Title

layout: true
<slide-template title="Quantum Computing" subtitle="Introduction into Quantum Machine Learning: Data Encoding, Model, Measurement"></slide-template>

---
class: title-slide, center, middle

<title-slide titletype="Lecture" title="Quantum Computing" modulenumber="(CS5070)" subtitle="Introduction into Quantum Machine Learning: Data Encoding, Model, Measurement"></title-slide>

---
# .darkblue[Influences to Machine Learning]

<div class="textcenter padding-bottom">
<img src="../../img/qc/ml/InfluencesToML.svg" align="top" class="fullwidth"></img>
</div>
<div class="textcenter darkblue em">
"Learning patterns from data in order to draw inferences"
</div>

---
# .darkblue[Machine Learning] - Basics

--
.more-more-condensed[
- .darkblue[How to choose a model?]
- .darkblue[How to find optimal parameters?]
]

--
.more-more-condensed[
.darkblue[Follow-up questions] related to quantum computing:
- .darkblue[Are there quantum models?]
- .darkblue[Can quantum computing be utilized to find optimal parameters?]
]

---
# Machine Learning .darkblue[Ingredients: Data]

.more-more-condensed.darkblue[
- <span class="material-icons">photo_library</span> Pictures
- <span class="material-icons">video_library</span> Videos
- <span class="material-icons">backup_table</span> Spreadsheets
- <span class="material-icons">pin</span> Numerical/Categorical
- <span class="material-icons">description</span> Texts in Natural Language
   - <span class="material-icons padding-left">picture_as_pdf</span> Scientific Publications
   - <span class="material-icons padding-left">email</span> EMail
   - <span class="material-icons padding-left">chat</span> Chat Communications
  - ...
- <span class="material-icons">terminal</span> Source Code
- ... 
]

---
# Machine Learning .darkblue[Ingredients: Data]

.reference[Adapted from <a href="https://github.com/qiskit-community/qiskit-presentations/tree/master/2021-11-29_12-02_eqtc" target="_blank">A. Dekusar, J. Gacon, Qiskit Machine Learning: Quantum algorithms for supervised learning, EQTC, 2022</a> <a href="https://pixabay.com/de/illustrations/natur-tier-katze-inl%c3%a4ndisch-4779493/" target="_blank">Picture Source</a>]

.more-more-condensed[
<table>
<tbody>
<tr>
<td class="darkblue textcenter bold halfwidth">Picture</td>
<td class="darkblue textcenter bold halfwidth">Dataset</td>
</tr>
<tr>
<td><div class="textcenter">
<table>
<tbody>
<tr>
<td class="halfwidth"><div class="round-border blue-border"><img src="../../img/qc/ml/nature-gc83b4b57c_640.jpg" align="top" class="fullwidth"></img></div></td>
<td class="halfwidth padding-left" style="text-align:left">$n$ pixels</td>
</tr>
<tr>
<td>$m$ pixels</td></tr></tbody></table>
</div>
.small-font[
- Every pixel $\in[-1,1]$ from black to white 
- Every image as vector:<br/>
<table><tbody><tr><td>
$x_1 := \left[\begin{array}{c} -0.6\\ +0.32\\ ...\\ -0.99 \end{array}\right]$
</td><td>
$n\cdot m$ elements (in this context also called .em[features])
</td></tr></tbody></table>
]
</td>
<td style="vertical-align:top">
.small-font[
- A dataset $D$ consists of many images: $(x_1,...,x_N)$
- Every image is a sample with $N$ samples in total
- Huge matrix to represent all images:<br/>
$D=\left[\begin{array}{c} x_1 & x_2 & ...& x_N \end{array}\right]$<br/>
$=\left[\begin{array}{cccc} -0.6 & +0.12 & ... & +0.55 \\\\ +0.32 & +1 & ... & -0.38 \\\\ ...&...&...&...\\\\ -0.99 & +0.03 & ... & +0.22 \end{array}\right]$
<dic class="textcenter">Matrix of size $N\times n\cdot m$</div>
]
</td>
</tr>
</tbody>
</table>
]

---
# Machine Learning .darkblue[Ingredients: Model]

.more-more-condensed[
- A .darkblue[model is] a .darkblue[function that takes data, depends on some parameters and maps this data to] some .darkblue[prediction]<br/>
<div class="textcenter padding-top"><img src="../../img/qc/ml/Prediction.svg" align="top" style="width:75%"></img></div>
]

