Update week14.do.txt

Author: mhjensen

doc/src/week14/week14.do.txt

@@ -207,137 +207,6 @@ Advantage: Efficient sampling in complex probability distributions.
 
 
 
-!split
-===== Quantum SVMs =====
-
-The idea of a classical SVM is that we have a set of points
-that are in either one group or another and we want to find a line that
-separates these two groups. This line can be linear, but it can also be
-much more complex, which can be achieved by the use of Kernels.
-
-!split
-===== What will we need in the case of a quantum computer? =====
-
-!bblock
-We will have to translate the classical data point \(\vec{x}\)
-into a quantum datapoint \(\vert \Phi{(\vec{x})} \rangle\). This can
-be achieved by a circuit $\mathcal{U}_{\Phi(\vec{x})} \vert 0\rangle$.
-
-
-Here $\Phi()$ could be any classical function applied
-on the classical data $\vec{x}$.
-!eblock
-!bblock 
-We need a parameterized quantum circuit $W(\theta)$ that
-processes the data in a way that in the end we
-can apply a measurement that returns a classical value \(-1\) or
-\(1\) for each classical input \(\vec{x}\) that indentifies the label
-of the classical data.
-!eblock
-
-
-!split
-===== The most general ansatz =====
-
-Following these steps we can define an ansatz for this kind of problem
-which is
-!bt
-\[
-W(\theta) \mathcal{U}_{\Phi}(\vec{x}) \vert 0 \rangle.
-\]
-!et
-
-These kind of ansatzes are called quantum variational circuits.
-
-!split
-===== Quantum SVM =====
-
-In the case of a quantum SVM we will only use the quantum feature maps
-!bt
-\[
-\mathcal{U}_{\Phi(\vec{x})},
-\]
-!et
-to translate the classical data into
-quantum states and build the Kernel of the SVM out of these quantum
-states. After calculating the Kernel matrix on the quantum computer we
-can train the Quantum SVM the same way as the classical SVM.
-
-
-!split
-===== Defining the Quantum Kernel =====
-
-The idea of the quantum kernel is exactly the same as in the classical
-case. We take the inner product
-!bt
-\[
-K(\vec{x}, \vec{z}) = \vert \langle \Phi (\vec{x}) \vert \Phi(\vec{z}) \rangle \vert^2 = \langle 0^n \vert \mathcal{U}_{\Phi(\vec{x})}^{t} \mathcal{U}_{\Phi(\vec{z})} \vert 0^n \rangle,
-\]
-!et
-but now with the quantum feature maps
-!bt
-\[
-\mathcal{U}_{\Phi(\vec{x})}.
-\]
-!et
-
-The idea is that if we choose a quantum feature maps that is not easy
-to simulate with a classical computer we might obtain a quantum
-advantage.
-
-
-!split
-===== Side note =====
-
-There is no proof yet that the QSVM brings a quantum
-advantage, but the argument the authors of
-URL:"https://arxiv.org/pdf/1804.11326.pdf" make, is that there is
-for sure no advantage if we use feature maps that are easy to simulate
-classically, because then we would not need a quantum computer to
-construct the Kernel.
-
-!split
-===== Feature map =====
-
-For the feature maps we use the ansatz
-!bt
-\[
-\mathcal{U}_{\Phi(x)} = U_{\Phi(x)} \otimes H^{\otimes n},
-\]
-!et
-where
-!bt
-\[
-U_{\Phi(x)} = \exp \left( i \sum_{S \in n} \phi_S(x) \prod_{i \in S} Z_i \right),
-\]
-!et
-which simplifies a lot when we (like in URL:"https://arxiv.org/pdf/1804.11326.pdf") only consider
-$S \leq 2$ interactions, which means we only let two qubits interact
-at a time. For $S \leq 2$ the product $\prod_{i \in S}$ only leads
-to interactions $Z_i Z_j$ and non interacting terms $Z_i$. And the
-sum $\sum_{S \in n}$ over all these terms that are possible with $n$
-qubits.
-
-!split
-===== Classical functions =====
-
-Finally we define the classical functions $\phi_i(\vec{x}) = x_i$ and
-$\phi_{i,j}(\vec{x}) = (\pi - x_i)( \pi- x_j)$.
-
-If we write this ansatz for 2 qubits and $S \leq 2$ we see how it
-simplifies:
-!bt
-\[
-U_{\Phi(x)} = \exp \left(i \left(x_1 Z_1 + x_2 Z_2 + (\pi - x_1)( \pi- x_2) Z_1 Z_2 \right) \right).
-\]
-!et
-We won't get into details to much here, why we would take this ansatz.
-It is simply an ansatz that is simple enough an leads to good results.
-
-Finally we can define a depth of these circuits. Depth 2 means we repeat
-this ansatz two times. Which means our feature map becomes
-$U_{\Phi(x)} \otimes H^{\otimes n} \otimes U_{\Phi(x)} \otimes H^{\otimes n}$.
-
 
 !split
 ===== Plans for next week =====
@@ -508,7 +377,7 @@ evaluate such kernels systematically .
 !split
 ===== Quantum feature map =====
 
