Title

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<slide-template title="Quantum Computing" subtitle="Quantum Data Encoding Patterns"></slide-template>

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<title-slide titletype="Lecture" title="Quantum Computing" modulenumber="(CS5070)" subtitle="Quantum Data Encoding Patterns"></title-slide>

---
# Motivation .darkblue[Design Patterns]

.reference[Software Design Pattern: <a href="https://doi.org/10.1145/62139.62151" target="_blank">Panel</a> <a href="https://www.worldcat.org/title/design-patterns-elements-of-reusable-object-oriented-software/oclc/961356420" target="_blank">Book</a> <a href="https://refactoring.guru/design-patterns" target="_blank">Web</a> <a href="https://sourcemaking.com/design_patterns" target="_blank">Web</a> Quantum Patterns: <a href="https://doi.org/10.1007/978-3-030-14082-3_19" target="_blank">Paper</a> <a href="https://arxiv.org/abs/1906.03082" target="_blank">ArXiv</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Web</a> Hybrid Quantum: <a href="https://doi.org/10.1007/978-3-030-87568-8_2" target="_blank">Paper</a> <a href="https://www.iaas.uni-stuttgart.de/publications/Weigold2021_HybridPatterns.pdf" target="_blank">Preprint</a>]

.more-more-condensed[
- .darkblue[Software design patterns]
  - .darkblue[structured document containing a]n abstract .darkblue[description of a] proven .darkblue[solution of a recurring problem]
  - .darkblue[refers to other patterns] that may jointly contribute to an encompassing solution of a complex problem<br/>.darkblue[$\Rightarrow$] network of related patterns, i.e. a .darkblue[pattern language]
  - .darkblue[.bold[Example:] Singleton], Definition: <a href="https://refactoring.guru/design-patterns/singleton" target="_blank">Link 1 </a> <a href="https://sourcemaking.com/design_patterns/singleton" target="_blank">Link 2 </a>
     - <span class="darkblue">creational design pattern</span> that lets you ensure <span class="darkblue">that a class has only one instance</span>, while providing a global access point to this instance
	 - .darkblue[.bold[Solution]]
	     - Make the .darkblue[default constructor private], to prevent other objects from using the new operator with the Singleton class
	     - Create a .darkblue[static creation method that acts as] a .darkblue[constructor]. Under the hood, this method calls the private constructor to create an object and saves it in a static field. All following calls to this method return the cached object.
- .darkblue[Quantum Patterns]
  - .darkblue[Design patterns using quantum algorithms]

]

---
# .darkblue[Pattern Format]

.more-more-condensed[
- .darkblue[differs depending on the domain] 
- .darkblue[Format] example:
  - .darkblue[Name and icon] that serves as a graphical representation of the pattern
  - .darkblue[Intent] that briefly summarizes the purpose of the pattern
  - .darkblue[Alias]: other used names of the pattern
  - .darkblue[Context]: problem and the circumstances of the pattern
  - .darkblue[Forces]: trade-offs or considerations that must be taken into account for solving the problem
  - .darkblue[Solution]: description in an abstract manner and often visualized by a solution sketch
  - .darkblue[Result]: consequences of the solution
  - optional section for .darkblue[variants] of the pattern
  - .darkblue[Related patterns]: Connections between patterns, as patterns are often applied in combination or solve similar problems
  - .darkblue[Known uses] of the pattern in quantum algorithms and concrete implementations
]

---
# .darkblue[Typical Structure of] a .darkblue[Quantum Calculation]

.more-more-condensed[
- Typical .darkblue[phases of quantum calculations]:
]

.more-more-condensed[
- .darkblue.bold[How to prepare the states for data loading?]
]

---
# <a href="https://quantumcomputingpatterns.org/#/patterns/15" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/initialization_icon.png" align="center" style="width:2em"></img></a> .darkblue[Initialization aka State Preparation] 1/2

.more-more-condensed[
- .darkblue[Intent]
  - .darkblue[Initialize the input of a quantum register], taking into account the prerequisites of the subsequent steps of the algorithm
- .darkblue[Context]
  - .darkblue[Specific parameters must be typically given as input data] to a quantum algorithm in order to solve a given problem
  - .darkblue[The first unitary transformations of a quantum circuit] typically .darkblue[encode] the .darkblue[input data] into the quantum register according to a defined encoding
- .darkblue[Solution]
  - .darkblue[.smaller-font[$\left|0\cdots 0\right\rangle$] and .smaller-font[$\left|0\cdots 01\right\rangle$] are frequently used as initialization] of quantum registers
  - .darkblue[Some qubits as ancilla bits], which may be used for the storage of intermediate results or quantum error correction
  - Example
      - for a function table of a Boolean function $f:\\{0, 1\\}^n \rightarrow \\{0, 1\\}^m$, the overall register is initialized as $\left| 0 \right\rangle^{\otimes n}\left| 0 \right\rangle^{\otimes m}$ (including $m$ ancilla bits)
]