---
# Machine Learning .darkblue[Ingredients: Data and Quantum Model]

.more-more-condensed[
- Possibilities of involving Quantum Computing:
<div class="padding-top">
<table class="result tablepadding textcenter">
<tbody>
<tr>
<th></th><th></th><th colspan="2">Data Processing</th>
</tr>
<tr>
<th></th><th></th><td class="darkgreen">Classical</td><td class="darkred">Quantum</td>
</tr>
<tr>
<th rowspan="2" style="border-radius: 0px 0px 0px 5px;">Representation of data</th><td class="darkgreen">Classical</td><td class="darkgreen">CC</td><td>.darkgreen[C].darkred[Q]</td>
</tr>
<tr>
<td class="darkred">Quantum</td><td>.darkred[Q].darkgreen[C]</td><td class="darkred">QQ</td>
</tr>
</tbody>
</table>
</div>
]

---
# How can we .darkblue[encode data and build] a .darkblue[quantum model]?

.reference[<a href="https://doi.org/10.1038/s41586-019-0980-2" target="_blank">Havlíček et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212, 2019</a>]

.more-more-condensed[
<div class="textcenter">
<img src="../../img/qc/ml/QuantumModel.svg" align="top" class="fullwidth"></img>
</div>
]

---
# How can we .darkblue[encode data and build] a .darkblue[quantum model]? - .darkblue[Data encoding]

.reference[<a href="https://doi.org/10.1038/s41586-019-0980-2" target="_blank">Havlíček et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212, 2019</a>]

.more-more-condensed[
<div class="textcenter">
<img src="../../img/qc/ml/QuantumModelData.svg" align="top" class="fullwidth"></img>
</div>
]

---
# .darkblue[How to encode data $x$?]

.reference[<a href="https://doi.org/10.1038/s41586-019-0980-2" target="_blank">Havlíček et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212, 2019</a>]

<div class="textcenter">
$x$<br/><svg-arrow orientation="down" color="darkblue" scale="0.1"></svg-arrow><br/>$|\psi\rangle = \alpha_0\cdot|0\rangle+\alpha_1\cdot|1\rangle+\cdots+\alpha_k\cdot|k\rangle$<br/>???
</div>

---
# How to encode data $x$? - .darkblue[Basis encoding]

.reference[<a href="https://doi.org/10.1038/s41586-019-0980-2" target="_blank">Havlíček et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212, 2019</a>]

.more-more-condensed[
<table>
<tbody>
<tr>
<td class="darkblue bold" style="width:50%">Idea</td>
<td class="darkblue bold" style="width:50%">Example</td>
</tr>
<tr>
<td class="darkblue small-font22">Encode .em[binary] representation of data in basis states</td>
<td class="darkblue small-font22">Data $x$ with 2 features</td>
</tr>
<tr>
<td class="small-font textcenter">$x=\left[\begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_k \end{array}\right]=\left[\begin{array}{c} b_{1,1}\cdots b_{1,n} \\ b_{2,1}\cdots b_{2,n} \\ \vdots \\  b_{k,1}\cdots b_{k,n} \end{array}\right]$</td>
<td class="small-font textcenter">$x=\left[\begin{array}{c} 1 \\ 3 \end{array}\right]=\left[\begin{array}{c} 01_b \\ 11_b \end{array}\right]$</td>
</tr>
<tr>
<td class="textcenter"><svg-arrow orientation="down" color="darkblue" scale="0.05"></svg-arrow></td>
<td class="textcenter"><svg-arrow orientation="down" color="darkblue" scale="0.05"></svg-arrow></td>
</tr>
<tr>
<td class="small-font textcenter">$|x\rangle=|b_{1,1}\cdots b_{1,n}\cdots b_{k,1}\cdots b_{k,n}\rangle$</td>
<td class="small-font textcenter">$|x\rangle=|0111\rangle$</td>
</tr>
<tr>
<td class="textcenter">Circuit: <img src="../../img/qc/ml/BasisEncodingGeneral.svg" align="top" style="width:6em"></img></td>
<td class="textcenter">Circuit: <img src="../../img/qc/ml/BasisEncodingExample.svg" align="top" style="width:4em"></img></td>
</tr>
<tr>
<td class="padding-right"><div class="round-border green-border">.darkgreen.bold[Pro:]<br/>.darkblue[Easy to load]</div></td>
<td class="padding-left"><div class="round-border red-border">.darkred.bold[Contra:]<br/>.darkblue[.small-font[$k\cdot n$] qubits for .small-font[$k\cdot n$] bits]</div></td>
</tr>
</tbody>
</table>
]

