some work
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@ -43,57 +43,80 @@ matrices. A general unitary on $n$ qbits would be a $2^{n} \times 2^{n}$ matrix
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which requires more space than a dense state vector. The Clifford group is
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a group preserving this tensor product property.
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When lifting the constraint that $\ket{\varphi}$ is stabilized by the multilocal Pauli
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group but using $n$ arbitrary commuting hermitians $\langle h_1, ..., h_n
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\rangle$ that are the tensor product of $2\times 2$ hermitians one quickly
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realizes that one could apply any single-qbit gate to the $h_i$ and preserve
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the tensor product property. Applying the $CX$ gate however will break this
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property in general.
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%{{{
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%When lifting the constraint that $\ket{\varphi}$ is stabilized by the multilocal Pauli
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%group but using $n$ arbitrary commuting hermitians $\langle h_1, ..., h_n
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%\rangle$ that are the tensor product of $2\times 2$ hermitians one quickly
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%realizes that one could apply any single-qbit gate to the $h_i$ and preserve
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%the tensor product property. Applying the $CX$ gate however will break this
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%property in general.
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%
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%Writing $h_j = \bigotimes\limits_{i=1}^{n} h_{j,i}$,
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%$A := \left(\bigotimes\limits_{l<j} I\right)$
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%and $B := \left(\bigotimes\limits_{l>i} I\right)$ this can be seen easily
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%by transforming a general $h_k$ with $CX_{i,j}$, $i = j+1$:
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%
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%
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%\begin{equation}
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%\begin{aligned}
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% CX_{i,j} h_k CX_{i,j}^\dagger &= \left( A\otimes |1\rangle\langle 1| \otimes X \otimes B
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% + A \otimes |0\rangle\langle 0| \otimes I \otimes B\right)\\
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% &h_k\\
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% &\left(A \otimes |1\rangle\langle 1| \otimes X \otimes B
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% + A \otimes |0\rangle\langle 0| \otimes I \otimes B\right) \\
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% &= h_{k,A} \otimes h_{k,j,11} |1\rangle\langle 1| \otimes Xh_{k,i}X \otimes h_{k,B}\\
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% &+ h_{k,A} \otimes h_{k,j,00}|0\rangle\langle 0| \otimes Ih_{k,i}I \otimes h_{k,B}\\
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% &+ h_{k,A} \otimes h_{k,j,01}|0\rangle\langle 1| \otimes Ih_{k,i}X\otimes h_{k,B}\\
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% &+ h_{k,A} \otimes h_{k,j,10}|1\rangle\langle 0| \otimes Xh_{k,i}I\otimes h_{k,B}\\
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%\end{aligned}
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%\end{equation}
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%
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%Searching for hermitians $h_1, h_2$ that fulfill
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%
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%\begin{equation}
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%CX_{1,2} (h_1 \otimes h_2) CX_{1,2} = h_1' \otimes h_2'
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%\end{equation}
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%
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%and
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%
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%\begin{equation}
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%CX_{2,1} (h_1 \otimes h_2) CX_{2,1} = h_1'' \otimes h_2''
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%\end{equation}
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%
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%might be a good step to find new classes of states that can be simulated
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%efficiently using this method. This property has to be fulfilled by all
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%elements of a group generated by such hermitian matrices. How computations and
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%measurements would work using this method is not clear at the moment as many
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%basic properties of the stabilizers are lost. One important property is that the
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%stabilization: The simulated state is the $+1$ eigenstate of the stabilizers.
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%This is another property that will have to be fulfilled by the hermitians as it
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%is a key property used in \ref{ref:dynamics_stabilizer}. To ensure that the
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%state is well defined one will have to demand that the eigenvalues fulfill
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%$\lambda_1 = 1$ and $\lambda_2 < 1$.
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%
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%One should also note that the sabilizer states do not form a Hilbert (sub)space.
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%Linear combinations of stabilizer states are (in general) no stabilizer states.
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%The superposition principle is quite essential to many quantum algorithms and
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%quantum physics which limits the use of the stabilizer formalism drastically.
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%}}}
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Writing $h_j = \bigotimes\limits_{i=1}^{n} h_{j,i}$,
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$A := \left(\bigotimes\limits_{l<j} I\right)$
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and $B := \left(\bigotimes\limits_{l>i} I\right)$ this can be seen easily
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by transforming a general $h_k$ with $CX_{i,j}$, $i = j+1$:
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The stabilizer formalism as introduced in \ref{ref:stab_states} has since been
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generalized to normalizers of a finite abelian group over the Hilbert space
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\cite{bermejovega_lin_vdnest2015}\cite{bermejovega_vdnest2018}\cite{vandennest2019}\cite{vandennest2018}.