.reference[Data Encoding: <a href="https://doi.org/10.1109/ICSA-C52384.2021.00025" target="_blank">Paper</a> <a href="https://www.iaas.uni-stuttgart.de/publications/Weigold2021_ExpandingDataEncodingPatterns.pdf" target="_blank">Preprint</a> <a href="https://doi.org/10.1049/qtc2.12032" target="_blank">Journal Paper</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Web</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Source of Icons: QC Patterns</a> <sup>1</sup> <a href="https://arxiv.org/abs/1803.01958" target="_blank">ArXiv</a> <sup>2</sup> <a href="https://doi.org/10.1119/1.1463744" target="_blank">Paper</a> <sup>3</sup> <a href="https://arxiv.org/abs/1802.08227" target="_blank">ArXiv</a> <sup>4</sup> <a href="https://arxiv.org/abs/1603.08675" target="_blank">ArXiv</a>]

.more-more-condensed[
- .darkblue[Result]
  - .darkblue[Preparing more advanced states] built on the previously described initialization techniques
  - .darkblue[Examples of loading into quantum registers]
     - <span class="darkblue">Loading classical bits</span><sup>1</sup>
     - <span class="darkblue">Loading of complex vectors</span><sup>2</sup>
     - <span class="darkblue">Loading of real-valued vectors</span><sup>3</sup>
     - <span class="darkblue">Loading of matrices</span> that are represented as sets of vectors<sup>4</sup>
- .darkblue[Related Patterns]
  - <a href="https://quantumcomputingpatterns.org/#/patterns/16" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/equal_superposition_icon.png" align="center" style="width:2em"></img></a> .darkblue[Uniform superposition]: <img src="../../img/qc/patterns/H_n.svg" align="center" style="width:15em"></img>
  - .darkblue[Refinements] of initialization: <a href="https://quantumcomputingpatterns.org/#/patterns/0" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/basis_encoding_icon.png" align="center" style="width:2em"></img></a> <a href="https://quantumcomputingpatterns.org/#/patterns/3" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/angle_encoding_icon.png" align="center" style="width:2em"></img></a> <a href="https://quantumcomputingpatterns.org/#/patterns/1" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/quam_icon.png" align="center" style="width:2em"></img></a> <a href="https://quantumcomputingpatterns.org/#/patterns/2" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/amplitude_encoding_icon.png" align="center" style="width:2em"></img></a> <a href="https://quantumcomputingpatterns.org/#/patterns/4" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/qram_icon.png" align="center" style="width:2em"></img></a> <a href="https://quantumcomputingpatterns.org/#/patterns/12" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/schmidt_icon.png" align="center" style="width:2em"></img></a>
  - Initialized register may be used to compute a <a href="https://quantumcomputingpatterns.org/#/patterns/18" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/function_table_icon.png" align="center" style="width:2em"></img></a> .darkblue[function table]<br/>
.blue-border.round-border.small-font.small-padding[For other patterns we do not present all items...]
]

---
# .darkblue[Data Encoding]

.more-more-condensed[
- .darkblue[define how data is represented by the state] of a quantum system
- .darkblue[Data encoding patterns] describe a particular encoding .darkblue[as] a .darkblue[trade-off] between:
  - .darkblue[Minimize \#qubits] needed for the encoding
     - because <span class="darkblue">current devices</span> are of intermediate size and thus only <span class="darkblue">support a limited \#qubits</span>
  - .darkblue[Minimize \#parallel operations] needed to realize the encoding 
     - to <span class="darkblue">minimize the width</span> of the quantum circuit
	 - the .darkblue[loading routine is] ideally .darkblue[of constant or logarithmic complexity]
  - .darkblue[Data must be represented] in a suitable manner .darkblue[for further calculations], e.g., arithmetic operations
- .darkblue[Patterns] for Data Encoding: <a href="https://quantumcomputingpatterns.org/#/patterns/0" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/basis_encoding_icon.png" align="center" style="width:2em"></img></a> <a href="https://quantumcomputingpatterns.org/#/patterns/3" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/angle_encoding_icon.png" align="center" style="width:2em"></img></a> <a href="https://quantumcomputingpatterns.org/#/patterns/1" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/quam_icon.png" align="center" style="width:2em"></img></a> <a href="https://quantumcomputingpatterns.org/#/patterns/2" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/amplitude_encoding_icon.png" align="center" style="width:2em"></img></a> <a href="https://quantumcomputingpatterns.org/#/patterns/4" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/qram_icon.png" align="center" style="width:2em"></img></a> <a href="https://quantumcomputingpatterns.org/#/patterns/12" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/schmidt_icon.png" align="center" style="width:2em"></img></a>
]

---
# <a href="https://quantumcomputingpatterns.org/#/patterns/0" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/basis_encoding_icon.png" align="center" style="width:2em"></img></a> .darkblue[Basis Encoding]