---
# How to encode data $x$? - .darkblue[Amplitude encoding]

.reference[<a href="https://doi.org/10.1038/s41586-019-0980-2" target="_blank">Havlíček et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212, 2019</a>]

.more-more-condensed[
<table>
<tbody>
<tr>
<td class="darkblue bold" style="width:50%">Idea</td>
<td class="darkblue bold" style="width:50%">Example</td>
</tr>
<tr>
<td class="darkblue">Encode .em[normalized] data in state amplitudes</td>
<td class="darkblue">Data $x$ with 4 features</td>
</tr>
<tr>
<td class="small-font textcenter">
 
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</td>
<td class="small-font textcenter">$x=\left[\begin{array}{c} 3.0 \\ 1.0 \\ 4.0 \\2.0 \end{array}\right]$</td>
</tr>
<tr>
<td class="textcenter"><svg-arrow orientation="down" color="darkblue" scale="0.05"></svg-arrow></td>
<td class="textcenter"><svg-arrow orientation="down" color="darkblue" scale="0.05"></svg-arrow></td>
</tr>
<tr>
<td class="small-font textcenter">$|\Phi_x\rangle=\sum_{i=0}^{2^n-1}x_{i+1}|i\rangle$</td>
<td class="small-font textcenter">$|\Phi_x\rangle=\frac{3.0\cdot|00\rangle+1.0\cdot|01\rangle+4.0\cdot|10\rangle+2.0\cdot|11\rangle}{\sqrt{3.0^2+1.0^2+4.0^2+2.0^2}}$</td>
</tr>
<tr>
<td class="padding-right"><div class="round-border green-border">.darkgreen.bold[Pro:]<br/>.darkblue[.small-font[$\lceil log(k)\rceil$] qubits for .small-font[$k$] features]</div></td>
<td class="padding-left"><div class="round-border red-border">.darkred.bold[Contra:]<br/>.darkblue[Difficult to load!]</div></td>
</tr>
</tbody>
</table>
]

---
# How to encode data $x$? - .darkblue[Angle encoding]

.reference[<a href="https://doi.org/10.1038/s41586-019-0980-2" target="_blank">Havlíček et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212, 2019</a> <a href="https://arxiv.org/abs/1803.07128" target="_blank">[SK'18]</a> <a href="https://arxiv.org/abs/1804.11326" target="_blank">[H+'18]</a>]

.more-more-condensed[
- .darkblue[.bold[Idea:] Encode] (normalized) .darkblue[data in rotation angles]<br/>
<table><tbody><tr><td>$x=\left[\begin{array}{c} x_1 \\\\ x_2 \\\\ \vdots \\\\ x_k \end{array}\right]$</td><td><svg-arrow orientation="right" color="darkblue" scale="0.05"></svg-arrow></td><td><img src="../../img/qc/ml/AngleEncoding_k.svg" align="top" style="width:23em"></img></td></tr></tbody></table>
<div class="textcenter">
<img src="../../img/qc/ml/CosSin.svg" align="top" style="width:90%"></img>
</div>
]

---
# .darkblue[Feature Map via Angle Encoding]

.more-more-condensed[
- .darkblue.em.bold[Kernel trick:] A .darkblue[feature map can transform data into] a .darkblue[space]
  - .darkblue[where it is .em[easier to process]]
  - .darkblue[where features are .em[easier to be classified]] (by e.g. linear functions)
  
<div class="textcenter">
<img src="../../img/qc/ml/FeatureMap.svg" align="top" class="fullwidth"></img>
</div>
]

