Title

layout: true
<slide-template title="Quantum Computing" subtitle="Introduction to Quantum Computing"></slide-template>

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

<title-slide titletype="Lecture" title="Quantum Computing" modulenumber="(CS5070)" subtitle="Introduction to Quantum Computing"></title-slide>

---
# .darkblue[Quantum Mechanics]

.reference[Picture: <a href="https://commons.wikimedia.org/wiki/File:Hydrogen_Wave.gif" target="_blank">Source</a>]

.more-more-condensed.no-margin[
<table>
<tbody>
<tr>
<td style="width:80%">
.more-more-condensed[
- .darkblue[Very small particles and light behave differently from objects in normal life] 
- .darkblue[Mechanics of light and matter at the atomic and subatomic scale are described by quantum theory]
  - forming the underlying principles of chemistry and most of physics
- .darkblue[Quantum theory has brought us the information age with] its disruptive .darkblue[technologies of] 
  - .darkblue[transistors], 
  - .darkblue[lasers], 
  - .darkblue[nuclear power], and 
  - .darkblue[superconductivity]...
  - .bold.darkred[...and now also quantum computers!]
]
</td>
<td style="width:20%;vertical-align:top;line-height:0.6;" class="padding-left">
<img src="../../img/qc/intro/Hydrogen_Wave.gif" align="top" class="fullwidth"></img><br/>
.smaller-font[.darkblue[Wavefunctions of the electron] in a hydrogen atom at different energy levels. .darkblue[Brighter areas represent a higher probability] of finding the electron.]
</td>
</tr>
</tbody>
</table>
]

---
# .darkblue[Superposition Experiments]

<div class="no-margin more-more-condensed small-font">
<div class="textcenter">
<img src="../../img/qc/intro/MirrorExperiment.svg" align="top" style="width:95%"></img>
</div>
<ul>
<table class="result tablesmallpadding">
<tr>
<th>B1</th>
<td colspan="2" class="textcenter">Not Blocking</td>
<td colspan="2" class="textcenter">Not Blocking</td>
<td colspan="2" class="textcenter">Blocking</td>
<td colspan="2" class="textcenter">Blocking</td>
</tr>
<tr>
<th>B2</th>
<td colspan="2" class="textcenter">Not Blocking</td>
<td colspan="2" class="textcenter">Blocking</td>
<td colspan="2" class="textcenter">Not Blocking</td>
<td colspan="2" class="textcenter">Blocking</td>
</tr>
<tr>
<th>Type</th>
<td>Classical</td>
<td class="darkblue">Quantum</td>
<td>Classical</td>
<td class="darkblue">Quantum</td>
<td>Classical</td>
<td class="darkblue">Quantum</td>
<td>Classical</td>
<td class="darkblue">Quantum</td>
</tr>
<tr>
<th>M1</th>
<td style="text-align:right">???</td>
<td style="text-align:right" class="darkblue">...</td>
<td style="text-align:right">???</td>
<td style="text-align:right" class="darkblue">...</td>
<td style="text-align:right">???</td>
<td style="text-align:right" class="darkblue">...</td>
<td style="text-align:right">???</td>
<td style="text-align:right" class="darkblue">...</td>
</tr>
<tr>
<th>M2</th>
<td style="text-align:right">???</td>
<td style="text-align:right" class="darkblue">...</td>
<td style="text-align:right">???</td>
<td style="text-align:right" class="darkblue">...</td>
<td style="text-align:right">???</td>
<td style="text-align:right" class="darkblue">...</td>
<td style="text-align:right">???</td>
<td style="text-align:right" class="darkblue">...</td>
</tr>
<tr>
<th>M3</th>
<td style="text-align:right">???</td>
<td style="text-align:right" class="darkblue">...</td>
<td style="text-align:right">???</td>
<td style="text-align:right" class="darkblue">...</td>
<td style="text-align:right">???</td>
<td style="text-align:right" class="darkblue">...</td>
<td style="text-align:right">???</td>
<td style="text-align:right" class="darkblue">...</td>
</tr>
</table>
</ul>
</div>

---
# .darkblue[Superposition Experiments]