.reference[Data Encoding: <a href="https://doi.org/10.1109/ICSA-C52384.2021.00025" target="_blank">Paper</a> <a href="https://www.iaas.uni-stuttgart.de/publications/Weigold2021_ExpandingDataEncodingPatterns.pdf" target="_blank">Preprint</a> <a href="https://doi.org/10.1049/qtc2.12032" target="_blank">Journal Paper</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Web</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Source of Icons: QC Patterns</a> <a href="https://doi.org/10.1088/2058-9565/abae7d" target="_blank">[LB'20]</a> <a href="https://doi.org/10.1103/PhysRevA.54.147" target="_blank">[VBE'96]</a> <a href="https://arxiv.org/abs/quant-ph/9511018v1" target="_blank">ArXiv</a> <a href="https://arxiv.org/abs/1803.01958" target="_blank">[CB'18]</a>]

.more-more-condensed[
- .darkblue[Intent]
  - .darkblue[Represent data in a quantum computer] for performing calculations
- .darkblue[Context]
  - .darkblue[Quantum algorithm requires numerical input data] .small-font[$X$] for further calculations
- .darkblue[Solution]
  - use the .darkblue[computational basis $|0...00\rangle, |0...01\rangle, \ldots, |1...11\rangle$]
     - <span class="darkblue">For example:</span> decimal $2$ in binary format $10$ encoded by $|10\rangle$
     - <span class="darkblue">Classical data: integer number $x := b_{n-1}\ldots b_1 b_0$</span> with $n$ bits $b_i$ 
	 - <span class="darkblue">Corresponding quantum data: $|x \rangle:=|b_{n-1} \ldots b_1 b_0\rangle$</span>
- .darkblue[Result]
  - .darkblue[suitable for arithmetic computations] .smaller-font[[LB'20][VBE'96][CB'18]] .darkblue[$\Rightarrow$ digital encoding]
  - .darkblue[Space requirements: $n$ qubits]
  - Initial .darkblue[$|0\rangle$ state] of qubits that represent a $1$ bit .darkblue[must be flipped into $|1\rangle$]<br/>.darkblue[$\Rightarrow O(n)$ NOT-gates in parallel $\Rightarrow O(1)$ preparation time]
]

---
# <a href="https://quantumcomputingpatterns.org/#/patterns/1" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/quam_icon.png" align="center" style="width:2em"></img></a> .darkblue[Quantum Associative Memory (QuAM)]

.reference[Data Encoding: <a href="https://doi.org/10.1109/ICSA-C52384.2021.00025" target="_blank">Paper</a> <a href="https://www.iaas.uni-stuttgart.de/publications/Weigold2021_ExpandingDataEncodingPatterns.pdf" target="_blank">Preprint</a> <a href="https://doi.org/10.1049/qtc2.12032" target="_blank">Journal Paper</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Web</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Source of Icons: QC Patterns</a> <a href="https://doi.org/10.1088/2058-9565/abae7d" target="_blank">[LB'20]</a> <a href="https://doi.org/10.1016/S0020-0255(99)00101-2" target="_blank">[VM'00]</a> <a href="https://arxiv.org/abs/quant-ph/9807053" target="_blank">ArXiv</a>]

.more-more-condensed[
- .darkblue[Context]
  - A quantum algorithm requires .darkblue[multiple numerical values .small-font[$X:=${$x_1,\ldots,x_k$}]] as input for further calculations.
- .darkblue[Solution]
  - Use a .darkblue[quantum associative memory (QuAM)] [VM'00] .darkblue[to prepare a superposition of basis encoded values in the same qubit register] [LB'20]<br/>.darkblue[$\Rightarrow$] quantum register is an .darkblue[equally weighted superposition .small-font[$\frac{1}{\sqrt{k}}\sum_{i=1}^k |x_i\rangle$] of all basis encoded values .small-font[$|x_1\rangle,\ldots,|x_k\rangle$]]<br/>
<table>
<tbody>
<tr>
<td><img src="../../img/qc/patterns/QuAM.svg" align="center" style="width:14.2em"></img>
</td>
<td>
.small-font[.darkblue[Basic steps] .smaller-font[[VM'00] (using modified parts of Grover)].darkblue[:].smaller-font[<br/>
.darkblue[(1) Additional element is prepared] in both branches.<br/> 
.darkblue[(2)] Processing branch is split in such a manner, that the new .darkblue[element gets a proper amplitude] such that<br/>(3) it can be brought .darkblue[into superposition with the already added elements].<br/>.darkblue[(4) Uncompute cleans for the next iteration].]]
</td>
</tr>
</tbody>
</table>
]

.reference[Data Encoding: <a href="https://doi.org/10.1109/ICSA-C52384.2021.00025" target="_blank">Paper</a> <a href="https://www.iaas.uni-stuttgart.de/publications/Weigold2021_ExpandingDataEncodingPatterns.pdf" target="_blank">Preprint</a> <a href="https://doi.org/10.1049/qtc2.12032" target="_blank">Journal Paper</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Web</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Source of Icons: QC Patterns</a> <a href="https://doi.org/10.1088/2058-9565/abae7d" target="_blank">[LB'20]</a> <a href="https://doi.org/10.1016/S0020-0255(99)00101-2" target="_blank">[VM'00]</a> <a href="https://arxiv.org/abs/quant-ph/9807053" target="_blank">ArXiv</a> <a href="http://dx.doi.org/10.1007/978-3-319-96424-9" target="_blank">[SP'18]</a>]