---
# .darkblue[Feature Maps] for other Class Positions than Quarter Circles 1/2

---
# .darkblue[Feature Maps] for other Class Positions than Quarter Circles 2/2

---
# .darkblue[Feature Maps] - Higher-Order Encodings

.more-more-condensed[
- Using .darkblue[non linear functions and entanglement] for feature map [Y+'20]<br/><img class="margin-top margin-bottom" src="../../img/qc/ml/HigherOrderFeatureMap.svg" align="top" style="width:75%"></img><div class="small-font">$u1(\phi)=diag(1,e^{-i\cdot\phi})=\left[\begin{array}{cc} 1 & 0 \\\\ 0 & e^{-i\cdot\phi} \end{array}\right],\phi_1(x) = x_1, \phi_2(x) = x_2$</div>
  <table class="result tablesmallpadding margin-top margin-bottom">
  <tbody>
  <tr>
  <th></th><th class="textcenter">1.</th><th class="textcenter">2.</th><th class="textcenter">3.</th><th class="textcenter">4.</th><th class="textcenter">5.</th>
  </tr>
  <tr>
  <th>$\phi_{1,2}(x)$</th>
  <td class="smaller-font">$\pi\cdot x_1\cdot x_2$</td>
  <td class="smaller-font">$\frac{\pi}{2}\cdot(1-x_1)\cdot(1-x_2)$</td>
  <td class="small-font">$e^{\frac{|x_1-x_2|^2\cdot ln(\pi)}{8}}$</td>
  <td class="smaller-font">$\frac{\pi}{3\cdot cos(x_1)\cdot cos(x_2)}$</td>
  <td class="smaller-font">$\pi\cdot cos(x_1)\cdot cos(x_2)$</td>
  </tr>
  </tbody>
  </table>

- More general forms like <a href="https://qiskit.org/documentation/stubs/qiskit.circuit.library.ZZFeatureMap.html" target="_blank">ZZFeatureMap</a> and <a href="https://qiskit.org/documentation/stubs/qiskit.circuit.library.PauliFeatureMap.html" target="_blank">PauliFeatureMap</a> (with a given number of repetitions) for more qubits
]

---
# .darkblue[Feature Maps] - Higher-Order Encodings

.more-more-condensed[
- .adrkblue[Classification accuracy] achieved .darkblue[by quantum SVM] [Y+'20]<br/>
.darkblue[Training] Accuracy .small-font[(= number of correct predictions / total number of predictions)]:<br/>
<table class="result tablepadding small-font margin-bottom" style="text-align:right">
<tbody>
<tr>
<th rowspan="2">Encoding Function</th><th colspan="4" class="textcenter">Dataset .em[(similar to)]</th>
</tr>
<tr>
<th>Circle</th><th>Exponential</th><th>Moon</th><th>XOR</th>
</tr>
<tr>
<th>1.</th><td>1.00</td><td>0.91</td><td>0.85</td><td>1.00</td>
</tr>
<tr>
<th>2.</th><td>1.00</td><td>0.93</td><td>0.96</td><td>0.97</td>
</tr>
<tr>
<th>3.</th><td>1.00</td><td>0.97</td><td>0.91</td><td>0.93</td>
</tr>
<tr>
<th>4.</th><td>1.00</td><td>0.98</td><td>1.00</td><td>0.95</td>
</tr>
<tr>
<th>5.</th><td>1.00</td><td>0.94</td><td>0.98</td><td>0.93</td>
</tr>
</tbody>
</table>
.darkblue[Test] Accuracy:<br/>
<table class="result tablepadding small-font" style="text-align:right">
<tbody>
<tr>
<th rowspan="2">Encoding Function</th><th colspan="4" class="textcenter">Dataset .em[(similar to)]</th>
</tr>
<tr>
<th>Circle</th><th>Exponential</th><th>Moon</th><th>XOR</th>
</tr>
<tr>
<th>1.</th><td>0.97</td><td>0.83</td><td>0.85</td><td>0.99</td>
</tr>
<tr>
<th>2.</th><td>0.96</td><td>0.89</td><td>0.87</td><td>0.96</td>
</tr>
<tr>
<th>3.</th><td>1.00</td><td>0.92</td><td>0.86</td><td>0.91</td>
</tr>
<tr>
<th>4.</th><td>1.00</td><td>0.88</td><td>0.92</td><td>0.89</td>
</tr>
<tr>
<th>5.</th><td>1.00</td><td>0.92</td><td>0.87</td><td>0.88</td>
</tr>
</tbody>
</table>
]

---
# Machine Learning .darkblue[Ingredients: Model]