<div class="no-margin more-more-condensed small-font">
<div class="textcenter">
<img src="../../img/qc/intro/MirrorExperiment.svg" align="top" style="width:95%"></img>
</div>
<ul>
<table class="result tablesmallpadding">
<tr>
<th>B1</th>
<td colspan="2" class="textcenter">Not Blocking</td>
<td colspan="2" class="textcenter">Not Blocking</td>
<td colspan="2" class="textcenter">Blocking</td>
<td colspan="2" class="textcenter">Blocking</td>
</tr>
<tr>
<th>B2</th>
<td colspan="2" class="textcenter">Not Blocking</td>
<td colspan="2" class="textcenter">Blocking</td>
<td colspan="2" class="textcenter">Not Blocking</td>
<td colspan="2" class="textcenter">Blocking</td>
</tr>
<tr>
<th>Type</th>
<td>Classical</td>
<td class="darkblue">Quantum</td>
<td>Classical</td>
<td class="darkblue">Quantum</td>
<td>Classical</td>
<td class="darkblue">Quantum</td>
<td>Classical</td>
<td class="darkblue">Quantum</td>
</tr>
<tr>
<th>M1</th>
<td style="text-align:right">50%</td>
<td style="text-align:right" class="darkblue">50%</td>
<td style="text-align:right">50%</td>
<td style="text-align:right" class="darkblue">50%</td>
<td style="text-align:right">50%</td>
<td style="text-align:right" class="darkblue">50%</td>
<td style="text-align:right">50%</td>
<td style="text-align:right" class="darkblue">50%</td>
</tr>
<tr>
<th>M2</th>
<td style="text-align:right">0%</td>
<td style="text-align:right" class="darkblue">0%</td>
<td style="text-align:right">0%</td>
<td style="text-align:right" class="darkred">12.5%</td>
<td style="text-align:right">0%</td>
<td style="text-align:right" class="darkred">12.5%</td>
<td style="text-align:right">0%</td>
<td style="text-align:right" class="darkblue">0%</td>
</tr>
<tr>
<th>M3</th>
<td style="text-align:right">50%</td>
<td style="text-align:right" class="darkblue">50%</td>
<td style="text-align:right">25%</td>
<td style="text-align:right" class="darkred">12.5%</td>
<td style="text-align:right">25%</td>
<td style="text-align:right" class="darkred">12.5%</td>
<td style="text-align:right">0%</td>
<td style="text-align:right" class="darkblue">0%</td>
</tr>
</table>
</ul>
</div>

---
# .darkblue[Superposition Experiments]

<div class="no-margin more-more-condensed small-font darkblue">
<div class="textcenter">
<img src="../../img/qc/intro/MirrorExperiment.svg" align="top" style="width:95%"></img>
</div>
<ul><li>Which route one single quantum particle (reaching either M2 or M3) takes?
<ul><li class="small-font">Does it take the route of B1? <span class="darkred"><span class="bold">No:</span> Blocking B2 $\Rightarrow$ Observation at M2/M3 changes!</span></li>
<li class="small-font">Does it take the route of B2? <span class="darkred"><span class="bold">No:</span> Blocking B1 $\Rightarrow$ Observation at M2/M3 changes!</span></li>
<li class="small-font">Does it take the routes of B1 and B2 together? <span class="darkred"><span class="bold">No:</span> Another experiment: Always 1 (full) quantum particle is observed either at B1 or at B2 when measuring at B1/B2!</span></li>
<li class="small-font">Does it take neither the route of B1 nor B2? <span class="darkred"><span class="bold">No:</span> Block.B1&B2$\Rightarrow$Nothing@M2/M3!</span></li>
</ul>
</li>
</ul>
</div>

---
# .darkblue[Superposition Experiments]

.reference[Allan Adams: Lecture Quantum Physics I <a href="https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/" target="_blank">Course@MIT</a> <a href="https://www.youtube.com/playlist?list=PLUl4u3cNGP61-9PEhRognw5vryrSEVLPr" target="_blank">Youtube</a> Aiello: Quantum field theory of single-photon states, J.of Optics 27(6),'25 <a href="https://doi.org/10.1088/2040-8986/addc7b" target="_blank">[A25]</a>]

.more-more-condensed.darkblue[
.bold[Allan Adams:] "<span style="font-style:italic">Apparantly, empirically, [quantum particles like] electrons have a way of moving, electrons have a way of being which is unlike anything that we are used to thinking about. And so do molecules. And so do bacteria. So does chalk. It's just harder to detect in those objects. So physicists have a name for this new mode of being. And we call it superposition. ... So to build a better definition of superposition than I have no idea what is going on is going to require a new language. And that language is quantum mechanics.</span>"<br/><br/>
.bold[Andrea Aiello] developed a quantum field theory to show that the electromagnetic field of a single photon spreads across both paths of a beam splitter, even though the photon is detected in only one [A25]
]

---
# .darkblue[Double-Slit Experiment with Classical Waves] (e.g., Water)

.reference[<a href="https://www.youtube.com/watch?v=D1s40lkdWRc" target="_blank">Video: Nils Berglund, "Young's double-slit experiment with waves, in 3D"</a> <a href="https://www.youtube.com/channel/UCHF8dGfnDzTO-8K5mYMohXA" target="_blank">Music: "Seductress" by Francis Preve@francispreve</a>]

<video width = "100%" controls = "controls" 
       preload = "preload">
  <source src = "../../img/qc/intro/DoubleSlitExperimentWithClassicalWaves.mp4"/>
  Your browser does not support the video element!
</video>

---
# .darkblue[Double-Slit Experiment] (e.g., Light/Laser)

.reference[<a href="https://commons.wikimedia.org/wiki/File:Interfere.gif" target="_blank">Image: And1mu, "simulation of interfering waves"</a>]

---
# .darkblue[Double-Slit Experiment with Electrons]

.reference[<a href="https://dx.doi.org/10.1088/1367-2630/15/3/033018" target="_blank">R. Bach, D. Pope, S.-H. Liou, H. Batelaan: Controlled double-slit electron diffraction, New J. Phys. 15, 2013, DOI 10.1088/1367-2630/15/3/033018</a>]