.more-more-condensed[
- .darkblue[Result]
  - .darkblue[digital encoding and] therefore .darkblue[suitable for arithmetic computations] [LB'20]
  - .darkblue[Space:] For $k$ input numbers each with $n$ bits, .darkblue[$n$ qubits] are needed
  - Each of encoded input values is represented by a basis vector with an amplitude of $\frac{1}{\sqrt{k}}$ and all other amplitudes of the register are $0$<br/>$\Rightarrow$ The .darkblue[amplitude vector] therefore .darkblue[often sparse] [SP'18]
  - .darkblue[Preparation time depends on number $k$] of input numbers 
]

---
# <a href="https://quantumcomputingpatterns.org/#/patterns/3" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/angle_encoding_icon.png" align="center" style="width:2em"></img></a> .darkblue[Angle Encoding] 1/3

.reference[Data Encoding: <a href="https://doi.org/10.1109/ICSA-C52384.2021.00025" target="_blank">[WBLS'21]</a> <a href="https://www.iaas.uni-stuttgart.de/publications/Weigold2021_ExpandingDataEncodingPatterns.pdf" target="_blank">Preprint</a> <a href="https://doi.org/10.1049/qtc2.12032" target="_blank">Journal Paper</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Web</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Source of Icons: QC Patterns</a> <a href="https://doi.org/10.1088/2058-9565/abae7d" target="_blank">[LB'20]</a> <a href="https://doi.org/10.1103/PhysRevA.102.032420" target="_blank">[LC'20]</a> <a href="https://arxiv.org/abs/2003.01695" target="_blank">ArXiv</a>]

.more-more-condensed[
- .darkblue[Intent]
  - "Represent each data point by a separate qubit" [WBLS'21]
- .darkblue[Alias]
  - .darkblue[Qubit Encoding]: since each qubit represents a single data point [LC'20]
  - .darkblue[(Tensor) Product Encoding]: since the resulting encoding of this pattern is not entangled [LB'20]
- .darkblue[Context]
  - .darkblue[Encoding is efficient in terms of operations to perform] more operations .darkblue[within the decoherence time of] noisy intermediate-scale quantum computers (.darkblue[NISQ]) after encoding the data
]

.more-more-condensed[
- .darkblue[Solution] [LB'20]
<ol><li>Each data point of the .darkblue[input is normalized to the interval $[0,\frac{\pi}{2}]$]</li>
<li>.darkblue[Encoding] the data points .darkblue[by rotating around the y-axis] with an angle depending on the value of the normalized data point</li></ol>
<img src="../../img/qc/patterns/AngleEncoding.svg" align="center" class="fullwidth"></img>
]

.more-more-condensed[
- .darkblue[Result]
  - .darkblue[Space requirements: $n+1$ qubits for $n+1$ data points]
  - Initial .darkblue[$|0\rangle$ state] of qubits that represent a $1$ data point .darkblue[must be rotated according to data point]<br/>.darkblue[$\Rightarrow O(n)$ Rotation-gates in parallel $\Rightarrow O(1)$ preparation time]
- .darkblue[Variants]
  - [LC'20] proposes to .darkblue[exploit the relative phase for a more dense encoding which requires only half of the qubits] for the same amount of data points
- .darkblue[Known Uses]
  - .darkblue[Classification algorithms] [LC'20,G+'18] based on angle encoding
  - .darkblue[Quantum image processing] <a href="https://doi.org/10.1007/s11128-015-1195-6" target="_blank">[YIV'15] </a>: angle encoding for a pixel's color information in the flexible representation of quantum image (FRQI) and an additional register for the position
  - In .darkblue[quantum neural networks] <a href="https://doi.org/10.1007/s11128-014-0809-8" target="_blank">[SSP14] </a> quantum neurons (quron) use angle encoding
]

---
# <a href="https://quantumcomputingpatterns.org/#/patterns/2" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/amplitude_encoding_icon.png" align="center" style="width:2em"></img></a> .darkblue[Amplitude Encoding]

.more-more-condensed[
- .darkblue[Intent]
  - Encode .darkblue[data in a compact manner that do not require calculations]
- .darkblue[Alias]
  - .darkblue[Wavefunction Encoding] [LC'20]
      - Every quantum system is described by its wavefunction $\psi$ defining also the measurement probabilities<br/>$\Rightarrow$ amplitudes of the quantum system represent data values
- .darkblue[Context]
  - Encoding of a .darkblue[numerical input data vector $(x_0,...,x_n)^T$] for a quantum algorithm.
]