.more-more-condensed[
- .darkblue[How good is your model] performing.darkblue[?]<br/>$\Rightarrow$ A .darkblue[.bold[cost function] .darkred.bold[$C(\hat{y},y)$] scores the model] based on model predictions and true value
  - .darkred.bold.small-font[$\hat{y}=f(x,\Theta)$] .darkblue[prediction] obtained from our model and
  - .darkred.bold.small-font[$y$] .darkblue[ground truth]
  - Alias: <span class="darkblue">loss function</span>, <span class="darkblue">error function</span>
- Usually: .darkblue[.bold[Minimize] cost function!]
- .darkblue[Widely used cost functions]
  - .darkblue[Mean Squared Error (MSE):] .darkred.bold.small-font[$MSE = (\hat{y}-y)^2$]
	 - being symmetric: an error above the target causes the same loss as the same magnitude of error below the target
  - .darkblue[Cross-Entropy Loss Function] 
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transform="translate(9015.5, 0)"><use xlink:href="#MJX-6-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(9404.5, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-6-TEX-I-70"></use></g><g data-mml-node="TeXAtom" transform="translate(503, -150) scale(0.707)"><g data-mml-node="mi"><use xlink:href="#MJX-6-TEX-I-6F"></use></g><g data-mml-node="mo" transform="translate(485, 0)"><use xlink:href="#MJX-6-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(763, 0)"><use xlink:href="#MJX-6-TEX-I-63"></use></g></g></g><g data-mml-node="mo" transform="translate(10803.2, 0)"><use xlink:href="#MJX-6-TEX-N-29"></use></g></g></g></svg>, where 
     - binary indicator (0 or 1) if label $c$ is the correct classification for observation $o$
	 - predicted probability observation $o$ is of class $c$
	 - Penalty is logarithmic in nature yielding a large score for large differences close to 1 and small score for small differences tending to 0
]

---
# Machine Learning .darkblue[Ingredients: Model]

.more-more-condensed[
- .darkblue[Classification]
  - .darkblue[Binary] Classification
     - <span class="darkblue"><span class="material-icons">email</span> $\overset{?}{\rightarrow}$ <span class="material-icons">delete</span> SPAM</span>-Detection: use emails seen so far to produce rules to predict whether future emails are spam or not
  - .darkblue[Multi-Class] Classification
     - <span class="darkblue"><span class="material-icons">scanner</span> <span class="material-icons">east</span> <span class="material-icons">description</span> Optical character recognition (OCR)</span> in handwritten text: Learn patterns from data and classify new data
  - .darkblue[Multi-Label] Classification
     - <span class="darkblue"><span class="material-icons">email</span> <span class="material-icons">east</span> <span class="material-icons">groups</span> Text Classification</span>: assign customer communications to employees in different departments
- .darkblue[Regression]
  - <span class="darkblue"><span class="material-icons padding-left">thermostat</span>Weather Forecast: Predicting values</span> based on relevant information
- .darkblue[Clustering]
  - <span class="darkblue"><span class="material-icons padding-left">shopping_cart</span>Identify consumers with similar behavior</span> (e.g. for marketing purposes)
- .darkblue[Generate data]
  - <span class="darkblue"><span class="material-icons padding-left">description</span><span class="material-icons">east</span><span class="material-icons">image</span>Learn patterns or structure from data and reproduce them</span> (e.g. generate pictures from descriptions, compose music)
]

---
# .darkblue[Supervised Learning: Data]

.reference[Adapted from <a href="https://github.com/qiskit-community/qiskit-presentations/tree/master/2021-11-29_12-02_eqtc" target="_blank">A. Dekusar, J. Gacon, Qiskit Machine Learning: Quantum algorithms for supervised learning, EQTC, 2022</a> Picture Sources: <a href="https://pixabay.com/de/vectors/katze-tier-die-silhouette-au%c3%9fer-1583462/" target="_blank">Cat</a> <a href="https://pixabay.com/de/vectors/hund-tier-silhouette-5890164/" target="_blank">Dog</a>]

<table>
<tbody>
<tr>
<td style="vertical-align:top;width:65%">
.more-more-condensed[
 - .darkblue[Classify] whether the picture contains a .darkblue[cat or dog] based on certain features of animals:<br/>1. Dataset: $\\{(x_1,y_1),...,(x_N,y_N)\\}$
   - .darkblue[$x_i$: features]
   - .darkblue[$y_i$: labels] (i.e., cat or dog)