<video width = "100%" controls = "controls" 
       preload = "preload">
  <source src = "../../img/qc/intro/njp458349movie2.mp4"/>
  Your browser does not support the video element!
</video>

---
# .darkblue[Double-Slit Experiment with "Soccer Balls"]

<div class="reference"><a href="https://doi.org/10.1038/44348" target="_blank">[A+99]</a> Image: <a href="https://de.wikipedia.org/wiki/Datei:Buckminsterfullerene_animated.gif" target="_blank">Cryberghost, spinning Buckminsterfullerene ($C_{60}$)</a></div>

--
<div class="more-more-condensed"</div>
<ul><li>.darkblue[Interference pattern] also visible for huge molecules like $C_{60}$ (i.e., consisting of 60 Carbon atoms) [A+99]</li></ul>
</div>

---
# .darkblue[Quantum Particle] (Schrödinger Equation)

.reference[<a href="https://www.youtube.com/watch?v=995uiiudmiI" target="_blank">Video: Nils Berglund, "Young's double-slit experiment with a quantum particle"</a> <a href="https://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100681" target="_blank">Music: "Consort for Brass - Classical Rousing" by Kevin MacLeod</a>]

<video width = "100%" controls = "controls" 
       preload = "preload">
  <source src = "../../img/qc/intro/DoubleSlitExperimentWithAQuantumParticle.mp4"/>
  Your browser does not support the video element!
</video>

---
# .darkblue[Timeline of important events]

.reference[<a href="https://doi.org/10.1007/BF02650179" target="_blank">[F'82]</a> <a href="https://doi.org/10.1038/s41586-019-1666-5" target="_blank">[A+'19]</a> <a href="https://www.ibm.com/blogs/research/2019/10/on-quantum-supremacy/" target="_blank">[P+'19]</a> <a href="https://doi.org/10.48550/arXiv.2111.03011" target="_blank">[PCZ'21]</a> <a href="https://doi.org/10.48550/arXiv.2304.11119" target="_blank">[G+'23]</a>
<a href="https://doi.org/10.1126/science.ado6285" target="_blank">[K+'25]</a>
<a href="https://siliconangle.com/2025/03/12/d-wave-claims-achieved-quantum-supremacy-last-others-disagree/" target="_blank">[W'25]</a>]

.more-more-condensed[
- .bold.darkblue[1982]: .darkblue[Feynman] .darkblue[proposes] the concept of .darkblue[quantum computers] [F'82]
- .bold.darkblue[2019]: .darkblue[Google announces "Quantum Supremacy"] by its 53-qubits chip "Sycamore" [A+'19]
  - .darkblue[200 seconds on Sycamore versus 10,000 years on the world’s fastest supercomputer] IBM Summit
  - .darkblue[IBM] [P+'19]: .darkblue[only 2.5 days on classical supercomputer] after deduction of the problem (i.e., using a better classical algorithm)
  - .darkblue[Pan et al.] [PCZ'21]: .darkblue[only 15 hours on 512 GPU-cluster] using another classical algorithm for obtaining a large number of uncorrelated samples
     - Estimation: a few dozens of seconds on ExaFLOPS supercomputer
  - .darkblue[Discussion intensified] the excessive .darkblue[hype about] .darkblue[quantum technology]
- .bold.darkblue[2023]: Next try: .darkblue[Google runs] Random Circuit Sampling .darkblue[experiments on its 70-qubits improved "Sycamore" in seconds instead of 47 years] (estimation for #1 classical supercomputer in 2023) [G+'23]
- .bold.darkblue[2025]: .darkblue[DWave solves] magnetic .darkblue[materials simulation] problems in<br/>20 min instead of 1 million years [K+25] (others disagree [W'25])
]

---
# Architectures of .darkblue[Emergent Hardware]
.reference[Extended from <a target="_blank" href="http://www.imprs-dynamics.mpg.de/pdfs/Plessl_talk.pdf">C. Plessl, Accelerating Scientific Computing with Massively Parallel Computer Architectures, IMPRS Winter School, Wroclaw, 2012</a>]

---
# <img src="../../img/keynotes/QuantumComputer.svg" align="top" width="10%"></img> .darkblue[Quantum Computer]

.reference[<a href="https://doi.org/10.1145/3410566.3410593" target="_blank">T. Bittner, S. Groppe. Avoiding blocking by scheduling transactions using quantum annealing. .em[IDEAS '20]</a>]

.more-more-condensed[
<table>
<tbody>
<tr>
<td style="width:65%">
<ul>
<li> use of .darkblue[quantum-mechanical phenomena such as superposition and entanglement] to perform computation</li>
<li> Different types of quantum computer, e.g.
  <ul><li> .darkblue[Universal Quantum Computer]
      <ul><li> uses <span class="darkblue">quantum logic gates</span> arranged in a circuit to do computation</li>
	      <li> <span class="darkblue">measurement</span> (sometimes called observation) assigns the observed variable to a single value</li>
      </ul></li>
  <li> .darkblue[Quantum Annealing]
      <ul><li> <span class="darkblue">metaheuristic for finding the global minimum</span> of a given objective function over a given set of candidate solutions</li>
          <li> i.e., some way to solve a special type of <span class="darkblue">mathematical optimization problem</span></li>
      </ul>
  </li>
</ul>
]
</td>
<td style="width:45%" class="textcenter">
<img src="../../img/qc/bloch/QCFullAdder.svg" align="top" class="fullwidth"></img><br/>
.smaller-font.darkblue[Quantum Circuit (Full Adder)]<br/>
<img src="../../img/ideas20/SimulatedVsQuantumAnnealing.svg" align="top" class="fullwidth"></img><br/>
.smaller-font.darkblue[Simulated versus Quantum Annealing]<br/>
</td>
</tr>
</tbody>
</table>

