258 lines
10 KiB
TeX
258 lines
10 KiB
TeX
% vim: ft=tex
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\section{Quantum Computing}
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\subsection{Qbits and Gates}
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\subsubsection{Single Qbits}
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\begin{definition}
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A qbit is a two level quantum mechanical system, i.e. it has the eigenbasis
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$ \{\ket{\uparrow} \equiv \ket{1}, \ket{\downarrow} \equiv \ket{0}\} $
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with $\braket{\uparrow}{\downarrow} = 0$. In the following this basis will be called
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the $Z$ basis in analogy to the conventions used in spin systems ($\sigma_Z$). For some computations
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it can be useful to have component vectors; $\ket{\uparrow} \equiv \left(\begin{array}{c} 0 \\ 1\end{array} \right)$ and
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$\ket{\downarrow} \equiv \left(\begin{array}{c} 1 \\ 0 \end{array} \right)$
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are used in these cases.
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\end{definition}
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A gate acting on a qbit is a unitary operator $G \in SU(2)$. One can show that
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$\forall G \in SU(2)$ $G$ can be approximated arbitrarily good as a product of unitary generator matrices
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\cite[Chapter 4.3]{kaye_ea2007}\cite[Chapter 2]{marquezino_ea_2019};
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common choices for the generators are $ X, H, R_{\phi}$ or $Z, H, R_{\phi}$ with
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\label{ref:singleqbitgates}
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\begin{equation}
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\begin{aligned}
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& X := \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right) \\
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& Z := \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right) \\
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& H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right) \\
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& R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)\\
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& I := \left(\begin{array}{cc} 1 & 0 \\ 0 & 1\end{array}\right). \\
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\end{aligned}
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\end{equation}
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Note that $X = HZH$ and $Z = R_{\pi}$, so the set of $H, R_\phi$ is sufficient.
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Further note that the basis vectors are chosen s.t. $Z\ket{0} = +\ket{0}$ and $Z\ket{1} = -\ket{1}$;
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transforming to the other Pauli eigenstates is done using $H$ and $SH$
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\begin{equation}
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S := R_{\frac{\pi}{2}} = \left(\begin{array}{cc} 1 & 0 \\ 0 & i\end{array}\right)
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\end{equation}
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\begin{equation}
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S H Z H^\dagger S^\dagger = S X S^\dagger = Y.
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\end{equation}
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The following states are the $\pm 1$ eigenstates of the $X,Y,Z$ operators
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and will be used in some calculations later:
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\begin{equation}
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\begin{aligned}
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&\ket{+_X} \equiv \ket{+} := \frac{1}{\sqrt{2}} (\ket{0} + \ket{1}) \\
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&\ket{-_X} \equiv \ket{-} := \frac{1}{\sqrt{2}} (\ket{0} - \ket{1}) \\
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&\ket{+_Y} := \frac{1}{\sqrt{2}} (\ket{0} + i\ket{1}) \\
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&\ket{-_Y} := \frac{1}{\sqrt{2}} (\ket{0} - i\ket{1}) \\
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&\ket{+_Z} := \ket{0} \\
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&\ket{-_Z} := \ket{1} \\
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\end{aligned}
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\end{equation}
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\subsubsection{Many-Qbit Systems}
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\label{ref:many_qbits}
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\begin{postulate}
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A $n$-qbit quantum mechanical state is the tensor product of the $n$ one-qbit
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states \cite[Postulate 4]{nielsen_chuang_2010}.
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\end{postulate}
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Let $\ket{0}_s := \left(\begin{array}{c} 1 \\ 0 \end{array} \right)$ and $\ket{1}_s := \left(\begin{array}{c} 0 \\ 1 \end{array} \right)$
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be the basis of the one-qbit systems. Then two-qbit basis states are
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\begin{equation}
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\ket{0} := \ket{0b00} := \ket{0}_s \otimes \ket{0}_s := \left(\begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right)
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\end{equation}
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\begin{equation}
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\ket{1} := \ket{0b01} := \ket{0}_s \otimes \ket{1}_s := \left(\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array} \right)
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\end{equation}
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\begin{equation}
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\ket{2} := \ket{0b10} := \ket{1}_s \otimes \ket{0}_s := \left(\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array} \right)
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\end{equation}
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\begin{equation}
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\ket{3} := \ket{0b11} := \ket{1}_s \otimes \ket{1}_s := \left(\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right).
