Skip to content

Commit 3bb7ba6

Browse files
mhjensenmhjensen
committed
update week 13
1 parent 4b4315a commit 3bb7ba6

File tree

9 files changed

+1524
-47
lines changed

9 files changed

+1524
-47
lines changed

‎doc/pub/week13/html/week13-bs.html‎

Lines changed: 308 additions & 8 deletions
Original file line numberDiff line numberDiff line change
@@ -8,8 +8,8 @@
88
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
99
<meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
1010
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
11-
<meta name="description" content="Quantum Computing, Quantum Machine Learning and Quantum Information Theories">
12-
<title>Quantum Computing, Quantum Machine Learning and Quantum Information Theories</title>
11+
<meta name="description" content="Quantum Computing and Quantum Machine Learning">
12+
<title>Quantum Computing and Quantum Machine Learning</title>
1313
<!-- Bootstrap style: bootstrap -->
1414
<!-- doconce format html week13.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week13-bs --no_mako -->
1515
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -39,10 +39,61 @@
3939
'sections': [('Plans for the week of April 21-25',
4040
2,
4141
None,
42-
'plans-for-the-week-of-april-21-25')]}
42+
'plans-for-the-week-of-april-21-25'),
43+
('What is Machine Learning?',
44+
2,
45+
None,
46+
'what-is-machine-learning'),
47+
('What is Quantum Machine Learning?',
48+
2,
49+
None,
50+
'what-is-quantum-machine-learning'),
51+
('Quantum Speedups in ML', 2, None, 'quantum-speedups-in-ml'),
52+
('Challenges in Quantum Machine Learning',
53+
2,
54+
None,
55+
'challenges-in-quantum-machine-learning'),
56+
('Classical Support Vector Machines (SVMs)',
57+
2,
58+
None,
59+
'classical-support-vector-machines-svms'),
60+
('First mathematical formulation of SVMs (formal details below)',
61+
2,
62+
None,
63+
'first-mathematical-formulation-of-svms-formal-details-below'),
64+
('Kernels and more', 2, None, 'kernels-and-more'),
65+
('Quantum Support Vector Machines (QSVMs)',
66+
2,
67+
None,
68+
'quantum-support-vector-machines-qsvms'),
69+
('Quantum Neural Networks (QNNs)',
70+
2,
71+
None,
72+
'quantum-neural-networks-qnns'),
73+
('Quantum Boltzmann Machines (QBMs)',
74+
2,
75+
None,
76+
'quantum-boltzmann-machines-qbms'),
77+
('Back to math of SVMs', 2, None, 'back-to-math-of-svms')]}
4378
end of tocinfo -->
4479