---
# .darkblue[Classical versus Quantum Computing]

.more-more-condensed.small-font[
<table class="result tablesmallpadding no-margin">
<tr>
<th style="width:15%"></th>
<th style="width:40%">Classical</th>
<th style="width:45%">Quantum</th>
</tr>
<tr>
<th>Information Unit</th>
<td style="vertical-align:top;">.bold[Bi]nary Digi.bold[t] (.darkblue[Bit]):
<ul>
<li>basis of a .darkblue[2-level system]</li>
<li>can be .darkblue[in state $0$ or $1$]</li>
</ul>
</td>
<td style="vertical-align:top;">.bold[Qu]antum .bold[Bit] (.darkblue[Qubit]):
<ul>
<li>basis of a .darkblue[2-level quantum system]</li>
<li>can be .darkblue[in state $|0\rangle$, $|1\rangle$ or in] a .darkblue[linear combination of both states]</li>
</ul>
</td>
</tr>
<tr>
<th>Operation</th>
<td style="vertical-align:top;">.darkblue[Logic Gate:]
<ul>
<li>performs .darkblue[on 1 or more bits to produce] a single .darkblue[bit output]</li>
</ul>
</td>
<td style="vertical-align:top;">.darkblue[Quantum Logic Gate:]
<ul>
<li>performs .darkblue[on 1 or more qubits to change the quantum state of] a .darkblue[single qubit]</li>
</ul>
</td>
</tr>
<tr>
<th>Example Operation</th>
<td style="vertical-align:top;">.darkblue[NOT]/Inverter:<br/>
<table><tbody><tr><td>Digital Circuit:</td><td><img src="../../img/qc/intro/ClassicalNotCircuit.svg" align="top" width="150em"></img></td></tr></tbody></table><br/>
<table style="text-align:center;line-height:0.9;" class="smaller-font" class="tablepadding">
<tbody>
<tr>
<td style="background-color:#d9ffb3">$In$</td>
<td style="background-color:#ffb3d1">$Out$</td>
</tr>
<tr>
<td style="background-color:#d9ffb3">$0$</td>
<td style="background-color:#ffb3d1">$1$</td>
</tr>
<tr>
<td style="background-color:#d9ffb3">$1$</td>
<td style="background-color:#ffb3d1">$0$</td>
</tr>
</tbody>
</table>

</td>
<td style="vertical-align:top;">.darkblue[NOT]/Pauli<sub>$x$</sub>-Gate:<br/>
<table class="no-padding no-margin"><tbody><tr><td>Quantum Circuit:</td><td><img src="../../img/qc/bloch/NOT.svg" align="top" width="150em"></img></td></tr>
<tr><td>Alternatively:</td><td><img src="../../img/qc/bloch/XGate.svg" align="top" width="150em"></img></td></tr></tbody></table>
<table style="text-align:center;line-height:0.9;" class="smaller-font" class="tablepadding">
<tbody>
<tr>
<td style="background-color:#d9ffb3">$In$</td>
<td style="background-color:#ffb3d1">$Out$</td>
</tr>
<tr>
<td style="background-color:#d9ffb3">$|0\rangle$</td>
<td style="background-color:#ffb3d1">$|1\rangle$</td>
</tr>
<tr>
<td style="background-color:#d9ffb3">$|1\rangle$</td>
<td style="background-color:#ffb3d1">$|0\rangle$</td>
</tr>
<tr>
<td style="background-color:#d9ffb3">$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$</td>
<td style="background-color:#ffb3d1">$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$</td>
</tr>
<tr>
<td style="background-color:#d9ffb3">$\frac{3\cdot i}{5}|0\rangle+\frac{4}{5}|1\rangle$</td>
<td style="background-color:#ffb3d1">$\frac{4}{5}|0\rangle+\frac{3\cdot i}{5}|1\rangle$</td>
</tr>
</tbody>
</table>

</td>
</tr>
</table>
]

---
# .darkblue[Digital versus Quantum Circuits]

.reference[<sup>1</sup>The dotted square marks a superfluous gate if uncomputation to restore the B output is not required. <a href="https://doi.org/10.1007%2Fbf01886518" target="_blank">[Feynman, 1986]</a>]