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\end{equation}
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The $n$-qbit basis states can be constructed in a similar manner.
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A general $n$-qbit state is now a superposition of the
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integer states
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\begin{equation}
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\label{eq:ci}
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\ket{\psi} = \sum\limits_{i = 0}^{2^n - 1} c_i \ket{i} .
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\end{equation}
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with the normation condition
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\begin{equation}
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\sum\limits_{i = 0}^{2^n - 1} |c_i|^2 = 1.
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\end{equation}
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The states $\ket{i}$ for $i = 0, ..., 2^{n}-1$ are called integer states. Note
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that integer states are eigenstates of the $Z$ operators. The computational basis
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is $\{\ket{i_0} \otimes ... \otimes \ket{i_{n-1}} | i_0, ..., i_{n-1} = 0, 1\}$.
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\begin{definition}
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For a single-qbit gate $U$ and a qbit $j = 0, 1, ..., n - 1$
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\begin{equation}
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U_j := \left(\bigotimes\limits_{i = 0}^{j - 1} I\right)
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\otimes U
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\otimes \left(\bigotimes\limits_{i = j + 1}^{n - 1} I \right)
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\end{equation}
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is acting on qbit $j$.
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\end{definition}
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% XXX
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\newpage
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\begin{definition}\label{def:CU}
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For two qbits $i,j = 0, 1, ..., n - 1$, $i \neq j$ and a gate $U_i$ acting on $i$
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the controlled version of $U$ is defined by
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\begin{equation}
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\begin{aligned}\label{eq:CU}
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CU_{i, j} &:= &\ket{0}\bra{0}_j\otimes I_i
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+ \ket{1}\bra{1}_j \otimes U_i \\
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&:= &\left(\bigotimes\limits_{l < j} I\right)
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\otimes \ket{0}\bra{0}
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\otimes \left(\bigotimes\limits_{j < l < i} I\right)
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\otimes I
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\otimes \left(\bigotimes\limits_{j > l} I \right)\\
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& &+ \left(\bigotimes\limits_{l < j} I\right)
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\otimes \ket{1}\bra{1}
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\otimes \left(\bigotimes\limits_{j < l < i} I\right)
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\otimes U
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\otimes \left(\bigotimes\limits_{j > l} I \right)\\
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\end{aligned}
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\end{equation}
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where without loss of generality $j < i$; the other case is analogous.
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In particular for $X, Z$
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\begin{equation}\label{eq:CX_pr}
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CX(i, j) = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes X_i
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\end{equation}
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\begin{equation}\label{eq:CZ_pr}
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CZ(i, j) = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes Z_i .
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\end{equation}
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\end{definition}
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In Definition \ref{def:CU} $i$ is called the act-qbit and $j$ the control-qbit. In words
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$CU$ applies the gate $U$ to the act-qbit if the control-qbit is in its $\ket{1}$ state.
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One can show that the gates in \ref{ref:singleqbitgates} together with an entanglement gate, such as $CX$ or $CZ$ are enough
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to generate an arbitrary $n$-qbit gate \cite[Chapter 4.3]{kaye_ea2007}\cite{barenco_ea_1995}.
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The matrix representation of $CX$ and $CZ$ for two qbits is given by
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\begin{equation}
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CX_{1, 0} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{array}\right)
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\end{equation}
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\begin{equation}
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CZ_{1, 0} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1 \end{array}\right).
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\end{equation}
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\subsubsection{Measurements}
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\begin{postulate}
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Let
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$$\ket{\psi} = \alpha\ket{\phi_1} \otimes \ket{1}_j + \beta\ket{\phi_0} \otimes \ket{0}_j$$
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be a $n$-qbit state
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where $\ket{1}_j, \ket{0}_j$ denote the $j$-th qbit state and $|\alpha|^2 + |\beta|^2 = 1$.
