 $$
 <p>&nbsp;<br>
 
@@ -349,7 +349,7 @@ <h2 id="quantum-circuit-implementation">Quantum Circuit Implementation </h2>
 
 <p>&nbsp;<br>
 $$
-\frac{1}{\sqrt{2^n}} \sum_{k=0}^{2^n - 1} \exp{2\pi i xk / 2^n} \vert k\rangle
+\frac{1}{\sqrt{2^n}} \sum_{k=0}^{2^n - 1} \exp{(2\pi i xk / 2^n)} \vert k\rangle
 $$
 <p>&nbsp;<br>
 
@@ -509,7 +509,7 @@ <h2 id="introduction-to-quantum-phase-estimation-qpe">Introduction to Quantum Ph
 
 <p>&nbsp;<br>
 $$
-U | \psi \rangle = \exp{2\pi i \phi} | \psi \rangle
+U | \psi \rangle = \exp{(2\pi i \phi)} | \psi \rangle
 $$
 <p>&nbsp;<br>
 
@@ -560,7 +560,7 @@ <h2 id="mathematical-background">Mathematical Background </h2>
 <p>The goal of QPE is to estimate the phase \( \phi \in [0,1) \) where:</p>
 <p>&nbsp;<br>
 $$
-    U\vert \psi\rangle = \exp{2\pi i \phi} \vert\psi\rangle.
+    U\vert \psi\rangle = \exp{(2\pi i \phi)} \vert\psi\rangle.
 $$
 <p>&nbsp;<br>
 
@@ -901,161 +901,6 @@ <h2 id="code-which-implements-the-qpe-using-pennylane">Code which implements the
 </div>
 </section>
 