.more-more-condensed[
<table class="result">
<tr>
<th style="width:5%"></th>
<th style="width:45%">Digital Circuit</th>
<th style="width:55%">Quantum Circuit</th>
</tr>
<tr>
<th>.small-font[Building Blocks]</th>
<td class="textcenter">.small-font[Logic Gates]</td>
<td class="textcenter">.small-font[Quantum Logic Gates]</td>
</tr>
<tr>
<th>.small-font[Full Adder Example]</th>
<td class="textcenter"><img src="../../img/qc/bloch/Full_Adder_using_NAND_gates.svg" align="top" width="98%"></img><br/>.small-font[consists of NAND gates]</td>
<td class="textcenter"><img src="../../img/qc/bloch/QCFullAdder.svg" align="top" width="50%"></img><br/>.small-font[consists of Toffoli and CNOT gates<sup>1</sup>]</td>
</tr>
<tr>
<th>.small-font[In- and Output Full Adder]</th>
<td>

<table style="text-align:center;line-height:0.9;" class="smaller-font">
<tbody><tr>
<td colspan="3" style="background-color:#d9ffb3"><b>Inputs</b></td>
<td colspan="2" style="background-color:#ffb3d1"><b>Outputs</b>
</td></tr>
<tr>
<td style="background-color:#d9ffb3;width:3em;"><b>A</b></td>
<td style="background-color:#d9ffb3;width:3em;"><b>B</b></td>
<td style="background-color:#d9ffb3;width:3em;"><b>C</b><sub>in</sub></td>
<td style="background-color:#ffb3d1;width:3em;"><b>C</b><sub>out</sub></td>
<td style="background-color:#ffb3d1;width:3em;"><b>S</b>
</td></tr>
<tr>
<td style="background-color:#d9ffb3">0</td>
<td style="background-color:#d9ffb3">0</td>
<td style="background-color:#d9ffb3">0</td>
<td style="background-color:#ffb3d1">0</td>
<td style="background-color:#ffb3d1">0
</td></tr>
<tr>
<td style="background-color:#d9ffb3">0</td>
<td style="background-color:#d9ffb3">0</td>
<td style="background-color:#d9ffb3">1</td>
<td style="background-color:#ffb3d1">0</td>
<td style="background-color:#ffb3d1">1
</td></tr>
<tr>
<td style="background-color:#d9ffb3">0</td>
<td style="background-color:#d9ffb3">1</td>
<td style="background-color:#d9ffb3">0</td>
<td style="background-color:#ffb3d1">0</td>
<td style="background-color:#ffb3d1">1
</td></tr>
<tr>
<td style="background-color:#d9ffb3">0</td>
<td style="background-color:#d9ffb3">1</td>
<td style="background-color:#d9ffb3">1</td>
<td style="background-color:#ffb3d1">1</td>
<td style="background-color:#ffb3d1">0
</td></tr>
<tr>
<td style="background-color:#d9ffb3">1</td>
<td style="background-color:#d9ffb3">0</td>
<td style="background-color:#d9ffb3">0</td>
<td style="background-color:#ffb3d1">0</td>
<td style="background-color:#ffb3d1">1
</td></tr>
<tr>
<td style="background-color:#d9ffb3">1</td>
<td style="background-color:#d9ffb3">0</td>
<td style="background-color:#d9ffb3">1</td>
<td style="background-color:#ffb3d1">1</td>
<td style="background-color:#ffb3d1">0
</td></tr>
<tr>
<td style="background-color:#d9ffb3">1</td>
<td style="background-color:#d9ffb3">1</td>
<td style="background-color:#d9ffb3">0</td>
<td style="background-color:#ffb3d1">1</td>
<td style="background-color:#ffb3d1">0
</td></tr>
<tr>
<td style="background-color:#d9ffb3">1</td>
<td style="background-color:#d9ffb3">1</td>
<td style="background-color:#d9ffb3">1</td>
<td style="background-color:#ffb3d1">1</td>
<td style="background-color:#ffb3d1">1
</td></tr>
</tbody></table>

</td>
<td style="line-height:0.9;">

.small-font[.bold[$|0\rangle$ and $|1\rangle$ as input:] Output is $|0\rangle$ and $|1\rangle$ analogous to digital circuit.<br/>.bold[Superpositions as input:] Superpositions as output with corresponding probabilities for basic quantum states, e.g.:<br/>]

<table style="text-align:center;line-height:0.9;" class="smaller-font">
<tbody>
<tr>
<td style="background-color:#d9ffb3;"><b>A</b></td>
<td style="background-color:#d9ffb3;"><b>B</b></td>
<td style="background-color:#d9ffb3;width:3em;"><b>C</b><sub>in</sub></td>
<td style="background-color:#ffb3d1;width:3em;"><b>C</b><sub>out</sub></td>
<td style="background-color:#ffb3d1;"><b>S</b>
</td></tr>
<tr>
<td style="background-color:#d9ffb3">$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$</td>
<td style="background-color:#d9ffb3">$|0\rangle$</td>
<td style="background-color:#d9ffb3">$|0\rangle$</img></td>
<td colspan="2" style="background-color:#ffb3d1">.bold[ABC<sub>out</sub>S]: $\frac{1}{\sqrt{2}}(|0000\rangle+|1001\rangle)$</td>
</tr>
<tr>
<td style="background-color:#d9ffb3">$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$</td>
<td style="background-color:#d9ffb3"><img src="../../img/qc/bloch/AntiCorrelatedBellState.svg" align="top" width="80em"></img></td>
<td style="background-color:#d9ffb3">$|0\rangle$</img></td>
<td style="background-color:#ffb3d1">$|0\rangle$</td>
<td style="background-color:#ffb3d1">$|1\rangle$</td>
</tr>
</tbody></table>

