Merge pull request #51 from marekyggdrasil/response

Minor changes for Alexeys response

Author: marekyggdrasil

manuscript.tex

@@ -164,19 +164,19 @@ \subsection{The classical algorithm}
 Initially the weight components are chosen randomly 
 and topological distances between neurons given. 
 We then can train our SOFM adjusting the components through the learning process which occur in the two basic procedures of 
-selecting a winning cluster vector, also called the best matching unit (BMU), and updating its weights (Fig.~\ref{fig:sofm_fitting}). 
+selecting a winning cluster vector, also called the best matching unit, and updating its weights (Fig.~\ref{fig:sofm_fitting}). 
 More specifically, they consist of four step process: 
 \begin{enumerate*}
 \item selecting an input vector randomly from the set of all input vectors; 
-\item finding a cluster vector which is closest to the input vector (BMU); 
-\item adjusting the weights of the BMU and neurons close to it on feature map in such a way 
+\item finding a cluster vector which is closest to the input vector; 
+\item adjusting the weights of the best matching unit and neurons close to it on feature map in such a way 
 	that these vectors  becomes even closer to the input vector; 
 \item repeating this process for many iterations until it converges.
 \end{enumerate*}
 
 
-On a step $t$ when the BMU $\vec{w}_{c}$ for a input $\vec{x}(t)$ is selected, 
-the weights $\vec{w}_{i}$ of the BMU and its neigbours on feature map are adjusted according to
+On a step $t$ when the best matching unit $\vec{w}_{c}$ for a input $\vec{x}(t)$ is selected, 
+the weights $\vec{w}_{i}$ of the best matching unit and its neigbours on feature map are adjusted according to
 %
 \begin{equation}
     \label{eq:learning}
@@ -224,7 +224,7 @@ \subsection{Optimized quantum scheme for Hamming distance calculation}
 \begin{figure}[t]
 	\includegraphics[width=\columnwidth]{qcircuit.eps}
 	\caption{
-		Quantum circuit for the quantum parallelized Hamming distance calculating  between all pairs of binary vectors from two sets ${X}$ and ${Y}$ encoded \cite{trugenberger2001} in $X$ and $Y$ quantum registers   respectively.
+		Quantum circuit for the quantum parallelized Hamming distance calculating  between all pairs of binary vectors from two sets ${X}$ and ${Y}$ encoded in $X$ and $Y$ quantum registers   respectively.
 		First, we encoded information about pairwise different qubits in a quantum state of the $X$-register with applying the CNOT gates. 
 		Second, Hamming distance values are extracted into the amplitudes of superposition with the controled rotation around $z$-axis gate~(\ref{eq:controled_rotation}) and Hadamard gates. 
 		Finally, a quantum state of the $X$-register returned to the initial basis for information retrieval. 	
@@ -447,7 +447,7 @@ \subsection{Optimized quantum scheme for Hamming distance calculation}
 In this special case scenario the circuit depth complexity is matching with \cite{schuld2014}.
 
 In the general case when multiple input vectors are present in the register, the ``Decoding'' stage still needs to be included leading to larger circuit depth and less attractive complexity in terms of number of controlled gate operations.
-The number of controlled gate operations in this general case of multiple input vectors is matching the number of controlled gate operations in \cite{trugenberger2001}.  
+The number of controlled gate operations in this general case of multiple input vectors is matching the number of controlled gate operations in \cite{trugenberger2001} but the number of remaining gates is reduced compared to [80], leading to less deep circuit, which is significant for NISQ devices.  
 
 
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