some work
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@ -56,6 +56,7 @@ and will be used in some calculations later.
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\end{equation}
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\end{equation}
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\subsubsection{Many Qbits}
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\subsubsection{Many Qbits}
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\label{ref:many_qbits}
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\begin{postulate}
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\begin{postulate}
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A $N$ qbit quantum mechanical state is the tensor product\cite[Definition 14.3]{wuest1995} of the $N$ single qbit
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A $N$ qbit quantum mechanical state is the tensor product\cite[Definition 14.3]{wuest1995} of the $N$ single qbit
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@ -174,12 +175,14 @@ to time as a measurement can be treated similarely while doing numerics.
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\subsection{Quantum Circuits}
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\subsection{Quantum Circuits}
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\textbf{FIXME: citation needed}
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As mentioned in \ref{ref:many_qbits} one can approximate an arbitrary $n$
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qbit gate $U$ as a product of some single qbit gates and either $CX$ or $CZ$.
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Quantum circuits are a simple and well-readable way to express the application
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Writing (possibly huge) products of matrices is quite unpractical and very much
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of several gates on a state. Qbits are represented by horizontal line on which
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unreadable. To address this problem quantum circuits have been introduced.
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time goes from left to right. On these lines single qbit gates are
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These represent the qbits as a horizontal line, a gate acting on a qbit is
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represented by a box:
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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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\[
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\Qcircuit @C=1em @R=.7em {
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\Qcircuit @C=1em @R=.7em {
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@ -198,4 +201,6 @@ the gate, for instance the circuit for $CZ_{2, 1}CX_{2,0}$ is
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}
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}
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\]
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\]
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\textbf{TODO: more info about quantum circuits}
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\subsection{Quantum Algorithms}
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@ -92,7 +92,7 @@ properties to understand the properties of the group.
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and $m$ is the smallest integer for which these statements hold.
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and $m$ is the smallest integer for which these statements hold.
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\end{definition}
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\end{definition}
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In the following discussions $\rangle S^{(i)} \rangle_{i=0, ..., n-1}$ will be used as
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In the following discussions $\langle S^{(i)} \rangle_{i=0, ..., n-1}$ will be used as
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the properties of a set of stabilizers that are used in the discussions can be studied using only its
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the properties of a set of stabilizers that are used in the discussions can be studied using only its
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generators.
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generators.
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@ -117,4 +117,91 @@ can be diagonalized simultaneously. This motivates and justifies the following d
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It is clear that it is sufficient to show the stabilization property for the generators of
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It is clear that it is sufficient to show the stabilization property for the generators of
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$S$, as all the generators forming an element in $S$ can be absorbed into $\ket{\psi}$.
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$S$, as all the generators forming an element in $S$ can be absorbed into $\ket{\psi}$.
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The dimension of $V_S$ is not immediately
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The dimension of $V_S$ is not immediately clear. One can however show that
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for a set of stabilizers $\langle S^{(i)} \rangle_{i=1, ..., n-m}$ the dimension
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$dim V_S = 2^m$ \cite[Chapter 10.5]{nielsen_chuang_2010}. This yields the following important
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result:
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\begin{theorem}
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For a $n$ qbit system and a set $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ the stabilizer
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space $V_S$ has $dim V_S = 1$, in particular there exists an up to a trivial phase unique
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state $\ket{\psi}$ that is stabilized by $S$.
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Without proof.
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\end{theorem}
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In the following discussions for $n$ qbits a set $S = \langle S^{(i)} \rangle_{i=1,...,n}$
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of $n$ independent stabilizers will be assumed.
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\subsubsection{Dynamics of Stabilizer States}
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Consider a $n$ qbit state $\ket{\psi}$ that is the $+1$ eigenstate of $S = \langle S^{(i)} \rangle_{i=1,...,n}$
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and a unitary transformation $U$ that describes the dynamics of the system, i.e.
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\begin{equation}
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\ket{\psi'} = U \ket{\psi}
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\end{equation}
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It is clear that in general $\ket{\psi'}$ will not be stabilized by $S$ anymore. There are
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however some statements that can still be made:
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\begin{equation}
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\begin{aligned}
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\ket{\psi'} &= U \ket{\psi} \\
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&= U S^{(i)} \ket{\psi} \\
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&= U S^{(i)} U^\dagger U\ket{\psi} \\
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&= U S^{(i)} U^\dagger \ket{\psi'} \\
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&= S^{\prime(i)} \ket{\psi'} \\
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\end{aligned}
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\end{equation}
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Note that in \ref{def:stabilizer} it has been demanded that stabilizers are a
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subgroup of the multilocal Pauli operators. This does not hold true for an arbitrary
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$U$ but there exists a group for which $S'$ will be a set of stabilizers.
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\begin{definition}
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For $n$ qbits
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\begin{equation}
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C_n := \left\{U \in SU(n) | UpU^\dagger \in P_n \forall p \in P_n\right\}
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\end{equation}
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is called the Clifford group. $C_1 =: C_L$ is called the local Clifford group.
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\end{definition}
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\begin{theorem}
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\begin{enumerate}
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\item{$C_L$ can be generated using only $H$ and $S$.}
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\item{$C_L$ can be generated from $\sqrt{iZ} = \exp(\frac{i\pi}{4}) S^\dagger$
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and $\sqrt{-iX} = \frac{1}{\sqrt{2}} \left(\begin{array}{cc} 1 & -i \\ -i & 1 \end{array}\right)$.
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Also $C_L$ is generated by a product of at most $5$ matrices $\sqrt{iZ}$, $\sqrt{-iX}$.
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}
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\item{$C_n$ can be generated using $C_L$ and $CZ$ or $CX$.}
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\end{enumerate}
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\end{theorem}
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\begin{proof}
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\begin{enumerate}
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\item{See \cite[Theorem 10.6]{nielsen_chuang_2010}}
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\item{
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One can easily verify that $\sqrt{iZ} \in C_L$ and $\sqrt{-iX} \in C_L$.
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Further one can show easily that (up to a global phase)
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$H = \sqrt{iZ} \sqrt{-iX}^3 \sqrt{iZ}$ and $S = \sqrt{iZ}^3$.
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The length of the product can be seen when explicitly calculating
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$C_L$.
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}
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\item{See \cite[Theorem 10.6]{nielsen_chuang_2010}}
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\end{enumerate}
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\end{proof}
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This is quite an important result: As under a transformation $U \in C_n$ $S'$ is a set of
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$n$ independent stabilizers and $\ket{\psi'}$ is stabilized by $S'$ one can consider
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the dynamics of the stabilizers instead of the actual state. This is considerably more
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efficient as only $n$ stabilizers have to be modified, each being just the tensor
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product of $n$ Pauli matrices. This has led to the simulation using stabilizer tableaux
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\cite{gottesman_aaronson2008}.
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Interestingly also measurements are dynamics covered by the stabilizers.
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When an observable $g_i \in \{\pm X_i, \pm Y_i \pm Z_i\}$ is measured
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