</td>
</tr>
</table>
]

---
# .darkblue[Timeline of Quantum Computers]
.reference[<a href="https://en.wikipedia.org/wiki/List_of_quantum_processors" target="_blank">Main Data Source</a> Roadmaps: <a href="https://research.ibm.com/blog/ibm-quantum-roadmap" target="_blank">IBM</a> <a href="https://www.computerwoche.de/a/2025-kommt-der-erste-4-000-qubit-quantencomputer" target="_blank">IBM'25</a> <a href="https://quantumai.google/research" target="_blank">Google</a> <a href="https://doi.org/10.1007/s42354-021-0342-8" target="_blank">Quantum Brilliance</a> <a href="https://www.dwavesys.com/media/xvjpraig/clarity-roadmap_digital_v2.pdf" target="_blank">D-Wave</a> <a href="https://education.jlab.org/qa/mathatom_05.html" target="_blank">#Atoms on Earth</a> <a href="https://www.popularmechanics.com/space/a27259/how-many-particles-are-in-the-entire-universe/" target="_blank">#Particles in Universe</a>]

---
# .darkblue[Noisy Intermediate-Scale Quantum (NISQ)]
.reference[<a target="_blank" href="https://doi.org/10.22331/q-2018-08-06-79">[P'18]</a>]

.more-more-condensed.no-margin.darkblue[
<div class="textcenter">
<img src="../../img/qc/intro/NISQ.svg" align="top" width="70%"></img>
</div>

- quantum computers with 50-100 qubits: noise in quantum gates limits the size of quantum circuits that can be executed reliably
- Such NISQ devices may be able to perform tasks which surpass the capabilities of today’s classical digital computers with application areas like quantum chemistry, optimization & machine learning
- 100-qubit quantum computers are (only) intermediate technologies
]

---
# .darkblue[Suspected Shape of BQP]

.reference[Michael Nielsen and Isaac Chuang. Quantum Computation and Quantum Information. Cambridge University Press. ISBN 0-521-63503-9. 2000]

.more-mode-condensed.small-font.no-margin[
<table>
<tbody>
<tr>
<td style="width:40%">
<img src="../../img/qc/intro/BQP_complexity_class_diagram.svg" align="top" class="fullwidth"></img>
</td>
<td style="width:60%">
<ul>
<li><span class="darkblue">PSPACE</span> is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space</li>
<li><span class="darkblue">NP: nondeterministic polynomial time</span> is the set of problems that can be solved in polynomial time by a nondeterministic Turing machine</li>
<li>A problem is said to be <span class="darkblue">NP-hard</span> if everything in NP can be transformed in polynomial time into it even though it may not be in NP</li>
</ul>
</td>
</tbody>
</table>
<ul>
<li>A problem is <span class="darkblue">NP-complete</span> if it is both in NP and NP-hard</li>
<li><span class="darkblue">P: PTIME</span> contains all decision problems that can be solved by a deterministic Turing machine in polynomial time</li>
<li><span class="darkblue">BQP: bounded-error quantum polynomial time</span> is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances</li>
<li>Note: .darkblue[Neither P≠NP nor P≠PSPACE has been proven], and if either of these are equal, the shape of BQP would be different</li>
</ul>
]

---
# .darkblue[Potential of Quantum Algorithms]

.more-more-condensed[
- <a href="https://quantumalgorithmzoo.org/" target="_blank">Quantum Algorithm Zoo</a> .darkgray[as example of collection of .em[important] quantum algorithms]
<table class="result tablesmallpadding small-margin-bottom">
<tbody>
<tr>
<th>Covered Years</th>
<td>1974-today</td>
</tr>
<tr>
<th>#Investigated References</th>
<td>430 .darkgray.smaller-font[(visited on October 2021)]</td>
</tr>
<tr>
<th>#Algorithms</th>
<td>.bold.darkred[64]</td>
</tr>
<tr>
<th>Speedups</th>
<td>
.darkred.bold[Superpolynomial: 31],<br/>
.darkred[Polynomial: 27],<br/>
.darkblue[Constant factor]: 1, .darkblue[Varies]: 3, .darkblue[Various]: 1,<br/>
.darkblue[Unknown]: 1
</td></tr>
</tbody>
</table>
.legend.small-font[.bold[Terminology:]<br/>
$\alpha$: positive constant<br/>
$C(n)$: runtime of the best known classical algorithm<br/>
$Q(n)$: runtime of the quantum algorithm<br/>
Superpolynomial Speedup: $C = 2^{\Omega(Q^\alpha)}$<br/>
Polynomial Speedup: otherwise
]
]

---
# Very Important Quantum Algorithms 1/2: .darkblue[Shor]'s Algorithm<sup>1</sup>