<table class="small-font">
<tbody>
<tr>
<td></td>
<td class="textcenter"><img src="../../img/qc/ml/cat-g88698f70e_640.jpg" align="top" style="width:3em"></img></td>
<td></td>
<td></td>
<td></td>
<td class="textcenter"><img src="../../img/qc/ml/dog-g5f8fc564b_640.jpg" align="top" style="width:4em"></img></td>
<td></td>
</tr>
<tr>
<td>$x_i=$</td>
<td>$\left[\begin{array}{c} x_{i,1} \\ x_{i,2} \\ \vdots \\  x_{i,k} \end{array}\right]$</td>
<td>$y_i=1$</td>
<td style="width:1em"></td>
<td>$x_j=$</td>
<td>$\left[\begin{array}{c} x_{j,1} \\ x_{j,2} \\ \vdots \\  x_{j,k} \end{array}\right]$</td>
<td>$y_i=-1$</td>
</tr>
<tr>
<td colspan="7" class="textcenter"><span class="darkblue">Dataset with $k$ features</span> like weight, tail length, ...</td>
</tr>
</tbody>
</table>

]
</td>
<td style="vertical-align:top" style="width:35%">
.more-more-condensed[
- .darkblue[Split] the dataset .darkblue[into] the .darkblue[training and test datasets]<br/>
<table style="border-collapse:collapse">
<tbody>
<tr>
<td style="background-color:DeepSkyBlue;width:3em;height:7em"></td>
<td class="small-font padding-left">70% Training dataset with<br/>$t$ samples<td>
</tr>
<tr>
<td style="background-color:SteelBlue;width:3em;height:3em"></td>
<td class="small-font padding-left">30% Test dataset with<br/>$u$ samples</td>
</tr>
<tr>
<td colspan="2">$N=t+u$</td>
</tr>
</tbody>
</table>
]
</td>
</tr>
</tbody>
</table>

---
# .darkblue[Supervised Learning: Model]

.more-more-condensed[
- .darkblue[Classify] whether the picture contains a .darkblue[cat or dog] based on certain features of animals:
<ol start="2"><li class="small-font32"><span class="darkblue">Linear Model<br/>$\hat{y}=f(x,\Theta)=\Theta^T\cdot x$</span><br/><span class="small-font28">Dimensionality of the vectors: $k$ (=number of features)<div class="round-border blue-border darkblue">Estimating the predicted label $\hat{y}$ by feeding a sample from the dataset and comparing $\hat{y}$ versus ground truth value $y$ from the dataset</div></span></li>
]

---
# .darkblue[Supervised Learning: Cost]

.more-more-condensed[
- .darkblue[Classify] whether the picture contains a .darkblue[cat or dog] based on certain features of animals:
<ol start="3"><li class="small-font28"><span class="darkblue">For the $t$ samples in the training dataset compute cost with
<ol style="list-style-type:lower-latin"><li>Mean Squared Error (MSE): $C_{MSE}=\frac{1}{t}\cdot\sum_{i=1}^t(\hat{y}_1-y_i)^2$</li>
<li>Binary classification (e.g., "cat" or "dog") using cross-entropy loss:<br/><img src="../../img/qc/ml/BinaryCrossEntropyLoss.svg" align="top" class="fullwidth padding-top"></img></li></ol></li><li>...</li></ol>
</span></li></ol>

.darkblue[.bold[Minimize] the] chosen .darkblue[cost function to have a good model $f(x_i,\Theta)$]
]

---
# How can we .darkblue[build] a .darkblue[quantum model]?

.reference[<a href="https://doi.org/10.1038/s41586-019-0980-2" target="_blank">Havlíček et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212, 2019</a>]

.more-more-condensed[
<div class="textcenter">
<img src="../../img/qc/ml/QuantumModelModel.svg" align="top" class="fullwidth"></img>
</div>
]

---
# How can we .darkblue[build] a .darkblue[quantum model]?