.more-more-condensed[
- .darkblue[Solution]
  - .darkblue[Encoding of the input vector in the amplitudes] of the quantum state: .darkblue[$|\psi\rangle=\sum_{i=0}^n x_i|i\rangle$]
     - Squared moduli of the amplitudes of a quantum state must sum up to 1<br/>$\Rightarrow$ <span class="darkblue">input vector needs to be normalized to length 1</span><br/>
	 <img src="../../img/qc/patterns/AmplitudeEncoding.svg" align="center" style="width:50%"></img>
     - Vector space of an $n$ qubit register has dimension $2^n$<br/>$\Rightarrow$ <span class="darkblue">input vector can be padded with additional zeros if dimension is not a power of 2</span> 
     - Amplitudes depend on the data $\Rightarrow$ <span class="darkblue">process of encoding the data</span> <span class="darkgray">(but not the encoding itself)</span> <span class="darkblue">is often referred to as <span class="em">arbitrary state preparation</span></span>
]

.reference[Data Encoding: <a href="https://doi.org/10.1109/ICSA-C52384.2021.00025" target="_blank">Paper</a> <a href="https://www.iaas.uni-stuttgart.de/publications/Weigold2021_ExpandingDataEncodingPatterns.pdf" target="_blank">Preprint</a> <a href="https://doi.org/10.1049/qtc2.12032" target="_blank">Journal Paper</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Web</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Source of Icons: QC Patterns</a> <a href="https://doi.org/10.1103/PhysRevA.102.032420" target="_blank">[LC'20]</a> <a href="https://arxiv.org/abs/2003.01695" target="_blank">ArXiv</a> <a href="http://dx.doi.org/10.1007/978-3-319-96424-9" target="_blank">[SP'18]</a>]

.more-more-condensed[
- .darkblue[Result]
  - .darkblue[Compact] representation: .darkblue[$\lceil log_2(n+1)\rceil$ qubits]
     - more compact (in terms of qubits) than Basis, Angle Encoding or Quantum Random Access Memory (QRAM) Encoding
  - For an arbitrary state represented by .darkblue[$k$ qubits] (i.e., $2^k$ data values), .darkblue[at least $2^k$ parallel operations] for initialization are needed [SP'18] (nearly reached in state-of-the-art approaches)
  - For special cases logarithmic runtime or $O(1)$ (e.g., Uniform Superposition)
  - Sparse data vectors can also be prepared more efficiently [SP'18]
  - Output is also encoded in the amplitude<br/>$\Rightarrow$ .darkblue[Multiple measurements to obtain a good estimate] of the output result<br/>$\Rightarrow$ .darkblue[Number of measurements scales with the number of amplitudes $2^k$] for $k$ qubits [SP'18]
]

.reference[Data Encoding: <a href="https://doi.org/10.1109/ICSA-C52384.2021.00025" target="_blank">Paper</a> <a href="https://www.iaas.uni-stuttgart.de/publications/Weigold2021_ExpandingDataEncodingPatterns.pdf" target="_blank">Preprint</a> <a href="https://doi.org/10.1049/qtc2.12032" target="_blank">Journal</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Web</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Source of Icons: QC Patterns</a> <a href="https://doi.org/10.1103/PhysRevA.102.032420" target="_blank">[LC'20]</a> <a href="https://arxiv.org/abs/2003.01695" target="_blank">ArXiv</a> <a href="https://doi.org/10.1103/PhysRevLett.103.150502" target="_blank">[HHL'09]</a> <a href="https://arxiv.org/abs/0811.3171" target="_blank">ArXiv</a> <a href="https://doi.org/10.1209/0295-5075/119/60002" target="_blank">[SFP'17]</a> <a href="https://arxiv.org/abs/1703.10793" target="_blank">ArXiv</a>]

.more-more-condensed[
- .darkblue[Known Uses]
  - Amplitude Encoding can be used in many .darkblue[quantum machine learning] algorithms [LC'20] 
  - Algorithm of Harrow, Hassidim and Lloyd [HHL'09] (.darkblue[HHL algorithm]) for solving linear equations
  - .darkblue[Data values are typically normalized in machine learning] [SFP'17], e.g. in support vector machine.
  - Various ways to construct a state preparation routine for amplitude encoding via e.g. Schmidt Decomposition <a href="https://doi.org/10.1103/PhysRevA.83.032302" target="_blank">[PB'11] </a> <a href="https://arxiv.org/abs/1003.5760" target="_blank">(ArXiv) </a> <a href="https://doi.org/10.1103/PhysRevA.93.032318" target="_blank">[I+16] </a> <a href="https://arxiv.org/abs/1501.06911" target="_blank">(ArXiv) </a> 
      - <span class="darkblue">Mathematica</span>: <a href="https://arxiv.org/abs/1904.01072" target="_blank">[I+'19]</a> 
	  - <span class="darkblue">Qiskit</span>: <a href="https://doi.org/10.1109/TCAD.2005.855930" target="_blank">[SBM'06]</a> <a href="https://arxiv.org/abs/quant-ph/0406176" target="_blank">(ArXiv)</a> <a href="https://qiskit.org/documentation/" target="_blank">QisKit Documentation</a>
	  - <span class="darkblue">PennyLane</span>: <a href="https://pennylane.readthedocs.io/en/stable/code/api/pennylane.AmplitudeEmbedding.html" target="_blank">qml.AmplitudeEmbedding</a> using the algorithm proposed by <a href="https://arxiv.org/abs/quant-ph/0504100" target="_blank">[MV'05]</a> requiring an exponential number of operations to encode $2^k$ data values 
	  - <span class="darkblue">Q#</span>: approximates the desired amplitude encoding <a href="https://docs.microsoft.com/en-us/qsharp/api/" target="_blank">Q# API reference</a>
]

