updating files

Author: mhjensen

doc/src/week11/gradientvqe.py doc/src/week11/prograns/gradientvqe.pydoc/src/week11/gradientvqe.py renamed to doc/src/week11/prograns/gradientvqe.py

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+\documentclass{article}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{hyperref}
+\usepackage{qcircuit}
+
+\title{Lecture Notes on Shor's Algorithm}
+\author{MHJ}
+\date{\today}
+
+\begin{document}
+
+\maketitle
+
+\tableofcontents
+\newpage
+
+\section{Introduction to Shor's Algorithm}
+
+\subsection{Historical Context}
+Shor's algorithm, introduced by Peter Shor in 1994, revolutionized the field of cryptography. It demonstrated that a quantum computer could solve integer factorization and the discrete logarithm problem efficiently, posing a significant threat to classical cryptosystems such as RSA.
+
+\subsection{Problem Statement}
+The objective of Shor's algorithm is to factorize a large composite integer \( N \) into its prime factors. Given \( N = p \times q \), where \( p \) and \( q \) are unknown distinct primes, the problem of factorization can be reduced to finding an integer \( a \) such that \( 1 < a < N \) and \( \gcd(a, N) = 1 \).
+
+\section{Classical Component}
+
+\subsection{Reduction to Order-Finding}
+The key reduction in Shor's algorithm lies in transforming factorization into an order-finding problem:
+\begin{itemize}
+    \item Choose a random integer \( a \) such that \( 1 < a < N \) and \( \gcd(a, N) = 1 \).
+    \item Determine the smallest positive integer \( r \) such that \( a^r \equiv 1 \pmod{N} \).
+\end{itemize}
+The integer \( r \) is known as the \textit{order} of \( a \) modulo \( N \), and it helps in discovering the factors of \( N \).
+
+\subsection{Continued Fraction Expansion}
+After obtaining the rational approximation from the quantum algorithm, classical computation using continued fraction expansion aids in deducing the correct order \( r \).
+
+\section{Quantum Component}
+
+\subsection{Quantum Fourier Transform (QFT)}
+The Quantum Fourier Transform is crucial to Shor's algorithm. It leverages quantum parallelism and interference to efficiently estimate the periodicity of a function.
+
+\[ \text{QFT}: \ket{x} \mapsto \frac{1}{\sqrt{2^n}} \sum_{y=0}^{2^n-1} e^{2\pi ixy/2^n} \ket{y} \]
+
+\subsection{Quantum Order Finding Procedure}
+The quantum part of Shor's algorithm is designed to find the period \( r \) of the modular exponentiation function \( f(x) = a^x \bmod N \):
+
+\[ \Qcircuit @C=1em @R=1em {
+    \lstick{\ket{0^{n}}} & \qw & \gate{\mathrm{H^{\otimes n}}} & \qw & \multigate{1}{U_f} & \qw & \measureD{} \\
+    \lstick{\ket{0^{n}}} & \gate{\mathrm{H^{\otimes m}}} & \qw & \qw & \ghost{U_f} & \gate{\mathrm{QFT}^{-1}} & \measureD{}
+}\]
+
+\subsubsection{Details of the Circuit}
+\begin{itemize}
+    \item Start with two registers. Apply Hadamard gates to obtain superposition.
+    \item Implement the unitary operation \( U_f \) for modular exponentiation \( f(x) = a^x \bmod N \).
+    \item Perform an inverse QFT to extract the order \( r \) from the phase information.
+\end{itemize}
+
+\section{Putting It All Together}
+
+\subsection{Algorithm Steps}
+1. Choose a random integer \( a \).
+2. Use a quantum circuit to find the order \( r \).
+3. If \( r \) is even and \( a^{r/2} \not\equiv -1 \pmod{N} \), compute \( \gcd(a^{r/2} \pm 1, N) \) to obtain non-trivial factors.
+4. If unsuccessful, repeat with new \( a \).
+
+\section{Error Analysis and Success Probability}
+
+\subsection{Probability of Success}
+The probability that a randomly chosen \( a \) leads to successful factoring is generally greater than 50\%. The quantum component requires carefully managed resources for high precision.
+
+\subsection{Error Sources}
+Error correction is crucial, as any errors in operations or QFT can significantly impact the accuracy of period estimation and the eventual factorization.
+
+\section{Conclusion}
+
+Shor's algorithm is an exemplary quantum algorithm demonstrating exponential speedup over classical counterparts. It ignites both excitement for potential quantum advancements and a reconsideration of current cryptographic standards.
+
+\end{document}