-<section>
-<h2 id="pennylane-implementation-of-qpe">PennyLane Implementation of QPE </h2>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #eeeedd">
-  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">pennylane</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">qml</span>
-<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">pennylane</span> <span style="color: #8B008B; font-weight: bold">import</span> numpy <span style="color: #8B008B; font-weight: bold">as</span> np
-
-n_estimation = <span style="color: #B452CD">3</span>
-dev = qml.device(<span style="color: #CD5555">&quot;default.qubit&quot;</span>, wires=n_estimation + <span style="color: #B452CD">1</span>)
-
-<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">apply_controlled_unitaries</span>(U, wires):
-   <span style="color: #8B008B; font-weight: bold">for</span> i, ctrl <span style="color: #8B008B">in</span> <span style="color: #658b00">enumerate</span>(wires[:-<span style="color: #B452CD">1</span>]):
-       power = <span style="color: #B452CD">2</span>**i
-       <span style="color: #8B008B; font-weight: bold">for</span> _ <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(power):
-           qml.ctrl(U, control=ctrl)(wires[-<span style="color: #B452CD">1</span>])
-
-<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">inverse_qft</span>(wires):
-   swap_registers(wires)
-   <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">reversed</span>(<span style="color: #658b00">range</span>(<span style="color: #658b00">len</span>(wires))):
-       <span style="color: #8B008B; font-weight: bold">for</span> j <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(i):
-           qml.ctrl(qml.PhaseShift, control=wires[j])(-np.pi / <span style="color: #B452CD">2</span>**(i - j), wires=wires[i])
-       qml.Hadamard(wires=wires[i])
-
-<span style="color: #707a7c">@qml</span>.qnode(dev)
-<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">qpe</span>():
-   target_wire = n_estimation
-   <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_estimation):
-       qml.Hadamard(wires=i)
-
-   qml.RZ(<span style="color: #B452CD">2</span> * np.pi * <span style="color: #B452CD">0.375</span>, wires=target_wire)  <span style="color: #228B22"># Assume phase = 0.375</span>
-
-   apply_controlled_unitaries(qml.RZ, <span style="color: #658b00">list</span>(<span style="color: #658b00">range</span>(n_estimation)) + [target_wire])
-   inverse_qft(<span style="color: #658b00">list</span>(<span style="color: #658b00">range</span>(n_estimation)))
-   <span style="color: #8B008B; font-weight: bold">return</span> qml.probs(wires=<span style="color: #658b00">range</span>(n_estimation))
-
-probs = qpe()
-<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Measured phase state:&quot;</span>, probs)
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
-</section>
-
-<section>
-<h2 id="accuracy-and-readout">Accuracy and Readout </h2>
-
-<p>The accuracy of phase estimation improves with the number of qubits in
-the control register. For \( n \) qubits, we obtain the \( n \) most
-significant bits of \( \phi \).
-</p>
-</section>
-
-<section>
-<h2 id="hamiltonian-simulation">Hamiltonian simulation </h2>
-
-<p>Let \( H \) be a Hamiltonian. The evolution \( U = \exp{-iHt} \) has
-eigenvectors \( \vert u\rangle \) and eigenvalues \( \exp{-iEt} \). QPE yields
-estimates of \( E \).
-</p>
-
-<p>Typical applications are</p>
-<ol>
-<p><li> Electronic structure problems</li>
-<p><li> Dynamics of molecules</li>
-<p><li> Energy spectrum estimation</li>
-</ol>
-</section>
-
-<section>
-<h2 id="amplitude-estimation">Amplitude estimation </h2>
-
-<p>Quantum Amplitude Estimation (QAE) uses QFT for estimating
-probabilities (amplitudes) more efficiently than classical sampling.
-</p>
-
-<p>Prepare a state:</p>
-<p>&nbsp;<br>
-$$
-\vert \psi\rangle = \sqrt{a}\vert1\rangle + \sqrt{1 - a}\vert 0\rangle.
-$$
-<p>&nbsp;<br>
-
-<p>The QAE estimates \( a \) using Grover-like iterations and QFT.</p>
-</section>
-
-<section>
-<h2 id="pennylane-example-energy-estimation">PennyLane Example: Energy Estimation </h2>
-
-<p>We simulate a simple diagonal Hamiltonian using QPE.</p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #eeeedd">
-  <pre style="font-size: 80%; line-height: 125%;">  <span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">pennylane</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">qml</span>
-  <span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">pennylane</span> <span style="color: #8B008B; font-weight: bold">import</span> numpy <span style="color: #8B008B; font-weight: bold">as</span> np
-
-  n_qubits = <span style="color: #B452CD">2</span>
-  dev = qml.device(<span style="color: #a61717; background-color: #e3d2d2">``</span>default.qubit<span style="color: #CD5555">&#39;&#39;</span>, wires=n_qubits + <span style="color: #B452CD">1</span>)
-
-  <span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">controlled_unitary</span>(t):
-  <span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">unitary</span>():
-  qml.PhaseShift(<span style="color: #B452CD">2</span> * np.pi * <span style="color: #B452CD">0.3</span> * t, wires=n_qubits)
-  <span style="color: #8B008B; font-weight: bold">return</span> unitary
-
-  <span style="color: #707a7c">@qml</span>.qnode(dev)
-  <span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">energy_estimation</span>():
-  <span style="color: #228B22"># Apply Hadamards to estimation qubits</span>
-  <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_qubits):
-  qml.Hadamard(wires=i)
-  <span style="color: #228B22"># Prepare eigenstate for target qubit</span>
-  qml.PauliX(wires=n_qubits)
-  <span style="color: #228B22"># Apply controlled-U^{2^j}</span>
-  <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_qubits):
-  unitary = controlled_unitary(<span style="color: #B452CD">2</span>**i)
-  qml.ctrl(unitary, control=i)()
-  <span style="color: #228B22"># Inverse QFT</span>
-  inverse_qft(<span style="color: #658b00">range</span>(n_qubits))
-  <span style="color: #8B008B; font-weight: bold">return</span> qml.probs(wires=<span style="color: #658b00">range</span>(n_qubits))
-
-  probs = energy_estimation()
-  <span style="color: #658b00">print</span>(<span style="color: #a61717; background-color: #e3d2d2">``</span>Measured energy phase:<span style="color: #CD5555">&#39;&#39;</span>, probs)
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
-</section>
-
 <section>
 <h2 id="applications-of-qpe">Applications of QPE </h2>