\documentclass[a4paper]{article} \usepackage{amsmath,amssymb,amsthm} \usepackage{braket} \usepackage[ps2pdf,bookmarks=true,bookmarksnumbered=true]{hyperref} \input{qcircuit} \hypersetup{ pdfauthor={Michal Charemza}, pdftitle={Quantum Circuit Diagrams}, pdfsubject={Quantum Computing}, pdfkeywords={quantum, computing, quantum circuit diagram} } \begin{document} \author{Michal Charemza} \title{Examples of Quantum Circuit Diagrams} \date{April 2006} \maketitle Below some examples of quantum circuit diagrams are given. They are typset using a modified version of the \LaTeX{} package \emph{QCircuit}. The source for the present document, the modified \emph{QCircuit} package, and the project \emph{An Introduction to Quantum Computing} from which the diagrams are taken are available online at http://go.warwick.ac.uk/mtcharemza/pastprojects . \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { \lstick{\ket{0}} & \gate{H} & \rstick{\frac{1}{\sqrt{2}}(\ket{0} + \ket{1}) } \qw } } \caption{Example of Hadamard gate acting on one qubit.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { \lstick{\ket{0}} & \targ & \rstick{\ket{1}} \qw } } \caption{Example of a \emph{not} gate acting on one qubit.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { \lstick{\ket{1}} & \ctrl{1} & \rstick{\ket{1}} \qw \\ \lstick{\ket{0}} & \targ & \rstick{\ket{1}} \qw } } \caption{Example of a controlled-\emph{not} gate.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { \lstick{\frac{1}{\sqrt{2}}\left(\ket{0}+\ket{1}\right)} & \measureD{M} & \rstick{\text{?}} \cw } } \caption{Example of a measurement. Note that for the input mixed state $\frac{1}{\sqrt{2}}\left(\ket{0}+\ket{1}\right)$, it is unknown what the result of the measurement will be. All that is known is that the result has equal probability of being $\ket{0}$ or $\ket{1}$.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { \lstick{\ket{1}} & \ctrl{1} & \rstick{\ket{1}} \qw \\ \lstick{\ket{0}} & \targ & \rstick{\ket{1}} \qw } } \caption{Example of a controlled-\emph{not} gate.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { \lstick{\frac{1}{\sqrt{2}}\left(\ket{0}+\ket{1}\right)} & \measureD{M} & \rstick{\text{?}} \cw } } \caption{Example of a measurement. Note that for the input mixed state $\frac{1}{\sqrt{2}}\left(\ket{0}+\ket{1}\right)$, it is unknown what the result of the measurement will be. All that is known is that the result has equal probability of being $\ket{0}$ or $\ket{1}$.} \end{figure} \clearpage \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=0.1em { & & & \multigate{13}{\rule{2em}{0em}C\rule{2em}{0em}} & & \\ & & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & & \\ & & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & \qw & \\ & \lstick{\raisebox{1.2em}{ $\mathbf{x} \in \mathbb{F}^2_n$} } & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & \qw & \rstick{\raisebox{1.4em}{$f(\mathbf{x}) \in \mathbb{F}^2_m$} } \\ & & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & & \\ & & & \ghost{\rule{2em}{0em}C\rule{2em}{0em}} & & \push{\rule{0.1em}{0em}} \gategroup{1}{3}{14}{3}{1.5em}{\{} \gategroup{3}{5}{12}{5}{1.5em}{\}} } } \caption{Boolean circuit performing function $f:\mathbb{F}^2_n \rightarrow \mathbb{F}^2_m$.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=1em @!R { \lstick{\ket{x}} & \qw & \ctrl{1} & \qw & \rstick{\ket{x}} \qw \\ \lstick{\ket{y}} & \qw & \ctrl{1} & \qw & \rstick{\ket{y}} \qw \\ \lstick{\ket{z}} & \qw & \targ & \qw & \rstick{\ket{x \oplus (y \wedge z})} \qw } } \caption{Toffoli gate.