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This allows to simulate more classes of circuits efficiently on classical computers
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including the Quantum Fourier Transforms which is often believed to be
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responsible for exponential speedups. One must note that in the definition of
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the QFTs as in \cite{vandennest2018} the QFT depends on the Abelian group it
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acts on. In particular the QFT on the group that decomposes the Hilbert space
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as seen in this paper ($Z_2^n$) is just the tensor product of the $H$ gates.
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The QFT as used in \ref{ref:quantum_algorithms} for the phase estimation
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however normalizes the group $Z_{2^n}$ \cite{vandennest2018}. This yields two
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interesting results: It is not the tensor product property of the multilocal
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Pauli group that makes computations efficient but the normalization property of
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the Clifford group \cite{vandennest2018}.
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\begin{equation}
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\begin{aligned}
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CX_{i,j} h_k CX_{i,j}^\dagger &= \left( A\otimes |1\rangle\langle 1| \otimes X \otimes B
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+ A \otimes |0\rangle\langle 0| \otimes I \otimes B\right)\\
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&h_k\\
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&\left(A \otimes |1\rangle\langle 1| \otimes X \otimes B
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+ A \otimes |0\rangle\langle 0| \otimes I \otimes B\right) \\
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&= h_{k,A} \otimes h_{k,j,11} |1\rangle\langle 1| \otimes Xh_{k,i}X \otimes h_{k,B}\\
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&+ h_{k,A} \otimes h_{k,j,00}|0\rangle\langle 0| \otimes Ih_{k,i}I \otimes h_{k,B}\\
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&+ h_{k,A} \otimes h_{k,j,01}|0\rangle\langle 1| \otimes Ih_{k,i}X\otimes h_{k,B}\\
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&+ h_{k,A} \otimes h_{k,j,10}|1\rangle\langle 0| \otimes Xh_{k,i}I\otimes h_{k,B}\\
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\end{aligned}
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\end{equation}
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Searching for hermitians $h_1, h_2$ that fulfill
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\begin{equation}
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CX_{1,2} (h_1 \otimes h_2) CX_{1,2} = h_1' \otimes h_2'
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\end{equation}
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and
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\begin{equation}
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CX_{2,1} (h_1 \otimes h_2) CX_{2,1} = h_1'' \otimes h_2''
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\end{equation}
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might be a good step to find new classes of states that can be simulated
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efficiently using this method. This property has to be fulfilled by all
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elements of a group generated by such hermitian matrices. How computations and
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measurements would work using this method is not clear at the moment as many
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basic properties of the stabilizers are lost. One important property is that the
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stabilization: The simulated state is the $+1$ eigenstate of the stabilizers.
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This is another property that will have to be fulfilled by the hermitians as it
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is a key property used in \ref{ref:dynamics_stabilizer}. To ensure that the
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state is well defined one will have to demand that the eigenvalues fulfill
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$\lambda_1 = 1$ and $\lambda_2 < 1$.
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One should also note that the sabilizer states do not form a Hilbert (sub)space.
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Linear combinations of stabilizer states are (in general) no stabilizer states.
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The superposition principle is quite essential to many quantum algorithms and
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quantum physics which limits the use of the stabilizer formalism drastically.
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The exponential speedup of quantum computing is often attributed to
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entanglement, superposition and interference effects
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\cite{uwaterloo}\cite{21732}\cite{vandennest2018}. Stabilizer states
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however show both entanglement, superposition and interference effects;
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as do computations done using general normalizers \cite{vandennest2018}.
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@ -296,7 +296,7 @@ interpreted as qbit indices and are measured in the order they appear.
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If the keyword argument \lstinline{keep_states} is \lstinline{True} the
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sampling function will include the resulting states in the result. At the
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moment this works for dense vectors only. Checking for equality on graphical
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states has yet to be implemented but can be done in polynomial time
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states has yet to be implemented but has $NP$ computational hardness
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\cite{dahlberg_ea2019}.
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Writing circuits out by hand can be rather painful. The function\\
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@ -439,6 +439,13 @@ regimes:
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higher than in the low-linear regime.}
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\end{description}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{../performance/regimes/scaling_circuits_measurements_linear.png}
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\caption[Circuit Execution Timer for Scaling Circuit Length with Random Measurements]{Circuit Execution Timer for Scaling Circuit Length with Random Measurements}
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\label{fig:scaling_circuits_measurements_linear}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{../performance/regimes/graph_low_linear_regime.png}
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@ -469,6 +476,21 @@ Figure \ref{fig:graph_high_linear_regime}. The latter is hardly visible; this is
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to the great amount of edges in this regime. Further the regimes are not clearly
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visibe for $n>30$ qbits so choosing smaller graphs is not possible.