.reference[Adapted from <a href="https://github.com/qiskit-community/qiskit-presentations/tree/master/2021-11-29_12-02_eqtc" target="_blank">A. Dekusar, J. Gacon, Qiskit Machine Learning: Quantum algorithms for supervised learning, EQTC, 2022</a> <a href="https://doi.org/10.1088/2058-9565/ab4eb5" target="_blank">[B+'19]</a> <a href="https://doi.org/10.1038/s41467-021-22539-9" target="_blank">[H+'21]</a>]

.more-more-condensed[
<table class="tablesmallpadding">
<tbody>
<tr>
<td style="width:30%">.darkblue.bold[Solution Space]</td>
<td style="width:70%">.darkblue[.bold[Hardware-efficient] Ansatz]</td>
</tr>
<tr>
<td><img src="../../img/qc/ml/ChoosingAnsatz.svg" align="top" class="fullwidth"></img></td>
<td><img src="../../img/qc/ml/HardwareEfficientAnsatz.svg" align="top" class="fullwidth"></td>
</tr>
<tr>
<td colspan="2">.darkblue[.bold[Problem-specific] ansatz]</td>
</tr>
<tr>
<td colspan="2" class="small-font24">.darkblue[Use knowledge about problem to choose ansatz!]</td>
</tr>
<tr>
<td colspan="2" class="small-font24">Example: Does the .darkblue[wave function of the quantum states only] have .darkblue[real amplitudes]?</td>
</tr>
<tr>
<td colspan="2" class="small-font24">.darkblue[Many different quantum circuits] for machine learning models, see [B+'19]</td>
</tr>
</tbody>
</table>
]

---
# .darkblue[Variants]

.reference[<a href="https://doi.org/10.1002/qute.201900070" target="_blank">[SJA'19]</a>]

.more-more-condensed[
<table class="tablesmallpadding result">
<tbody>
<tr>
<th style="width:20%">Entanglement Layer</th>
<th style="width:20%">Linear</th>
<th style="width:24%">Circular</th>
<th style="width:36%">Fully Connected</th>
</tr>
<tr>
<th>Circuit</th>
<td><img src="../../img/qc/ml/LinearEntanglementLayer.svg" align="top" class="fullwidth"></img></td>
<td><img src="../../img/qc/ml/CircularEntanglementLayer.svg" align="top" class="fullwidth"></td>
<td><img src="../../img/qc/ml/FullyConnectedEntanglementLayer.svg" align="top" class="fullwidth"></td>
</tr>
<tr>
<th>#Gates for $n$ qubits</th>
<td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height: 0.43056em;"></span><span class="strut bottom" style="height: 0.43056em; vertical-align: 0em;"></span><span class="base"><span class="mord mathit">n</span></span></span></span> − 1</td>
<td>$n$</td>
<td>$\binom{n}{2}$</td>
</tr>
</tbody>
</table>

- .darkblue[Different combinations of x ,y and z] rotation gates in the rotation layer
- In variants .darkblue[rotation layer and entanglement layer are combined into] one by using .darkblue[controlled rotation gates]
]

---
# .darkblue[Retrieving the result of a Quantum Model] - Measurement

---
# .darkblue[Result of a Quantum Model] - Measurement

.more-more-condensed[
<table>
<tbody>
<tr>
<td style="width:70%">
<div class="textcenter">
<img src="../../img/qc/bloch/MeasurementSuperposition.svg" align="top" class="fullwidth"></img>
</div>
</td>
<td class="darkblue" style="width:30%">
.small-font[$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$]<br/>.small-font[$|\alpha|^2+|\beta|^2=1$]
</td>
</tbody>
</table>
<table class="tablesmallpadding small-font fullwidth">
<tbody>
<tr>
<td class="bold darkblue textcenter" style="width:30%">Measure $x$ samples:</td>
<td class="bold darkblue textcenter" style="width:70%">Expectation value:</td>
</tr>
<tr>
<td class="textcenter"><table><tbody><tr><td>e.g., $x$=1000:</td><td><img src="../../img/qc/ml/SampleCount.svg" align="top" style="width:6em"></img></td></tr></tbody></table></td>
<td class="darkblue textcenter" style="vertical-align:top">.em[Average value of all results of a quantum operator] $Z$ (with eigenvectors $\phi_i$ and eigenvalues $a_i$) applied to a state $\psi$:<br/>$\langle Z\rangle_\psi=lim_{n\rightarrow\infty}\sum\frac{N_i}{N}a_i=\sum|\langle \phi_i|\psi\rangle|^2 a_i=\langle\psi|Z|\psi\rangle$<br/>e.g.: $\langle Z\rangle=1\cdot P_0+(-1)\cdot P_1$</td>
</tr>
</tbody>
</table>
]