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=1em @!R { \lstick{\ket{x}} & \qw & \qw & \ctrl{1} & \qw & \ctrl{1} & \ctrl{2} & \qw & \rstick{\ket{x}} \qw \\ \lstick{\ket{y}} & \qw & \ctrl{1} & \targ & \ctrl{1} & \targ & \qw & \qw & \rstick{\ket{y}} \qw \\ \lstick{\ket{z}} & \gate{H} & \gate{F_{\pi / 2}} & \qw & \gate{F_{3\pi / 2}} & \qw & \gate{F_{\pi / 2}} & \gate{H} & \rstick{\ket{x \oplus (y \wedge z})} \qw } } \caption{Decomposition of a Toffoli gate.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { & & & & & & \\ \lstick{\ket{x}} & \qw & \qw & \ctrl{1} & \rstick{\ket{x}} \qw & & \\ \lstick{\ket{y}} & \qw & \qw & \ctrl{1} & \rstick{\ket{y}} \qw & & \\ & & \lstick{\ket{0}} & \targ & \qw & \qw & \rstick{\ket{x \wedge y}} \qw \\ & & & & & & \push{\rule{0.2em}{0em}} \gategroup{1}{2}{5}{6}{1.5em}{.} } } \caption{Toffoli gate as an \emph{and} gate.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { & & & & & & & \\ \lstick{\ket{x}} & \qw & \qw & \targ & \ctrl{1} & \rstick{\ket{\neg x}} \qw & & \\ \lstick{\ket{y}} & \qw & \qw & \targ & \ctrl{1} & \rstick{\ket{\neg y}} \qw & & \\ & & \lstick{\ket{0}} & \qw & \targ & \targ & \qw & \rstick{\ket{x \vee y}} \qw \\ & & & & & & & \push{\rule{0.2em}{0em}} \gategroup{1}{2}{5}{7}{1.5em}{.} } } \caption{A Toffoli gate as an \emph{or} gate.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { & & & & & & & \\ \lstick{\ket{x}} & \qw & \qw & \qw & \ctrl{1} & \qw & \qw & \rstick{\ket{x}} \qw \\ & & \lstick{\ket{0}} & \targ & \ctrl{1} & \rstick{\ket{1}} \qw & & \\ & & \lstick{\ket{0}} & \qw & \targ & \qw & \qw & \rstick{\ket{x}} \qw \\ & & & & & & & \push{\rule{0.2em}{0em}} \gategroup{1}{2}{5}{7}{1.5em}{.} } } \caption{Toffoli gate as \emph{fanout}.} \end{figure} \clearpage \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=0.1em { & & & \multigate{21}{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \rstick{\raisebox{1.4em}{$\ket{f(\mathbf{x})}$}} \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & \lstick{\raisebox{1.4em}{$\ket{\mathbf{x}}$}} & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghostnowire{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghostnowire{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \rstick{\raisebox{1.4em}{garbage bits}} \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & \lstick{\raisebox{1.4em}{$\ket{\mathbf{0}}$ (ancilla qubits)}} & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \\ & & & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \push{\rule{0.1em}{0em}} \gategroup{1}{3}{14}{3}{1.5em}{\{} \gategroup{17}{3}{22}{3}{1.5em}{\{} \gategroup{1}{5}{8}{5}{1.5em}{\}} \gategroup{11}{5}{22}{5}{1.5em}{\}} } } \caption{Quantum circuit emulating boolean circuit that performs function $f:\mathbb{F}^2_n \rightarrow \mathbb{F}^2_m$.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=0.4em @R=0.1em { & & & \qw & \multigate{21}{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \ctrl{23} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \multigate{21}{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \ctrl{23} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \ctrl{23} & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \ctrl{23} & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \ctrl{23} & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \ctrl{23} & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ctrl{23} & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ \lstick{\raisebox{1.4em}{$\ket{\mathbf{x}}$}} & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ctrl{23} & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \rstick{\raisebox{1.4em}{$\ket{\mathbf{x}}$}} \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & & & & & & & & & & & \ghostnowire{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & & & & & & & & & & & \ghostnowire{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & & \ghostnowire{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & & & & \\ & & & & \ghostnowire{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & & & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ \lstick{\raisebox{1.4em}{$\ket{\mathbf{0}}$}} & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \rstick{\raisebox{1.4em}{$\ket{\mathbf{0}}$}} \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & \qw & \ghost{\rule{2em}{0em}R\rule{2em}{0em}} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ghost{\rule{1.8em}{0em}R^{-1}\rule{1.8em}{0em}} & \qw & \qw & & \\ & & & & \push{\rule{0em}{2em}} & & & & & & & & & & & \push{\rule{0em}{1em}} & & & & \\ & & & \qw & \qw & \qw & \targ & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & & \\ & & & \qw & \qw & \qw & \qw & \targ & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & & \\ & & & \qw & \qw & \qw & \qw & \qw & \targ & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & & \\ & & & \qw & \qw & \qw & \qw & \qw & \qw & \targ & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & & \\ \lstick{\raisebox{1.4em}{$\ket{\mathbf{0}}$}} & & & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \targ & \qw & \qw & \qw & \qw & \qw & \qw & \qw & & \rstick{\raisebox{1.4em}{$\ket{f(\mathbf{x})}$}} \\ & & & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \targ & \qw & \qw & \qw & \qw & \qw & \qw & & \\ & & & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \targ & \qw & \qw & \qw & \qw & \qw & & \\ & & & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \targ & \qw & \qw & \qw & \qw & & \push{\rule{0.01em}{0em}} \gategroup{1}{3}{14}{3}{1.5em}{\{} \gategroup{17}{3}{22}{3}{1.5em}{\{} \gategroup{24}{3}{31}{3}{1.5em}{\{} \gategroup{1}{18}{14}{18}{1.5em}{\}} \gategroup{17}{18}{22}{18}{1.5em}{\}} \gategroup{24}{18}{31}{18}{1.5em}{\}} } } \caption{Quantum circuit emulating boolean circuit that performs function $f:\mathbb{F}^2_n \rightarrow \mathbb{F}^2_m$. Note that this circuit preserves input and ancilla qubits.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { & & & & & & & & \mbox{Alice} \\ \lstick{a\rule{0.5em}{0em}} & \control \cw \qwx[2] & \cw & \cw & \cw & \cw & \cw & \cw & \lstick{a} \\ \lstick{b\rule{0.5em}{0em}} & \cw & \control \cw \qwx[1] & \cw & \cw & \cw & \cw & \rstick{\rule{0.5em}{0em}} \cw & \lstick{b} \\ \lstick{A\rule{0.5em}{0em}} & \gate{F} & \targ & \qw \qwx[1] & & & & \push{\rule{0em}{1.5em}} & \\ & & & \qwx[1] & & & & & \\ & & & \qwx[1] & & & & & \mbox{Bob} \\ \push{\rule{0em}{1.5em}} & & & & \ctrl{1} & \gate{H} & \measureD{M} & \cw & \lstick{a} \\ \lstick{B\rule{0.5em}{0em}} & \qw & \qw & \qw & \targ & \qw & \measureD{M} & \push{\rule{0em}{1.5em}} \cw & \lstick{b} \gategroup{1}{1}{4}{8}{.7em}{.} \gategroup{6}{1}{8}{8}{.7em}{.} } } \caption{Superdense coding.