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The Figure \ref{fig:scaling_circuits_measurements_linear} brings more substance
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to this interpretation. In this simulation the Pauli $X$ gate has been replaced
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by the measurement gate $M$, .i.e. in every gate drawn from the probability
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space a qbit is measured with probability $\frac{1}{4}$. As described in
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\cite{hein_eisert_briegel2008} the Schmidt measure for entropy is bounded from
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above by Pauli persistency, i.e. the minimal amount of Pauli measurements
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required to disentangle a state. This Pauli persistency is closely related to
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the amount (and structure of) vertices in the graph
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\cite{hein_eisert_briegel2008}. In particular Pauli measurements decrease the
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entanglement (and the amount of edges) in a state
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\cite{hein_eisert_briegel2008}\cite{li_chen_fisher2019}. The frequent
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measurements in the simulation therefore keeps the amount of edges low thus
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preventing a transition from the low linear regime to the intermediate regime.
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Because states with more qbits reach the intermediate regime at higher circuit
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lengths it is important to account for this virtual performance boost when
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comparing with other simulation methods. This explains why the circuit length
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@ -232,6 +232,7 @@ Several qbits can be abbreviated by writing a slash on the qbit line:
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\subsection{Quantum Algorithms}
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\label{ref:quantum_algorithms}
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The great hope behind quantum computing is that it will speed up some
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computations exponentially using algorithms that utilize the laws of quantum
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@ -194,7 +194,7 @@
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}
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@online{
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intelqc,
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url={https://newsroom.intel.com/press-kits/quantum-computing/#gs.2s0dux},
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url={https://newsroom.intel.com/press-kits/quantum-computing/\#gs.2s0dux},
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urldate={19.09.2019},
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title={Intel Press Kit: Quantum Computing},
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author={Intel},
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@ -217,7 +217,7 @@
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}
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@online{
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lrzqc,
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url={https://www.lrz.de/wir/newsletter/2019-08/#LRZ_bereit_fuer_bayerische_Quantechnologie},
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url={https://www.lrz.de/wir/newsletter/2019-08/\#LRZ_bereit_fuer_bayerische_Quantechnologie},
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urldate={19.09.2019},
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title={LRZ-Newsletter Nr. 08/2019 vom 01.08.2019},
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author={S. Vieser},
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@ -231,3 +231,53 @@
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author={Arne Grävemeyer},
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year=2018
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}
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@article{
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li_chen_fisher2019,
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year=2019,
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author={Yaodong Li, Xiao Chen, Matthew P. A. Fisher},
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title={Measurement-driven entanglement transition in hybrid quantum circuits},
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note={https://arxiv.org/abs/1901.08092}
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}
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@article{
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bermejovega_lin_vdnest2015,
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year=2015,
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author={Juan Bermejo-Vega, Cedric Yen-Yu Lin, Maarten Van den Nest},
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title={Normalizer circuits and a Gottesman-Knill theoremfor infinite-dimensional systems},
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note={https://arxiv.org/pdf/1409.3208.pdf}
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}
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@article{
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vandennest2019,
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year=2019,
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author={Maarten Van den Nest},
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title={Classical simulation of quantum computation,the Gottesman-Knill theorem, and slightly beyond},
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note={https://arxiv.org/pdf/0811.0898v1.pdf}
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}
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@article{
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vandennest2018,
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year=2018,
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author={Maarten Van den Nest},
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title={Efficient classical simulations of quantum Fourier transforms and normalizer circuits over Abelian groups},
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note={https://arxiv.org/abs/1201.4867v1}
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}
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@article{
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bermejovega_vdnest2018,
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year=2018,
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author={Juan Bermejo-Vega, Maarten Van den Nest},
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title={Classical simulations of Abelian-group normalizer circuits with intermediate measurements},
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note={https://arxiv.org/abs/1210.3637v2}
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}
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@MISC {21732,
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TITLE = {Why and how is a quantum computer faster than a regular computer?},
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AUTHOR = {babou (https://cs.stackexchange.com/users/8321/babou)},
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HOWPUBLISHED = {Computer Science Stack Exchange},
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NOTE = {URL:https://cs.stackexchange.com/q/21732 (version: 2014-02-17)},
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EPRINT = {https://cs.stackexchange.com/q/21732},
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URL = {https://cs.stackexchange.com/q/21732}
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}
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@MISC {uwaterloo,
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TITLE = {Quantum computing 101},
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AUTHOR = {University of Waterloo: Institute for Quantum Computing},
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NOTE = {https://uwaterloo.ca/institute-for-quantum-computing/quantum-computing-101\#Quantum-effects-matter},
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URL = {https://uwaterloo.ca/institute-for-quantum-computing/quantum-computing-101\#Quantum-effects-matter}
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}
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