---
# <a href="https://quantumcomputingpatterns.org/#/patterns/4" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/qram_icon.png" align="center" style="width:2em"></img></a> .darkblue[QRAM Encoding] 1/4

.reference[Data Encoding: <a href="https://doi.org/10.1109/ICSA-C52384.2021.00025" target="_blank">[W+'21]</a> <a href="https://www.iaas.uni-stuttgart.de/publications/Weigold2021_ExpandingDataEncodingPatterns.pdf" target="_blank">Preprint</a> <a href="https://doi.org/10.1049/qtc2.12032" target="_blank">Journal Paper</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Web</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Source of Icons: QC Patterns</a> <a href="https://www.oreilly.com/library/view/programming-quantum-computers/9781492039679/" target="_blank">[JHG'19]</a>]

.more-more-condensed[
- .darkblue[Intent]
  - ".darkblue[Use a quantum random access memory to access a superposition of data values at once]" [W+'21]
- .darkblue[Context]
  - Accessing the .darkblue[values of input] data .darkblue[via random access memory]
- .darkblue[Solution]
  - .darkblue[.bold[Classical] random access memory (RAM) transfers the data] value .darkblue[stored at a given address into a specified output] register
  - .darkblue[.bold[Quantum] random access memory (QRAM)]: similar to RAM, but the .darkblue[registers are] not classical but .darkblue[quantum registers] [JHG'19]
  - Consequently, .darkblue[address and output registers can be in superposition] instead of classical values
]

.more-more-condensed[
<div class="textcenter">
<img src="../../img/qc/patterns/QRAM.svg" align="top" class="fullwidth"></img>
<div>$\sum_a c_a|a\rangle \xrightarrow{\text{QRAM}} \sum_a c_a|a\rangle|D_a\rangle$</div>
<div>with $c_a$ amplitude, $|a\rangle$ address and $|D_a\rangle$ data value of address $|a\rangle$</div>
</div>
]

.more-more-condensed[
- .darkblue[Result]
  - .darkblue[Data values consuming $n$ bits: $n$ qubits] for basis encoded data
  - .darkblue[Address] register: .darkblue[additional .small-font[$\lceil log(k) \rceil$ qubits]] for up to .small-font[$k$] addresses
  - .darkblue[Basis Encoding] is used .darkblue[for data values: computational properties<br/>$\approx$ other digital encodings] (e.g., QuAM and Basis Encoding):
     - Since <span class="darkblue">data values</span> are represented <span class="darkblue">in superposition</span>, 
	     - .darkblue[data values can be manipulated at once] (using quantum parallelism)
	     - .darkblue[multiple arithmetic operations] (e.g., $+, \cdot$) can be applied
  - .darkblue[State preparation via the QRAM is efficient and of logarithmic runtime] [SP'18]: QRAM queries .small-font[$N$] adresses in .small-font[$O(log(N))$] [KKP'20]
     - <span class="darkblue">Exponential speed-up</span> of an algorithm using QRAM:<br/><span class="darkblue">only possible if filling QRAM is efficient</span>
  - To our best knowledge, there are currently .darkblue[.bold[no] commercial .bold[hardware] implementations for QRAM]
     - State preparation routine must be used for loading the QRAM,<br/>but <span class="darkblue"><span class="bold">no</span> routine for arbitrary input data exists that is as efficient as QRAM</span>
]

.reference[Data Encoding: <a href="https://doi.org/10.1109/ICSA-C52384.2021.00025" target="_blank">[W+'21]</a> <a href="https://www.iaas.uni-stuttgart.de/publications/Weigold2021_ExpandingDataEncodingPatterns.pdf" target="_blank">Preprint</a> <a href="https://doi.org/10.1049/qtc2.12032" target="_blank">Journal Paper</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Web</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Source of Icons: QC Patterns</a> <a href="https://arxiv.org/abs/1803.01958" target="_blank">[CB'18]</a> <a href="https://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-211.pdf" target="_blank">[P'14]</a> <a href="https://arxiv.org/abs/1805.11250" target="_blank">[MKF'19]</a> <a href="https://arxiv.org/abs/0708.1879" target="_blank">[GLM'08]</a>]

.more-more-condensed[
- .darkblue[Known Uses]
  - .darkblue[Alternative state preparation] to realize QRAM Encoding can be found in [CB'18] (circuit family #3) or [P'14]
  - Algorithms for .darkblue[solving semi-definite programs] [MKF'19] use QRAM Encoding.
  - QRAM is required or assumed .darkblue[in various other algorithms] [GLM'08], <a href="https://arxiv.org/abs/1307.0471" target="_blank">[RML'14] </a>, <a href="https://arxiv.org/abs/1412.3489" target="_blank">[WKS'14] </a>, <a href="https://arxiv.org/abs/1307.0401" target="_blank">[LMR'13] </a>
  - .darkblue[HHL algorithm for solving linear equations] <a href="https://arxiv.org/abs/0811.3171" target="_blank">[HHL'09] </a> uses QRAM Encoding as an intermediate representation for eigenvalues [MKF'19]
]