.reference[<sup>1</sup> <a href="https://doi.org/10.1109/SFCS.1994.365700" target="_blank">[Shor, 1994]</a> <sup>2</sup> <a href="https://doi.org/10.1103%2FPhysRevA.54.1034" target="_blank">[Beckman et al., 1996]</a>]

.more-more-condensed.no-margin[
<table>
<tbody>
<tr style="vertical-align:top">
<td style="width:32%" style="vertical-align:top;"><img src="../../img/qc/intro/FactorizationIsHard.svg" align="top" style="vertical-align:top;" class="fullwidth"></img></td>
<td style="width:68%">
<ul>
<li>.darkblue[factoring integers in polynomial time]
  <ul><li>.darkblue[Depth of quantum circuit<sup>2</sup>] to factor integer $N$: $O((log\ N)^2 (log\ log\ N) (log\ log\ log\ N))$</li>
  <li>superpolynomial speedup, i.e., .darkblue[almost exponentially faster than] the most efficient known .darkblue[classical factoring algorithm] (general number field sieve): $O(e^{1.9(log\ N)^{\frac{1}{3}}(log\ log\ N)^{\frac{2}{3}}})$</li>
  </ul>
</li>
</ul>
</td>
</tr>
</tbody>
</table>
- Important for cryptography .darkblue[$\rightarrow$ Post-Quantum Cryptography]
- Most quantum algorithms with .darkblue[superpolynomial speedup] like Shor's algorithm .darkblue[are based on quantum Fourier transforms] .darkgray[(quantum analogue of inverse discrete Fourier transform)]
]

---
# Very Important Quantum Algorithms 2/2: .darkblue[Grover]'s Search Algorithm

.reference[<a href="https://doi.org/10.1145/237814.237866" target="_blank">Lov K. Grover. A fast quantum mechanical algorithm for database search. .em[STOC'96], 1996</a>]

.more-more-condensed[
- .darkblue[Black box] function (oracle) $f:${$0,...,2^b -1$} $\mapsto$ {$true, false$}
- .darkblue[Grover's search] algorithm finds one $x \in${$0,...,2^b -1$},<br/>such that $f(x)=true$
  - if there is only .darkblue[one solution]: $\frac{\pi}{4}\cdot\sqrt{2^b}$ basic steps each of which calls $f$<br/>Let $f'(b)$ be runtime complexity of $f$ for testing $x$ to be true:<br/>.darkblue[$\Rightarrow O(\sqrt{2^b}\cdot f'(b))$]
  - if there are .darkblue[$k$] possible .darkblue[solutions]: .darkblue[$O(\sqrt{\frac{2^b}{k}}\cdot f'(b))$]
- .darkblue[Basis of many other quantum algorithms and applications]
]

---
# .darkblue[Algorithms] .small-font30[(used e.g. in Query Optimization)] .darkblue[and their Quantum Counterparts]

.more-more-condensed.small-font[
<table class="result tablesmallpadding margin-bottom">
<tbody>
<tr>
<th>Query Optimization Approach</th>
<th>Basic Algorithm</th>
<th>Quantum Computing Counterpart</th>
</tr>
<tr>
<td><a href="https://doi.org/10.1145/582095.582099" target="_blank">[S+79]</a></td>
<td>Dynamic Programming <a href="https://doi.org/10.1038/nbt0704-909" target="_blank">[E04]</a></td>
<td><a href="https://arxiv.org/pdf/1906.02229.pdf" target="_blank">[R19]</a> <a href="https://epubs.siam.org/doi/pdf/10.1137/1.9781611975482.107" target="_blank">[A+19]</a></td>
</tr>
<tr>
<td><a href="https://doi.org/10.1145/38713.38722" target="_blank">[IW87]</a>, QA: <a href="https://doi.org/10.14778/2947618.2947621" target="_blank">[TK16]</a></td>
<td>Simulated Annealing <a href="https://doi.org/10.1126%2Fscience.220.4598.671" target="_blank">[KGV83]</a></td>
<td><a href="https://doi.org/10.1038/nature10012" target="_blank">[J+11]</a></td>
</tr>
<tr>
<td><a href="https://doi.org/10.1145/3211954.3211957" target="_blank">[MP18]</a> <a href="https://doi.org/10.1109/ICDE48307.2020.00116" target="_blank">[Y+20]</a> <a href="https://doi.org/10.1145/3329859.3329875" target="_blank">[W+19]</a> <a href="http://arxiv.org/abs/1905.06425" target="_blank">[O+19]</a></td>
<td>Reinforcement Learning <a href="https://doi.org/10.1007/BF00453370" target="_blank">[BSB81]</a></td>
<td><a href="https://doi.org/10.1038/s41586-021-03242-7" target="_blank">[S+21]</a> <a href="https://doi.org/10.1007/11539117_97" target="_blank">[DCC05]</a></td>
</tr>
<tr>
<td><a href="http://www.vldb.org/conf/1994/P085.PDF" target="_blank">[GPK94]</a></td>
<td>Random Walk <a href="https://books.google.com/books?hl=de&lr=&id=WnSoBtMiU2oC&oi=fnd&pg=PA1&dq=M.N.+Barber,+B.W.+Ninham,+Random+and+Restricted+Walks:+Theory+and+Applications,++Gordon+and+Breach,+New+York,+1970&ots=AxF58L4AL1&sig=PArEXcaS26QhyTZiYKPc5ekxEsA" target="_blank">[BN70]</a></td>
<td><a href="https://doi.org/10.1103/PhysRevA.48.1687" target="_blank">[ADZ93]</a> <a href="https://doi.org/10.1145/380752.380757" target="_blank">[A+01]</a></td>
</tr>
<tr>
<td><a href="https://minds.wisconsin.edu/bitstream/handle/1793/59438/TR1004.pdf?sequence=1" target="_blank">[BFI91]</a></td>
<td>Genetic Algorithm <a href="http://dx.doi.org/10.1038/scientificamerican0792-66" target="_blank">[H92]</a></td>
<td><a href="https://doi.org/10.1155/2013/730749" target="_blank">[W+13]</a></td>
</tr>
<tr>
<td><a href="https://doi.org/10.1007/978-981-13-2285-3_45" target="_blank">[TC19]</a></td>
<td>Ant Colony Optimization <a href="https://www-public.imtbs-tsp.eu/~gibson/Teaching/Teaching-ReadingMaterial/ColorniDorigoManiezzo91.pdf" target="_blank">[CDM91]</a> <a href="https://doi.org/10.1109/MCI.2006.329691" target="_blank">[DBS06]</a></td>
<td><a href="https://doi.org/10.1007/978-3-540-74769-7_31" target="_blank">[WNF07]</a> <a href="https://arxiv.org/abs/2010.07413" target="_blank">[G+20]</a></td>
</tr>
</tbody>
</table>
.legend.darkgray[This list is not complete...]
]

