Modifying quantum perceptron

Author: Ikiga1

algpseudocode/TEMPLATE.tex

@@ -3,7 +3,7 @@
 \usepackage{algorithm}
 \usepackage{algpseudocode}
 \usepackage{braket}
-\usepackage{amsmath}
+\usepackage{amsmath, amsfonts, amssymb}
 
 \newcommand{\norm}[1]{\left\lVert#1\right\rVert}
 
@@ -20,7 +20,7 @@
 
 		\Require
 		\Ensure
-		\vspace{10pt}
+		\vspacae{10pt}
 		\Statex
 		\State
 

algpseudocode/online_quantum_perceptron.tex

@@ -0,0 +1,45 @@
+\documentclass{article}
+\usepackage[utf8]{inputenc}
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+\usepackage{braket}
+\usepackage{amsmath, amsfonts, amssymb}
+
+\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
+
+\makeatletter
+\renewcommand{\fnum@algorithm}{\fname@algorithm}
+\makeatother
+
+\begin{document}
+\pagestyle{empty}
+% #Classical perceptron algorithm
+\begin{algorithm}[ht]
+	\caption{Quantum online perceptron training} 
+	\begin{algorithmic}[1]
+	   \Require Quantum access to a dataset $\mathcal{D}=\{ (x^{(1)},y^{(1)}), (x^{(2)}, y^{(2)} ) ,\ldots, ( x^{(n)},y^{(n)})\} \in (\mathbb{R}^{m} \times\{0,1\})^{n}$ via a unitary $U: |j\rangle|{0}\rangle \rightarrow |j\rangle|x^{(j)}\rangle$. An oracle $\mathcal{F}_w$ that can be adapted to the weights $w$ at each iteration.
+    \Ensure A vector of weights $w$ such that $P\left(\exists j: f_{w}\left(x^{(j)}, y^{(j)}\right)=1\right) \leq \epsilon$
+
+    	\State Initialize the weights vector $w = 0 \in \mathbb{R}^{m+1}$.
+		\For {\(h=1, \ldots,\left\lceil \frac{1}{\gamma^{2}}\right\rceil\)} 
+
+			\For {\(k=1, \ldots,\left\lceil\log \left(\frac{1}{\gamma^{2} \epsilon}\right)\right\rceil\)} 
+		
+				\State Apply Grover's algorithm using $U$ and $\mathcal{F}_w$ on $\frac{1}{\sqrt{n}}\sum_{j=1}^{n} |j\rangle|0\rangle$.
+				
+				\State Measure the first register of the resulting state $\sum_{j=1}^{n} \alpha_j|j\rangle|0\rangle$.
+
+				\If{$f_{w}(x^{(j)}, y^{(j)})=1$ classically} 
+					\State Update $w^{\prime} \leftarrow w+\Delta w \phi$.
+					\State Update the oracle $\mathcal{F}_w = \mathcal{F}_{w^{\prime}}$.		
+				\EndIf
+
+			\EndFor
+		
+		\EndFor
+				
+		\Return $w^{\prime}=0$.
+	\end{algorithmic}
+\end{algorithm}
+
+\end{document}

algpseudocode/perceptronalgo.tex

@@ -0,0 +1,37 @@
+\documentclass{article}
+\usepackage[utf8]{inputenc}
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+\usepackage{braket}
+\usepackage{amsmath, amsfonts, amssymb}
+
+\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
+\algdef{SE}[DOWHILE]{Do}{doWhile}{\algorithmicdo}[1]{\algorithmicwhile\ #1}%
+
+\makeatletter
+\renewcommand{\fnum@algorithm}{\fname@algorithm}
+\makeatother
+
+\begin{document}
+\pagestyle{empty}
+% #Classical perceptron algorithm
+\begin{algorithm}[ht]
+	\caption{Classical online perceptron training} 	
+	\begin{algorithmic}[1]
+    \Require A dataset $\mathcal{D}=\{ (x^{(1)},y^{(1)}), (x^{(2)}, y^{(2)} ) ,\ldots, ( x^{(n)},y^{(n)})\} \in (\mathbb{R}^{m} \times\{0,1\})^{n}$
+    \Ensure A vector of weights $w$ such that $y^{(j)}(w^\top x^{(i)}) > 0$ for all $j \in \{1, \dots, n\}$
+		\State Initialize ${w}:={0} \in \mathbb{R}^{m+1}$
+		\State $\mathrm{continue}=0$
+		  \Do
+			\For {every data point $\left(x^{(j)}, y^{(j)}\right) \in \mathcal{D}$}
+				\State Compute the prediction $\hat{y}^{[i]}:=\sigma\left(\mathbf{x}^{[i]T} {w}_{i}+b\right)$
+        		\State Compute the $\mathrm{error}:=\left(\hat{y}^{(i)}-{y}^{(i)}\right)$
+        		\State $\mathrm{continue} = \mathrm{continue} + |\mathrm{error}|$
+        		\State Update the weights ${w}_{i}^{'}:={w}_{i}+\Delta{w}$
+        \EndFor
+        
+		\doWhile{$\mathrm{continue}!=0$}        
+	\end{algorithmic}
+\end{algorithm}
+
+\end{document}

algpseudocode/version_space_quantum_perceptron.tex

@@ -0,0 +1,41 @@
+\documentclass{article}
+\usepackage[utf8]{inputenc}
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+\usepackage{braket}
+\usepackage{amsmath, amsfonts, amssymb}
+
+\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
+
+\makeatletter
+\renewcommand{\fnum@algorithm}{\fname@algorithm}
+\makeatother
+
+\begin{document}
+\pagestyle{empty}
+\begin{algorithm}[ht]
+	\caption{Quantum version space perceptron training} 
+	\begin{algorithmic}[1]
+	\Require Quantum access to a set of weight vectors $\{w^{(1)}, \ldots, w^{(k)}\}$ sampled from a Gaussian spherical distribution, via a unitary $U: |j\rangle|{0}\rangle \rightarrow |j\rangle|w^{(j)}\rangle$. An oracle $\mathcal{F}$ such that $\mathcal{F}|w^{(j)}\rangle=(-1)^{1+\left(f\left(w^{(j)},x^{(1)}, y^{(1)}\right) \vee \cdots \vee f\left(w^{(j)},x^{(n)}, y^{(n)}\right)\right)}|w^{(j)}\rangle$.
+    \Ensure A vector of weights $w$ such that $P\left(\exists j: f_{w}\left(x^{(j)}, y^{(j)}\right)=1\right) \leq \epsilon$
+
+		\State Initialize $k = \frac{1}{\gamma}\log(1/\epsilon)$.
+		\For {\(h = 1, \ldots,\left\lceil\log \epsilon\right\rceil\)}
+		
+			\State Apply Grover's algorithm using $U$ and $\mathcal{F}$ on $\frac{1}{\sqrt{k}}\sum_{j=1}^{k} |j\rangle|0\rangle$.
+						
+			\State Measure the first register of the resulting state $\sum_{j=1}^{n} \alpha_j|j\rangle|0\rangle$ and call the output $j$.
+			
+			\If{\(f\left(w^{(j)}, x^{(i)}, y^{(i)}\right)=0\) for all \(i \in\{1, \ldots, n\}\)}
+			
+				\State Return $w_j$.
+			
+			\EndIf
+					
+		\EndFor
+		
+		\State \Return $w=0$.
+	\end{algorithmic}
+\end{algorithm}
+
+\end{document}