From 372187f1e93613a00fb0a3e2824dd41de651de71 Mon Sep 17 00:00:00 2001 From: matpia Date: Thu, 10 Mar 2022 18:42:47 +0100 Subject: [PATCH] Adding 1st lecture; correcting .gitignore and Makefile --- Makefile | 2 +- wykład-2022-02-24.pdf | Bin 0 -> 143771 bytes wykład-2022-02-24.tex | 97 +++++++++++++++++++++++++++++++++++++++++ wykład-2022-03-10.pdf | Bin 0 -> 68796 bytes 4 files changed, 98 insertions(+), 1 deletion(-) create mode 100644 wykład-2022-02-24.pdf create mode 100644 wykład-2022-02-24.tex create mode 100644 wykład-2022-03-10.pdf diff --git a/Makefile b/Makefile index 453be89..9df624f 100644 --- a/Makefile +++ b/Makefile @@ -7,4 +7,4 @@ all: $(PDFs) pdflatex $< clean: - rm -f *.log *.aux $(PDFs) + rm -f *.log *.aux diff --git a/wykład-2022-02-24.pdf b/wykład-2022-02-24.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b8c9ea7afac604e6ecfc8f2292ca610708c89880 GIT binary patch literal 143771 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+\linespread{1.5} +\usepackage{microtype} +\usepackage{geometry} +\usepackage{ragged2e} +\usepackage{mathtools} +\usepackage[T1]{fontenc} +\usepackage[polish]{babel} +\title{Algorytmy kwantowe - wykład 2022} + +\begin{document} +\section{Wykład 1 - 24 lutego 2022} + +\subsection{Przegląd zagadnień} + +\subsubsection{Tematyka zajęć} +\paragraph{}Lista zagadnień będacych przedmiotem wykładu (prawdopodbnie się powiększy) +\begin{enumerate} + \item przestrzeń liniowa: + \item przestrzeń unitarna; + \item liczby zespolone: + \item przekształcenia liniowe (macierze); + \item iloczyn skalarny; + \item iloczyn tensorowy; + \item kubit; + \item bramka kwantowa; + \item teoria grup; + \item notacja Diraca. +\end{enumerate} + +\subsubsection{Algorytmy} +\paragraph{}Po omówieniu teroii będziemy implementować następujące algorytmy: + +\begin{enumerate} + \item algorytm teleportacji kwantowej; + \item algorytm Deutscha; + \item algorytm Deutscha-Jozsy; + \item algorytm faktoryzacji Shor'a; + \item algorytm Grovera; + \item algorytm Simona; + \item być możę algorytmy grafowe. +\end{enumerate} + +\subsubsection{Literatura} +\paragraph{}Proponowana literatura do wykładu: + +\begin{enumerate} + \item Mika Hirvensalo - Algorytmy kwantowe +\end{enumerate} + +\subsection{Wykład merytoryczny} + +\subsubsection{Liczby zespolony} + +\paragraph{Postać algebraiczna} Liczba zespolona ma następującą postać algebraiczną: +\[ \alpha = a + bi \] +gdzie \(a\) oznacza część rzeczywistą, a \(bi\) część urojoną. + +\subparagraph{}Liczby zespolone są grupą abelową ze względu na dodwanie (operacja łączna przemienna, ma element neutralny, dla każdego elementu istnieje element przeciwny). Dla mnożenia liczby zespolone też są grupą, bez zera, inny natomiast jest element neutralny (1), nie ma elementu przeciwnego, występuje zaś element odwrotny. + +\subparagraph{}Zbiór z dwoma operacjami tworzy ciało, dochodzi do tego rozdzielność mnożenia względem dodawania. + +\paragraph{Własności liczb zespolonych} +\subparagraph{Sprzężenie zespolone} Sprzężenie zespolone można interpretować jako odbicie liczby względem osi wartości rzeczywistych. +\[ \bar{\alpha} = a - bi \] + +\subparagraph{Moduł liczby zespolonej} Moduł liczby zespolonej może być interpretowany jako odległość liczby od środka układu współrzędnych wyznaczanego przez oś wartości rzeczywistych i oś wartości urojonych: +\[ |\alpha| = a^2 + b^2 \] + +\subparagraph{Własności liczb zespolonych} Liczby zespolone posidają następujące własności: +\begin{enumerate} + \item \( \alpha + \bar{\alpha} = 2a \) + \item \( \alpha \bar{\alpha} = a^2 + b^2 = |\alpha|^2 \) + \item \( |\alpha| |\beta| = |\alpha \beta| \) + \item \( |\alpha + \beta| = |\alpha| + |\beta| \) +\end{enumerate} + +\paragraph{Postać trygonometryczna} + +\subparagraph{Argument zespolony}\(\varphi\) jest \textit{argumentem} liczby zespolonej. +\[ arg \alpha = \varphi + 2k\pi \] +Założenie, że \( arg \alpha \) jest z przedziału od \(-\pi\) do \(\pi\) jest \textit{głównym argumentem} liczby zespolonej (\( Arg \alpha \)) + +\[ \cos(\varphi) = \frac{a}{|\alpha|} \] +\[ \sin(\varphi) = \frac{b}{|\alpha|} \] +\[ a = |\alpha| \cos(\varphi) \] +\[ b = |\alpha| \sin(\varphi) \] +\[ \alpha = |\alpha|(\cos(\varphi) + i\sin(\varphi)) \] +\[ \alpha \beta = r_1r_2(\cos(\varphi_1 + \varphi_2) + i\sin(\varphi_1 + \varphi_2)) \] +\[ \frac{\alpha}{\beta} = \frac{r_1}{r_2}(\cos(\varphi_1 - \varphi_2) + i\sin(\varphi_1 - \varphi_2)) \] +\[ e^{i\varphi} = \cos(\varphi) + i\sin(\varphi) \] + +\subparagraph{Wzór de Moivre’a} Wzór na \(n-\)tą potęgę liczby zespolonej: +\[ z^n = |z|^n(\cos(n\varphi) + i\sin(n\varphi)) \] + +\end{document} diff --git 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