---
# .darkblue[Open Challenges] for QC for Databases

.more-more-condensed[
- .darkblue[Are QC counterparts] of basic algorithms used in query optimizations .darkblue[suitable for speeding up databases?]
- .darkblue[What should be the properties of a quantum computer] (e.g. #qubits, latencies of gates) to achieve certain speedups.darkblue[?]
- .darkblue[How to combine classical and quantum computing algorithms to achieve good speedups with few qubits?]<br/>.darkgray[(...for running database optimizations on current available quantum computers...)]
- .darkblue[What other database domains] besides query optimization benefit from quantum computers.darkblue[?]<br/>.darkgray[(In short: those based on mathematical optimization problems, but also other...?)]
]

---
# .darkblue[QC4DB: .small-font36[Accelerating Relational Database Management Systems via Quantum Computing]]

.more-more-condensed[
- <a href="https://www.quantentechnologien.de/forschung/foerderung/anwendungsnetzwerk-fuer-das-quantencomputing/qc4db.html" target="_blank">Project Website@Quantentechnologien</a>
- Project .darkblue[funded by BMBF]
  - Duration 3 years, 1.8M Euros
- Topics<ul><li>.darkblue[Query Optimization]</li><li>.darkblue[Optimizing Transaction Schedules]</li></ul>of an open source relational database management system
- Partners
  - .darkblue[University of Lübeck] (Coordinator Sven Groppe)
      - Hardware-Acceleration of Databases
	  - Website: <a href="https://www.ifis.uni-luebeck.de/~groppe/" target="_blank">https://www.ifis.uni-luebeck.de/~groppe/</a>
  - .darkblue[Quantum Brilliance] GmbH
      - Room Temperature Diamond Quantum Accelerators
	  - Website: <a href="https://quantumbrilliance.com/" target="_blank">https://quantumbrilliance.com/</a>
]

---
# .darkblue[Topics of this Lecture]

.more-more-condensed.darkblue.small-font22[
<ol>
<li>Introduction to Quantum Computing</li>
<li>Introduction to the Bloch Sphere</li>
<li>Quantum Logic Gates</li>
<li>Qiskit and Deutsch-Jozsa Algorithm</li>
<li>Introduction to Silq</li>
<li>Grover's Search</li>
<li>Quantum Annealing Versus Grover's Search: Optimizing Transaction Schedules</li>
<li>Quantum Data Encoding Patterns</li>
<li>Introduction into Quantum Machine Learning: Data Encoding, Model, Measurement</li>
<li>Quantum Machine Learning: Optimization</li>
<li>Quantum Cryptography: Shor, Quantum Key Distribution</li>
<li>Quantum Error Correction</li>
<li>Summary and What Else?</li>
</ol>
]

---
# .darkblue[Summary and Conclusions]

.more-more-condensed.darkblue[
 - Quantum Mechanics $\Rightarrow$ Quantum Computers
  - Universal Quantum Computing versus Quantum Annealing
- Quantum Computing as Hardware Accelerator
- Classical versus Quantum Computing
  - Bits versus Qubits
  - (Quantum) Logic Gate
  - Digital Circuits versus Quantum Circuits
- Timeline of Quantum Computers
- Noisy Intermediate-Scale Quantum (NISQ) technologies
- Bounded-error quantum polynomial time problems
- Potential of Quantum Algorithms
  - Shor (superpolynomial speedup)
  - Grover (quadratic speedup)
- Quantum Computing for databases
]