164 lines
3.7 KiB
TeX
164 lines
3.7 KiB
TeX
\documentclass{article}
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\usepackage[top=1.5cm,bottom=1.5cm]{geometry}
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\usepackage[utf8]{inputenc}
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\usepackage{amsfonts}
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\usepackage{hyperref}
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\usepackage{mathtools}
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\usepackage{multicol}
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\usepackage{polski}
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\usepackage{qcircuit}
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\newcommand\CC{\mathbb{C}}
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\newcommand\NN{\mathbb{N}}
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\DeclarePairedDelimiter\ket{\lvert}{\rangle}
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\DeclarePairedDelimiter\bra{\langle}{\rvert}
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\begin{document}
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\section{Bramki}
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% TODO Bramki X, Y, Z
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\subsection{Bramki odwracalne}
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Bramka $U$ działająca na $m$ kubitach jest odwracalna, jeśli liczba wejść bramki jest równa liczbie wyjść.
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\subsection{Bramka Hadamarda}
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$$ \Qcircuit @C=1em @R=1em { & \gate{H} & \qw } $$
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$$ H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} $$
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\begin{align*}
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H(|0\rangle) & = \frac{1}{\sqrt{2}} |0\rangle + \frac{1}{\sqrt{2}} |1\rangle =: |+\rangle \\
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H(|1\rangle) & = \frac{1}{\sqrt{2}} |0\rangle - \frac{1}{\sqrt{2}} |1\rangle =: |-\rangle \\
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H(|+\rangle) & = |0\rangle \\
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H(|-\rangle) & = |1\rangle
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\end{align*}
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\subsection{Bramka CNOT}
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Jest to bramka kontrolowanej negacji.
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\[
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\Qcircuit @C=1em @R=.7em {
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\lstick{\ket{\alpha}} & \ctrl{1} & \rstick{\ket{\alpha}} \qw \\
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\lstick{\ket{\beta}} & \targ & \rstick{\ket{\alpha \oplus \beta }} \qw
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}
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\]
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\begin{multicols}{2}
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\begin{align*}
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\ket{00} \rightarrow \ket{00} \\
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\ket{01} \rightarrow \ket{01} \\
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\ket{10} \rightarrow \ket{11} \\
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\ket{11} \rightarrow \ket{10}
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\end{align*}
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$$
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\text{CNOT} = \begin{bmatrix}
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1 & 0 & 0 & 0 \\
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0 & 1 & 0 & 0 \\
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0 & 0 & 0 & 1 \\
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0 & 0 & 1 & 0
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\end{bmatrix}
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$$
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\end{multicols}
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\subsection{Bramka SWAP}
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Zamienia kubity ze sobą.
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\[
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\Qcircuit @C=1em @R=2em {
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\lstick{\ket{\alpha}} & \qswap & \rstick{\ket{\beta}} \qw \\
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\lstick{\ket{\beta}} & \qswap \qwx & \rstick{\ket{\alpha}} \qw
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}
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\]
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\begin{multicols}{2}
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\begin{align*}
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\ket{00} \rightarrow \ket{00} \\
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\ket{01} \rightarrow \ket{10} \\
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\ket{10} \rightarrow \ket{01} \\
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\ket{11} \rightarrow \ket{11}
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\end{align*}
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$$
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\text{SWAP} = \begin{bmatrix}
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1 & 0 & 0 & 0 \\
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0 & 0 & 1 & 0 \\
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0 & 1 & 0 & 0 \\
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0 & 0 & 0 & 1
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\end{bmatrix}
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$$
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\end{multicols}
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Implementowalna przy pomocy bramki CNOT.
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\[
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\Qcircuit @C=1em @R=.7em {
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\lstick{\ket{\alpha}} & \ctrl{1} & \targ & \ctrl{1} & \rstick{\ket{\beta}} \qw \\
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\lstick{\ket{\beta}} & \targ & \ctrl{-1} & \targ & \rstick{\ket{\alpha}} \qw
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}
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\]
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\subsection{Bramka Toffoliego}
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\[
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\Qcircuit @C=1em @R=.7em {
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\lstick{\ket{a}} & \ctrl{1} & \rstick{\ket{a}} \qw \\
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\lstick{\ket{b}} & \ctrl{1} & \rstick{\ket{b}} \qw \\
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\lstick{\ket{c}} & \targ & \rstick{\ket{c \oplus (a \land b) }} \qw
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}
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\]
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\begin{multicols}{2}
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\begin{align*}
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\ket{000} \rightarrow \ket{000} \\
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\ket{001} \rightarrow \ket{001} \\
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\ket{010} \rightarrow \ket{010} \\
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\ket{011} \rightarrow \ket{011} \\
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\ket{100} \rightarrow \ket{100} \\
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\ket{101} \rightarrow \ket{101} \\
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\ket{110} \rightarrow \ket{111} \\
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\ket{111} \rightarrow \ket{110}
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\end{align*}
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W zależności od ustawienia kubitów, bramka implementuje następujące operacje:
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\begin{align*}
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c = 0 &\longrightarrow a \land b \\
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c = 1 &\longrightarrow \lnot (a\land b) \\
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a = b = 1 & \longrightarrow \lnot c \\
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a = 1, c = 0 & \longrightarrow \text{copy } b \text{ to } c
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\end{align*}
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\end{multicols}
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Zbiór bramek odwracalnych $R$ nazywamy uniwersalnym, jeśli przy użyciu bramek należących do $R$, można zbudować dowolny układ odwracalny.
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\section{Układy kwantowe}
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\subsection{Półsumator}
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Dodaje kubity $a$ i $b$, dając w wyniku bit sumy $s$ i bit przeniesienia (carry) $c$.
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\[
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\Qcircuit @C=1em @R=.7em {
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\lstick{\ket{a}} & \ctrl{1} & \ctrl{1} & \rstick{\ket{a}} \qw \\
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\lstick{\ket{b}} & \ctrl{1} & \targ & \rstick{\ket{s}} \qw \\
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\lstick{\ket{0}} & \targ & \qw & \rstick{\ket{c}} \qw
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}
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\]
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% TODO Teleportacja kwantowa
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\end{document}
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