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { & & & & & & & & & \mbox{Alice} \\ \lstick{\ket{\psi} \rule{0.5em}{0em}} & \ctrl{1} & \gate{H} & \measureD{M} & \cw & \cw \cwx[1] & & & & \\ \lstick{A\rule{0.5em}{0em}} & \targ & \qw & \measureD{M} & \cw \cwx[1] & \cwx[1] & & & \push{\rule{0em}{1.5em}} & \\ & & & & \cwx[1] & \cwx[1] & & & & \\ & & & & \cwx[1] & \cwx[1] & & & & \mbox{Bob} \\ & & & & \cwx[1] & & \cw & \control \cw \qwx[2] & \cw & \\ & & & & & \cw & \control \cw \qwx[1] & \cw & \cw & \\ \lstick{B\rule{0.5em}{0em}} & \qw & \qw & \qw & \qw & \qw & \targ & \gate{F} & \push{\rule{0em}{1.5em}} \qw & \ket{\psi} \gategroup{1}{1}{3}{9}{.7em}{.} \gategroup{5}{1}{8}{9}{.7em}{.} } } \caption{Quantum teleportation.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { & & \multigate{4}{QFT_{\mathbb{F}_{2}^{m}}} & \qw & \\ & & \ghost{QFT_{\mathbb{F}_{2}^{m}}} & \qw & \\ \lstick{\raisebox{1.2em}{ $\ket{\mathbf{x}}$}~~ } & & \ghost{QFT_{\mathbb{F}_{2}^{m}}} & \qw & \rstick{\raisebox{1.2em}{~ $\frac{1}{\sqrt{2^m}}\displaystyle\sum_{\mathbf{y}\in\mathbb{F}_{2}^{m}}(-1)^{\mathbf{x}.\mathbf{y}}\ket{\mathbf{y}}$} } \\ & \vdots & & \vdots & \\ & & \ghost{QFT_{\mathbb{F}_{2}^{m}}} & \qw & \push{\rule{0em}{0.1em}} \gategroup{1}{1}{5}{1}{0.5em}{\{} \gategroup{1}{5}{5}{5}{0.5em}{\}} } } \caption{Quantum Fourier transform in $\mathbb{F}_{2}^{m}$.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=1.5em { & & \gate{\rule{2em}{0em}H\rule{2em}{0em}} & \qw & \\ & & \gate{\rule{2em}{0em}H\rule{2em}{0em}} & \qw & \\ \lstick{\raisebox{1.2em}{ $\ket{\mathbf{x}}$}~~ } & & \gate{\rule{2em}{0em}H\rule{2em}{0em}} & \qw & \rstick{\raisebox{1.2em}{~ $\frac{1}{\sqrt{2^m}}\displaystyle\sum_{\mathbf{y}\in\mathbb{F}_{2}^{m}}(-1)^{\mathbf{x}.\mathbf{y}}\ket{\mathbf{y}}$} } \\ & \vdots & & \vdots & \\ & & \gate{\rule{2em}{0em}H\rule{2em}{0em}} & \qw & \push{\rule{0em}{0.1em}} \gategroup{1}{1}{5}{1}{0.5em}{\{} \gategroup{1}{5}{5}{5}{0.5em}{\}} } } \caption{Decomposition of QFT in $\mathbb{F}_{2}^{m}$.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=1.5em @R=1em { \lstick{\ket{\psi}} & \ctrl{1} & \targ & \ctrl{1} & \rstick{\ket{\phi}} \qw \\ \lstick{\ket{\phi}} & \targ & \ctrl{-1} & \targ & \rstick{\ket{\psi}} \qw } } \caption{Swap of two qubits.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=1em @R=0.75em { \lstick{\ket{x_{m-1}}} & \gate{H} & \gate{\phi} & \gate{\phi} & \qw & \gate{\phi} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \rstick{\ket{y_0}} \qw \\ \lstick{\ket{x_{m-2}}} & \qw & \ctrl{-1} & \qw & \qw & \qw & \gate{H} & \gate{\phi} & \qw & \gate{\phi} & \qw & \qw & \qw & \qw & \rstick{\ket{y_1}} \qw \\ \lstick{\ket{x_{m-3}}} & \qw & \qw & \ctrl{-2} & \qw & \qw & \qw & \ctrl{-1} & \qw & \qw & \gate{H} & \qw & \gate{\phi} & \qw & \rstick{\ket{y_2}} \qw \\ \lstick{\vdots } & & & & \ddots & & & & \ddots & & & \ddots & & & \rstick{\vdots } \\ \lstick{\ket{x_{0}}} & \qw & \qw & \qw & \qw & \ctrl{-4} & \qw & \qw & \qw & \ctrl{-3} & \qw & \qw & \ctrl{-2} & \gate{H} & \rstick{\ket{y_{m-1}}} \qw } } \caption{Decomposition of QFT in $\mathbb{Z}_{2^m}$.} \end{figure} \begin{figure}[ht] \centerline{ \Qcircuit @C=2em @R=0.9em { & & \multigate{4}{H_m} & \multimeasureD{4}{M} & \cw & \\ & & \ghost{H_m} & \ghost{M} & \cw & \\ \lstick{\raisebox{1.2em}{$\ket{0}$~~}} & & \ghost{H_m} & \ghost{M} & \cw & \rstick{\raisebox{1.2em}{~~$\ket{y}$} } \\ & \vdots & & & \vdots & \\ & & \ghost{H_m} & \ghost{M} & \cw & \push{\rule{0em}{0.5em}} \gategroup{1}{6}{5}{6}{.7em}{\}} \gategroup{1}{1}{5}{1}{.7em}{\{} } } \caption{Circuit that generates random numbers.} \end{figure} \end{document}