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Then the measurement of the $j$-th qbit will yield
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$$\ket{\phi_1} \otimes \ket{1}_j$$
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with probability $|\alpha|^2$ and
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$$\ket{\phi_0} \otimes \ket{0}_j$$
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with probability $|\beta|^2$. This is called collapse of the wave function.
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\end{postulate}
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Measuring a qbit will also yield a classical result $0$ or $1$ with the respective probabilities.
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Measurements are always performed in the computational basis, i.e. for a qbit
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$i$ $Z_i$ is measured. The $+1$ eigenvalue of $Z_i$ is $\ket{0}_i$, $\ket{1}_i$ for $-1$.
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\begin{corrolary}
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In general the measurement of a qbit is not invertible, in particular it cannot be represented as a
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unitary operator.
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\end{corrolary}
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\begin{proof}
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The measuerment is not injective: Measuring both
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$\ket{0}$ and $\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ (can) map to $\ket{0}$.
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Any unitary matrix $U$ has the inverse $U^\dagger \equiv U^{-1}$.
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\end{proof}
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Because a measurement is not unitary it is not a gate in the sense of the definition above.
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In the following discussion the term \textit{measurement gate} will be used from time
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to time as a measurement can be treated similarely while doing numerics.
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\subsection{Quantum Circuits}
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\label{ref:quantum_circuits}
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As mentioned in \ref{ref:many_qbits} one can approximate an arbitrary $n$-qbit
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gate $U$ as a product of some single-qbit gates and either $CX$ or $CZ$.
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Writing (possibly huge) products of matrices is quite unpractical and
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unreadable. To address this problem quantum circuits have been introduced.
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These represent the qbits as a horizontal line and a gate acting on a qbit is
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a box with a name on the respective line. Quantum circuits are read from
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left to right. This means that a gate $U_i = Z_i X_i H_i$ has the
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circuit representation
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\[
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\Qcircuit @C=1em @R=.7em {
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& \gate{H} & \gate{X} & \gate{Z} &\qw \\
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}.
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\]
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The controlled gates (such as $CX$ and $CZ$) have a vertical line from the control-qbit to
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the gate, for instance the circuit for $CZ_{2, 1}CX_{2,0}$ is
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\[
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\Qcircuit @C=1em @R=.7em {
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& \qw & \ctrl{2} & \qw & \qw &\qw \\
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& \qw & \qw & \qw & \ctrl{1} &\qw \\
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& \qw & \gate{X} & \qw & \gate{Z} &\qw &\rstick{.}\\
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}
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\]
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Several qbits can be abbreviated by writing a slash on the qbit line:
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\[
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\Qcircuit @C=1em @R=.7em {
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& {/} \qw & \qw & \gate{Z^{\otimes n}} & \qw \\
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}
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\]
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\subsection{Quantum Algorithms}
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The great hope behind quantum computing is that it will speed up some computations
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exponentially using algorithms that utilize the laws of quantum mechanics. Current
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algorithms are based upon quantum search and quantum fourier transform \cite{nielsen_chuang_2010}.
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The latter is particular interesting for physical problems as it is used in the phase estimation
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algorithm that can be used to analyze the spectrum of the transfer matrix:
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\begin{equation}
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T = \exp(itH)
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\end{equation}
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The eigenvalue of $T$ can now be estimated by using the phase estimation circuit
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\[
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\Qcircuit @C=1em @R=.7em {
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& \lstick{\ket{0}} & {/^N} \qw & \gate{H^{\otimes n}} & \ctrl{1} & \gate{FT^\dagger} & \qw & \meter & \rstick{x}\\
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& \lstick{\ket{\varphi}} & {/} \qw & \qw & \gate{T} & \qw & {/} \qw & \qw& \rstick{\ket{\varphi}.} \\
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}
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\]
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Where $T\ket{\varphi} = \exp(2\pi i\varphi) \ket{\varphi}$ and the measurement result $\tilde{\varphi} = \frac{x}{2^n}$ is an
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estimation for $\varphi$. If a success rate of $1-\epsilon$ and an accuracy of $| \varphi - \tilde{\varphi} | < 2^{-m}$
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is wanted, then $N = m + \log_2(2 + \frac{1}{2\epsilon})$ qbits are required \cite{nielsen_chuang_2010}\cite{lehner2019}.
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