---
# <a href="https://quantumcomputingpatterns.org/#/patterns/12" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/schmidt_icon.png" align="center" style="width:2em"></img></a> .darkblue[Schmidt Decomposition]

.more-more-condensed[
- .darkblue[Context]
  - A .darkblue[state $|s\rangle$ has to be prepared] on an empty $n$-qubit register
- .darkblue[Forces]
  - .darkblue[Small depth of] the constructed .darkblue[state preparation circuit]
  - .darkblue[Runtime for constructing the state preparation circuit on classical computer should not outweigh] the potential benefit of quantum computing
- .darkblue[Solution]
  - Express $|s\rangle$ in terms of two subspaces $V$ and $W$ that span $H^{\otimes n}$ 
     - Choose orthogonal basis <span class="smaller-font">$\\{f_1, …, f_k\\}\in V\wedge \\{g_1, …, g_k\\}\in W$</span>, such that:
	 - $|s\rangle=\sum b_{ij}\cdot f_i\otimes g_j$
  - $M:=\\{b_{ij}\\}$
  - Compute the singular value decomposition (SVD) $M=\biggl(U_1 U_2\biggr)\left(\\begin{array}{c} A\\\\ 0 \\end{array}\right)V^*$ of $M$ .darkgray[(see [OS'18] for detailed instructions)]
]

.reference[Data Encoding: <a href="https://doi.org/10.1109/ICSA-C52384.2021.00025" target="_blank">Paper</a> <a href="https://www.iaas.uni-stuttgart.de/publications/Weigold2021_ExpandingDataEncodingPatterns.pdf" target="_blank">Preprint</a> <a href="https://doi.org/10.1049/qtc2.12032" target="_blank">Journal Paper</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Web</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Source of Icons: QC Patterns</a> <a href="https://doi.org/10.1007/978-3-319-91041-3" target="_blank">[OS'18]</a> <a href="https://doi.org/10.1088/2058-9565/abae7d" target="_blank">[LB'20]</a> <a href="https://arxiv.org/abs/1804.03719" target="_blank">[A+20]</a>]

.more-more-condensed[
- .darkblue[Solution] .darkgray[(continued)]
  - Entries .small-font[$\\\\{\alpha_1, …,\alpha_m\\\\}$] of the diagonal matrix .small-font[$A$]: Schmidt decomposition
  -  With .small-font[$\alpha_1,…,\alpha_m$] Schmidt coefficients for the Schmidt basis .small-font[$\\\\{u_i\\\\}, \\\\{v_i\\\\}$]:
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  - Circuit for state preparation [A+20]
     - <span class="smaller-font">$B$</span> transforms the amplitude of the first register to the Schmidt coefficients
	 - Copy this state to the second register using CNOT operations
	 - .smaller-font[$U_1$] and .smaller-font[$V$] transform the basis states .smaller-font[$\\\\{e_i\\\\}$] into the Schmidt basis states<br/>
	 <table class="tablepadding">
	 <tbody>
	 <tr style="vertical-align:top">
	 <td class="small-font" style="width:48%">
	 <div class="textcenter">
<img src="../../img/qc/patterns/SchmidtDecomposition.svg" align="top" class="fullwidth"></img>
</div>
</td>
<td class="smaller-font" style="width:22%">
<span class="bold">Example 1:</span><br/>$|s\rangle=\frac{|00\rangle+|01\rangle}{\sqrt{2}}$<br/>$B=U_1=I$<br/>$V=H$
</td>
<td class="smaller-font" style="width:30%">
<span class="bold">Example 2:</span><br/>$|s\rangle=\frac{\sqrt{2}\cdot|00\rangle+|11\rangle}{\sqrt{3}}$<br/>$B=\frac{1}{\sqrt{3}}\left[\begin{array}{c} \sqrt{2} & 0 \\ 1 & 0 \end{array}\right]$<br/>$U_1=V=I$
</td>
</tr>
</tbody>
</table>
	 - For the execution on a quantum computer, the unitary matrices .smaller-font[$B, U_1, V$] must be further decomposed into one and two qubit gates [LB'20]
]

.reference[Data Encoding: <a href="https://doi.org/10.1109/ICSA-C52384.2021.00025" target="_blank">Paper</a> <a href="https://www.iaas.uni-stuttgart.de/publications/Weigold2021_ExpandingDataEncodingPatterns.pdf" target="_blank">Preprint</a> <a href="https://doi.org/10.1049/qtc2.12032" target="_blank">Journal Paper</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Web</a> <a href="https://quantumcomputingpatterns.org/" target="_blank">Source of Icons: QC Patterns</a> <a href="https://doi.org/10.1017/CBO9780511976667" target="_blank">[NC'10]</a> <a href="https://doi.org/10.1103/PhysRevA.83.032302" target="_blank">[PB'11]</a> <a href="https://arxiv.org/abs/1003.5760" target="_blank">ArXiv</a> <a href="https://arxiv.org/abs/1403.0270" target="_blank">[DGK'14]</a> <a href="https://arxiv.org/abs/1904.01072" target="_blank">[I+21]</a>]