-Aquantum feature map
+A quantum feature map
 $\bm{x}\mapsto\vert \phi(\bm{x})\rangle$ combined with the kernel
 $k(\bm{x},\bm{x}’)=\vert \langle\phi(\bm{x})\vert \phi(\bm{x}’)\rangle\vert ^2$
 defines a QSVM kernel.  The intuition is that the quantum device is
@@ -1060,4 +929,135 @@ In summary, these examples show how to apply QSVM end-to-end.  For each, the ste
 
 
 
+!split
+===== Quantum SVMs =====
+
+The idea of a classical SVM is that we have a set of points
+that are in either one group or another and we want to find a line that
+separates these two groups. This line can be linear, but it can also be
+much more complex, which can be achieved by the use of Kernels.
+
+!split
+===== What will we need in the case of a quantum computer? =====
+
+!bblock
+We will have to translate the classical data point \(\vec{x}\)
+into a quantum datapoint \(\vert \Phi{(\vec{x})} \rangle\). This can
+be achieved by a circuit $\mathcal{U}_{\Phi(\vec{x})} \vert 0\rangle$.
+
+
+Here $\Phi()$ could be any classical function applied
+on the classical data $\vec{x}$.
+!eblock
+!bblock 
+We need a parameterized quantum circuit $W(\theta)$ that
+processes the data in a way that in the end we
+can apply a measurement that returns a classical value \(-1\) or
+\(1\) for each classical input \(\vec{x}\) that indentifies the label
+of the classical data.
+!eblock
+
+
+!split
+===== The most general ansatz =====
+
+Following these steps we can define an ansatz for this kind of problem
+which is
+!bt
+\[
+W(\theta) \mathcal{U}_{\Phi}(\vec{x}) \vert 0 \rangle.
+\]
+!et
+
+These kind of ansatzes are called quantum variational circuits.
+
+!split
+===== Quantum SVM =====
+
+In the case of a quantum SVM we will only use the quantum feature maps
+!bt
+\[
+\mathcal{U}_{\Phi(\vec{x})},
+\]
+!et
+to translate the classical data into
+quantum states and build the Kernel of the SVM out of these quantum
+states. After calculating the Kernel matrix on the quantum computer we
+can train the Quantum SVM the same way as the classical SVM.
+
+
+!split
+===== Defining the Quantum Kernel =====
+
+The idea of the quantum kernel is exactly the same as in the classical
+case. We take the inner product
+!bt
+\[
+K(\vec{x}, \vec{z}) = \vert \langle \Phi (\vec{x}) \vert \Phi(\vec{z}) \rangle \vert^2 = \langle 0^n \vert \mathcal{U}_{\Phi(\vec{x})}^{t} \mathcal{U}_{\Phi(\vec{z})} \vert 0^n \rangle,
+\]
+!et
+but now with the quantum feature maps
+!bt
+\[
+\mathcal{U}_{\Phi(\vec{x})}.
+\]
+!et
+
+The idea is that if we choose a quantum feature maps that is not easy
+to simulate with a classical computer we might obtain a quantum
+advantage.
+
+
+!split
+===== Side note =====
+
+There is no proof yet that the QSVM brings a quantum
+advantage, but the argument the authors of
+URL:"https://arxiv.org/pdf/1804.11326.pdf" make, is that there is
+for sure no advantage if we use feature maps that are easy to simulate
+classically, because then we would not need a quantum computer to
+construct the Kernel.
+
+!split
+===== Feature map =====
+
+For the feature maps we use the ansatz
+!bt
+\[
+\mathcal{U}_{\Phi(x)} = U_{\Phi(x)} \otimes H^{\otimes n},
+\]
+!et
+where
+!bt
+\[
+U_{\Phi(x)} = \exp \left( i \sum_{S \in n} \phi_S(x) \prod_{i \in S} Z_i \right),
+\]
+!et
+which simplifies a lot when we (like in URL:"https://arxiv.org/pdf/1804.11326.pdf") only consider
+$S \leq 2$ interactions, which means we only let two qubits interact
+at a time. For $S \leq 2$ the product $\prod_{i \in S}$ only leads
+to interactions $Z_i Z_j$ and non interacting terms $Z_i$. And the
+sum $\sum_{S \in n}$ over all these terms that are possible with $n$
+qubits.
+
+!split
+===== Classical functions =====
+
+Finally we define the classical functions $\phi_i(\vec{x}) = x_i$ and
+$\phi_{i,j}(\vec{x}) = (\pi - x_i)( \pi- x_j)$.
+
+If we write this ansatz for 2 qubits and $S \leq 2$ we see how it
+simplifies:
+!bt
+\[
+U_{\Phi(x)} = \exp \left(i \left(x_1 Z_1 + x_2 Z_2 + (\pi - x_1)( \pi- x_2) Z_1 Z_2 \right) \right).
+\]
+!et
+We won't get into details to much here, why we would take this ansatz.
+It is simply an ansatz that is simple enough an leads to good results.
+
+Finally we can define a depth of these circuits. Depth 2 means we repeat
+this ansatz two times. Which means our feature map becomes
+$U_{\Phi(x)} \otimes H^{\otimes n} \otimes U_{\Phi(x)} \otimes H^{\otimes n}$.
+