4580
<body>
81+
82+
83+
84+
<script type="text/x-mathjax-config">
85+
MathJax.Hub.Config({
86+
TeX: {
87+
equationNumbers: { autoNumber: "AMS" },
88+
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
89+
}
90+
});
91+
</script>
92+
<script type="text/javascript" async
93+
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
94+
</script>
95+
96+
4697
<!-- Bootstrap navigation bar -->
4798
<div class="navbar navbar-default navbar-fixed-top">
4899
<div class="navbar-header">
@@ -51,14 +102,25 @@
51102
<span class="icon-bar"></span>
52103
<span class="icon-bar"></span>
53104
</button>
54-
<a class="navbar-brand" href="week13-bs.html">Quantum Computing, Quantum Machine Learning and Quantum Information Theories</a>
105+
<a class="navbar-brand" href="week13-bs.html">Quantum Computing and Quantum Machine Learning</a>
55106
</div>
56107
<div class="navbar-collapse collapse navbar-responsive-collapse">
57108
<ul class="nav navbar-nav navbar-right">
58109
<li class="dropdown">
59110
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
60111
<ul class="dropdown-menu">
61112
<!-- navigation toc: --> <li><a href="#plans-for-the-week-of-april-21-25" style="font-size: 80%;">Plans for the week of April 21-25</a></li>
113+
<!-- navigation toc: --> <li><a href="#what-is-machine-learning" style="font-size: 80%;">What is Machine Learning?</a></li>
114+
<!-- navigation toc: --> <li><a href="#what-is-quantum-machine-learning" style="font-size: 80%;">What is Quantum Machine Learning?</a></li>
115+
<!-- navigation toc: --> <li><a href="#quantum-speedups-in-ml" style="font-size: 80%;">Quantum Speedups in ML</a></li>
116+
<!-- navigation toc: --> <li><a href="#challenges-in-quantum-machine-learning" style="font-size: 80%;">Challenges in Quantum Machine Learning</a></li>
117+
<!-- navigation toc: --> <li><a href="#classical-support-vector-machines-svms" style="font-size: 80%;">Classical Support Vector Machines (SVMs)</a></li>
118+
<!-- navigation toc: --> <li><a href="#first-mathematical-formulation-of-svms-formal-details-below" style="font-size: 80%;">First mathematical formulation of SVMs (formal details below)</a></li>
119+
<!-- navigation toc: --> <li><a href="#kernels-and-more" style="font-size: 80%;">Kernels and more</a></li>
120+
<!-- navigation toc: --> <li><a href="#quantum-support-vector-machines-qsvms" style="font-size: 80%;">Quantum Support Vector Machines (QSVMs)</a></li>
121+
<!-- navigation toc: --> <li><a href="#quantum-neural-networks-qnns" style="font-size: 80%;">Quantum Neural Networks (QNNs)</a></li>
122+
<!-- navigation toc: --> <li><a href="#quantum-boltzmann-machines-qbms" style="font-size: 80%;">Quantum Boltzmann Machines (QBMs)</a></li>
123+
<!-- navigation toc: --> <li><a href="#back-to-math-of-svms" style="font-size: 80%;">Back to math of SVMs</a></li>
62124

63125
</ul>
64126
</li>
@@ -71,7 +133,7 @@
71133
<!-- ------------------- main content ---------------------- -->
72134
<div class="jumbotron">
73135
<center>
74-
<h1>Quantum Computing, Quantum Machine Learning and Quantum Information Theories</h1>
136+
<h1>Quantum Computing and Quantum Machine Learning</h1>
75137
</center> <!-- document title -->
76138