.more-more-condensed[
- .darkblue[Result]
  - The state $|s\rangle$ is created in the register for which the Schmidt coefficients $\alpha_i$ are known, which can be used to quantify entanglement [NC'10]
  - The .darkblue[state $|s\rangle$ is separable (i.e., not entangled) if and only if exactly one of the Schmidt coefficients is non-zero]
  - Depth of the circuit is $\frac{23}{48}2^n$ in the worst case [PB'11]
     - <span class="darkblue">Arbitrary state preparation is of exponential complexity</span> in general (in the worst case)
- .darkblue[Related Patterns]
  - Schmidt Decomposition can be used as a state preparation method .darkblue[for Amplitude or QRAM Encoding]
- .darkblue[Known Uses]
  - Schmidt Decomposition can be used to create random states with a controlled amount of entanglement [DGK'14]
  - Mathematica implementation: [I+21]
]

---
# .darkblue[Comparison Data Encoding] Patterns

.more-more-condensed[
- $k$ data points, one data point consumes $n$ bits

<table class="result tablepadding">
<tbody>
<tr>
<th>Encoding Pattern</th>
<th>Encoding</th>
<th>#Qubits</th>
<th>Prepa-<br/>ration</th>
<th class="small-font">Digital<br/>Enco-<br/>ding</th>
</tr>
<tr>
<td><a href="https://quantumcomputingpatterns.org/#/patterns/0" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/basis_encoding_icon.png" align="center" style="width:2em"></img></a> .darkblue[Basis]</td>
<td class="small-font">$x_i\approx\sum_i b_i\cdot 2^i$<br/>$\mapsto |x_i \rangle=|b_{n-1} \ldots b_1 b_0\rangle$</td>
<td class="small-font" style="text-align:right">$k\cdot n$</td>
<td class="small-font textcenter">$O(1)$</td>
<td class="textcenter">✓</td>
</tr>
<tr>
<td><a href="https://quantumcomputingpatterns.org/#/patterns/3" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/angle_encoding_icon.png" align="center" style="width:2em"></img></a> .darkblue[Angle]</td>
<td class="small-font">$x_i\mapsto cos(x_i)|0\rangle+sin(x_i)|1\rangle$</td>
<td class="small-font" style="text-align:right">$k$</td>
<td class="small-font textcenter">$O(1)$</td>
<td></td>
</tr>
<tr>
<td><a href="https://quantumcomputingpatterns.org/#/patterns/1" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/quam_icon.png" align="center" style="width:2em"></img></a> .darkblue[QuAM]</td>
<td class="small-font">$X\mapsto\frac{1}{\sqrt{k}}\sum_{i=1}^k |x_i\rangle$</td>
<td class="small-font" style="text-align:right">$n$</td>
<td class="small-font textcenter">$O(k)$</td>
<td class="textcenter">✓</td>
</tr>
<tr>
<td><a href="https://quantumcomputingpatterns.org/#/patterns/4" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/qram_icon.png" align="center" style="width:2em"></img></a> .darkblue[QRAM]</td>
<td class="small-font">$X\mapsto\sum_i c_i|i\rangle|x_i\rangle$</td>
<td class="small-font" style="text-align:right">$n+\lceil log_2(k)\rceil$</td>
<td class="small-font textcenter">$O(2^k)$ (Schmidt)</td>
<td class="textcenter">✓</td>
</tr>
<tr>
<td><table class="no-margin no-padding"><tbody><tr><td class="no-padding" style="border-collapse:collapse;border:none"><a href="https://quantumcomputingpatterns.org/#/patterns/2" target="_blank" class="icon"><img src="../../img/qc/patterns/patterns-icons/amplitude_encoding_icon.png" align="center" style="width:2em"></img></a></td><td class="no-padding"> .darkblue[Ampli-<br/> tude]</td></tr></tbody></table></td>
<td class="small-font">$X\mapsto\sum_{i=0}^{k-1} x_i|i\rangle$</td>
<td class="small-font" style="text-align:right">$\lceil log_2(k)\rceil$</td>
<td class="small-font textcenter">$O(2^k)$ (Schmidt)</td>
<td></td>
</tr>
</tbody>
</table>
]

---
# .darkblue[Summary & Conclusions]

.more-more-condensed[
- Quantum Patterns
  - Software design patterns using quantum algorithms
- In this lecture
  - Patterns for Data Encoding
     - State Preparations refined by
	     - Basis Encoding
		 - Quantum Associative Memory (QuAM)
		 - Angle Encoding
		 - Amplitude Encoding
		 - QRAM Encoding
	 - Schmidt Decomposition to be used as basic state preparation method for Amplitude and QRAM Encoding
]