---
# .darkblue[Post-Processing]

.more-more-condensed[
<table class="tablepadding">
<tbody>
<tr>
<td class="darkblue bold">Sampling-based</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td class="round-border blue-border padding textcenter"><img src="../../img/qc/ml/SampleCount3bits.svg" align="top" style="width:6em"></img></td>
<td><table class="textcenter fullwidth"><tbody><tr><td>.darkblue[mapping]</td></tr><tr><td><svg-arrow scale="0.1" orientation="right" color="darkblue"></svg-arrow></td></tr><tr><td>e.g. parity: even or<br/>odd number of 1s?</td></tr></tbody></table></td>
<td class="round-border blue-border padding">.darkblue[label]<br/><br/>0 with $P_{010}$<br/>1 with $P_{110}$</td>
</tr>
<tr>
<td class="darkblue bold padding-top">Expectation-based</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td class="round-border blue-border padding darkblue textcenter">Measure expectation<br/>value<br/><br/>$\langle z_1\cdots z_n\rangle=-0.5$</img></td>
<td><table class="textcenter fullwidth"><tbody><tr><td>.darkblue[mapping]</td></tr><tr><td><svg-arrow scale="0.1" orientation="right" color="darkblue"></svg-arrow></td></tr><tr><td>e.g. sign</td></tr></tbody></table></td>
<td class="round-border blue-border padding">.darkblue[label]<br/><br/>1 with $Pr=\frac{1+0.5}{2}$</td>
</tr>
</tbody>
</table>
]

---
# .darkblue[Join Ordering] with .darkblue[Quantum ML]

.reference[<a href="https://doi.org/10.1145/3579142.3594299" target="_blank">Winker,Çalıkyılmaz,Gruenwald,Groppe, Quantum Machine Learning for Join Order Optimization using V. Q. C., BiDEDE@SIGMOD'23</a>]

<div class="more-more-condensed">
<ul><li>Compared to ML: <a href="https://doi.org/10.1038/s41467-022-32550-3" target="_blank">Learning with fewer data, higher accuracy</a></li></ul>
<div class="textcenter">
<img src="../../img/qc/ml/QuantumModel.svg" align="top" style="width:97%"></img>
</div>

<table class="fullwidth textcenter">
<tbody>
<tr>
<td>.bold[Input:] .darkblue[Relations]</td><td>Trained Model</td><td>.bold[Output:] .darkblue[Join Order]</td>
</tr>
<tr>
<td></td><td></td><td></td>
</tr>
</tbody>
</table>
<div class="textcenter">
<img src="../../img/keynotes/bv/JOOQMLInOut.svg" align="top" class="fullwidth"></img>
</div>
</div>

---
# .darkblue[Join Ordering] with .darkblue[Quantum ML]

.reference[ErgastF1 benchmark joining 4 relations, <a href="https://doi.org/10.1145/3579142.3594299" target="_blank">Winker,Çalıkyılmaz,Gruenwald,Groppe, QML for Join Order Opt. using VQCs, BiDEDE@SIGMOD'23</a>]

---
# .darkblue[Is quantum ML better than classical ML?]

.reference[Adapted from <a href="https://doi.org/10.1038/s41467-021-22539-9" target="_blank">[H+'21]</a> Quantum speed-up in supervised machine learning: <a href="https://doi.org/10.1038/s41567-021-01287-z" target="_blank">[YAT'21]</a>]

---
# .darkblue[Summary & Conclusions]

.more-more-condensed[
- .darkblue[Machine Learning]
  - Basics: .darkblue[Data, Model, Cost]
- .darkblue[.bold[Quantum] Machine Learning]
  - .darkblue[Data Encoding/Feature Maps]
     - <span class="darkblue">Basis</span> Encoding
	 - .darkblue[Amplitude] Encoding
	 - .darkblue[Angle] Encoding
	 - .darkblue[Higher-Order] Encodings
  - .darkblue[Quantum Model]
     - <span class="darkblue">Hardware-efficient Ansatz</span>
  - .darkblue[Result] of a Quantum Model .darkblue[via Measurement]
     - <span class="darkblue">Sampling versus expectation value</span>
	 - .darkblue[Post-processing]
]