77139
<!-- author(s): Morten Hjorth-Jensen -->
@@ -98,13 +160,251 @@ <h2 id="plans-for-the-week-of-april-21-25" class="anchor">Plans for the week of
98160
<div class="panel-body">
99161
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
100162
<ol>
101-
<li> TBA</li>
102-
<li> <a href="https://youtu.be/" target="_self">Video of lecture TBA</a>
163+
<li> Quantum Machine Learning (QML)
164+
<ol type="a"></li>
165+
<li> Introduction to QML</li>
166+
<li> Support Vector Machines (SVM, classical machine learning approach)</li>
167+
<li> Quantum Support Vector Machine Learning (QSVM)
168+
<!-- o <a href="https://youtu.be/" target="_self">Video of lecture TBA</a> -->
103169
<!-- o <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/HandWrittenNotes/2024/NotesApril17.pdf" target="_self">Whiteboard notes</a> --></li>
104170
</ol>
171+
</ol>
172+
</div>
173+
</div>
174+
175+
176+
<!-- !split -->
177+
<h2 id="what-is-machine-learning" class="anchor">What is Machine Learning? </h2>
178+
179+
<p>Machine Learning (ML) is the study of algorithms that improve through data experience.</p>
180+
181+
<div class="panel panel-default">
182+
<div class="panel-body">
183+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
184+
<ol>
185+
<li> \textbf{Supervised Learning:} Labeled data for classification or regression.</li>
186+
<li> \textbf{Unsupervised Learning:} No labels; discover hidden patterns.</li>
187+
</ol>
188+
<p>o\textbf{Reinforcement Learning:} Learning through interaction with the environment.</p>
189+
</div>
190+
</div>
191+
192+
193+
<div class="panel panel-default">
194+
<div class="panel-body">
195+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
196+
<p>\textbf{ML Workflow:}</p>
197+
$$
198+
\text{Data} \rightarrow \text{Model Training} \rightarrow \text{Prediction}
199+
$$
200+
</div>
201+
</div>
202+
203+
204+
<!-- !split -->
205+
<h2 id="what-is-quantum-machine-learning" class="anchor">What is Quantum Machine Learning? </h2>
206+
207+
<p>\textbf{Quantum Machine Learning (QML)} integrates quantum computing with machine learning algorithms to exploit quantum advantages.</p>
208+
209+
<div class="panel panel-default">
210+
<div class="panel-body">
211+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
212+
<ol>
213+
<li> High-dimensional Hilbert spaces for better feature representation.</li>
214+
<li> Quantum parallelism for faster computation.</li>
215+
<li> Quantum entanglement for richer data encoding.</li>
216+
</ol>
217+
</div>
218+
</div>
219+
220+
221+
<!-- !split -->
222+
<h2 id="quantum-speedups-in-ml" class="anchor">Quantum Speedups in ML </h2>
223+
<p>\textbf{Why Quantum?}</p>
224+
<div class="panel panel-default">
225+
<div class="panel-body">
226+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
227+
<ol>
228+
<li> \textbf{Quantum Parallelism:} Process multiple states simultaneously.</li>
229+
<li> \textbf{Quantum Entanglement:} Correlated states for richer information.</li>
230+
<li> \textbf{Quantum Interference:} Constructive and destructive interference to enhance solutions.</li>
231+
</ol>
232+
</div>
233+
</div>
234+
235+
236+
<div class="panel panel-default">
237+
<div class="panel-body">
238+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
239+
<p>Example - Grover's Algorithm:</p>
240+
$$
241+
\text{Quantum Search Complexity: } O(\sqrt{N}) \text{ vs. } O(N)
242+
$$
243+
</div>
244+
</div>
245+
246+
247+
<p>Advantage: Speedups in high-dimensional optimization and linear algebra problems.</p>
248+
249+
<!-- !split -->
250+
<h2 id="challenges-in-quantum-machine-learning" class="anchor">Challenges in Quantum Machine Learning </h2>
251+
252+
<div class="panel panel-default">
253+
<div class="panel-body">
254+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
255+
<ol>
256+
<li> Noisy Intermediate-Scale Quantum (NISQ) devices.</li>
257+
<li> Decoherence and limited qubit coherence times.</li>
258+
</ol>
259+
</div>
260+
</div>
261+
262+
263+
<div class="panel panel-default">
264+
<div class="panel-body">
265+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
266+
<ol>
267+
<li> Efficient embedding of classical data into quantum states.</li>
268+
</ol>
269+
</div>
270+
</div>
271+
272+
273+
<div class="panel panel-default">
274+
<div class="panel-body">
275+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
276+
<ol>
277+
<li> Difficult to scale circuits to large datasets.</li>
278+
</ol>
279+
</div>
280+
</div>
281+
282+
283+
<!-- !split -->
284+
<h2 id="classical-support-vector-machines-svms" class="anchor">Classical Support Vector Machines (SVMs) </h2>
285+
286+
<p>Support Vector Machines (SVM) are supervised learning algorithms used
287+
for classification tasks, and regression tasks as well. The main goal
288+
of SVMs is to find the optimal separating hyperplane (in
289+
high-dimensional space) that provides a maximum margin between
290+
classes.
291+
</p>
292+
293+
<!-- !split -->
294+
<h2 id="first-mathematical-formulation-of-svms-formal-details-below" class="anchor">First mathematical formulation of SVMs (formal details below) </h2>
295+
296+
<p>For a dataset \((\mathbf{x}_i, y_i)\) where \(\mathbf{x}_i \in
297+
\mathbb{R}^n\) and \(y_i \in \{-1, 1\}\), the decision boundary is
298+
defined as:
299+
</p>
300+
$$
301+
f(\mathbf{x}) = \mathbf{w} \cdot \mathbf{x} + b = 0.
302+
$$
303+
304+
<p>The goal is to optimize</p>
305+
$$
306+
\min_{\mathbf{w}, b} \frac{1}{2} ||\mathbf{w}||^2,
307+
$$
308+
309+
<p>subject to</p>
310+
$$
311+
y_i (\mathbf{w} \cdot \mathbf{x}_i + b) \geq 1, \quad \forall i
312+
$$
313+
314+
315+
<!-- !split -->
316+
<h2 id="kernels-and-more" class="anchor">Kernels and more </h2>
317+
318+
<p>In classical SVMs, kernels help with non-linear data
319+
separations. Quantum computers can speed up the computation of complex
320+
kernel evaluations by efficiently simulating an inner product in an
321+
exponentially large Hilbert space.
322+
</p>
323+
324+
<!-- !split -->
325+
<h2 id="quantum-support-vector-machines-qsvms" class="anchor">Quantum Support Vector Machines (QSVMs) </h2>
326+
327+
<div class="panel panel-default">
328+
<div class="panel-body">
329+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
330+
<ol>
331+
<li> Maps classical data to a quantum Hilbert space.</li>
332+
<li> Quantum kernel measures similarity in high-dimensional space.</li>
333+
</ol>
334+
</div>
335+
</div>
336+
337+
338+
<div class="panel panel-default">
339+
<div class="panel-body">
340+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
341+
$$
342+
K(\boldsymbol{x}, \boldsymbol{x}') = |\langle \psi(\boldsymbol{x}) \vert \psi(\boldsymbol{x}')\rangle|^2
343+
$$
344+
</div>
345+
</div>
346+
347+
348+
<p>Here, \( \vert \phi(\boldsymbol{x})\rangle \) is the quantum state encoding of
349+
the classical data \( \boldsymbol{x} \).
350+
</p>
351+
352+
<p>Advantage: Potentially exponential speedup over classical SVMs.</p>
353+
354+
<!-- !split -->
355+
<h2 id="quantum-neural-networks-qnns" class="anchor">Quantum Neural Networks (QNNs) </h2>
356+
357+
<p>Quantum Neural Networks replace classical neurons with parameterized quantum circuits.</p>
358+
<div class="panel panel-default">
359+
<div class="panel-body">
360+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
361+
<ol>
362+
<li> Quantum Gates as Activation Functions.</li>
363+
<li> Variational Quantum Circuits (VQCs) for optimization.</li>
364+
</ol>
365+
</div>
366+
</div>
367+
368+
<div class="panel panel-default">
369+
<div class="panel-body">
370+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
371+
<p>\textbf{Parameterized Quantum Circuit:}</p>
372+
$$
373+
U(\theta) = \prod_i R_y(\theta_i) \cdot CNOT \cdot R_x(\theta_i)
374+
$$
375+
</div>
376+
</div>
377+
378+
<p>Advantage:Quantum gradients enable exploration of non-convex landscapes.</p>
379+
380+
<!-- !split -->
381+
<h2 id="quantum-boltzmann-machines-qbms" class="anchor">Quantum Boltzmann Machines (QBMs) </h2>
382+
383+
<p>Quantum Boltzmann Machines leverage quantum mechanics to sample from a probability distribution.</p>
384+
385+
<div class="panel panel-default">
386+
<div class="panel-body">
387+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
388+
<ol>
389+
<li> Quantum tunneling aids in escaping local minima.</li>
390+
<li> Quantum annealing for optimization problems.</li>
391+
</ol>
392+
</div>
393+
</div>
394+
395+
<div class="panel panel-default">
396+
<div class="panel-body">
397+
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
398+
$$
399+
H = -\sum_i b_i \sigma_i^z - \sum_{ij} w_{ij} \sigma_i^z \sigma_j^z
400+
$$
105401
</div>
106402
</div>
107403

404+
<p>Advantage: Efficient sampling in complex probability distributions.</p>
405+
406+
<!-- !split -->
407+
<h2 id="back-to-math-of-svms" class="anchor">Back to math of SVMs </h2>
108408

109409
<!-- ------------------- end of main content --------------- -->
110410
</div> <!-- end container -->
@@ -117,7 +417,7 @@ <h2 id="plans-for-the-week-of-april-21-25" class="anchor">Plans for the week of
117417
</footer>
118418
-->
119419
<center style="font-size:80%">
120-
<!-- copyright --> &copy; 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
420+
<!-- copyright --> &copy; 1999-2025, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
121421
</center>
122422
</body>
123423
</html>

0 commit comments

Comments
 (0)