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127
thesis/chapters/conclusion.tex
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@ -43,57 +43,80 @@ matrices. A general unitary on $n$ qbits would be a $2^{n} \times 2^{n}$ matrix
which requires more space than a dense state vector. The Clifford group is
a group preserving this tensor product property.
When lifting the constraint that $\ket{\varphi}$ is stabilized by the multilocal Pauli
group but using $n$ arbitrary commuting hermitians $\langle h_1, ..., h_n
\rangle$ that are the tensor product of $2\times 2$ hermitians one quickly
realizes that one could apply any single-qbit gate to the $h_i$ and preserve
the tensor product property. Applying the $CX$ gate however will break this
property in general.
%{{{
%When lifting the constraint that $\ket{\varphi}$ is stabilized by the multilocal Pauli
%group but using $n$ arbitrary commuting hermitians $\langle h_1, ..., h_n
%\rangle$ that are the tensor product of $2\times 2$ hermitians one quickly
%realizes that one could apply any single-qbit gate to the $h_i$ and preserve
%the tensor product property. Applying the $CX$ gate however will break this
%property in general.
%
%Writing $h_j = \bigotimes\limits_{i=1}^{n} h_{j,i}$,
%$A := \left(\bigotimes\limits_{l<j} I\right)$
%and $B := \left(\bigotimes\limits_{l>i} I\right)$ this can be seen easily
%by transforming a general $h_k$ with $CX_{i,j}$, $i = j+1$:
%
%
%\begin{equation}
%\begin{aligned}
% CX_{i,j} h_k CX_{i,j}^\dagger &= \left( A\otimes |1\rangle\langle 1| \otimes X \otimes B
% + A \otimes |0\rangle\langle 0| \otimes I \otimes B\right)\\
% &h_k\\
% &\left(A \otimes |1\rangle\langle 1| \otimes X \otimes B
% + A \otimes |0\rangle\langle 0| \otimes I \otimes B\right) \\
% &= h_{k,A} \otimes h_{k,j,11} |1\rangle\langle 1| \otimes Xh_{k,i}X \otimes h_{k,B}\\
% &+ h_{k,A} \otimes h_{k,j,00}|0\rangle\langle 0| \otimes Ih_{k,i}I \otimes h_{k,B}\\
% &+ h_{k,A} \otimes h_{k,j,01}|0\rangle\langle 1| \otimes Ih_{k,i}X\otimes h_{k,B}\\
% &+ h_{k,A} \otimes h_{k,j,10}|1\rangle\langle 0| \otimes Xh_{k,i}I\otimes h_{k,B}\\
%\end{aligned}
%\end{equation}
%
%Searching for hermitians $h_1, h_2$ that fulfill
%
%\begin{equation}
%CX_{1,2} (h_1 \otimes h_2) CX_{1,2} = h_1' \otimes h_2'
%\end{equation}
%
%and
%
%\begin{equation}
%CX_{2,1} (h_1 \otimes h_2) CX_{2,1} = h_1'' \otimes h_2''
%\end{equation}
%
%might be a good step to find new classes of states that can be simulated
%efficiently using this method. This property has to be fulfilled by all
%elements of a group generated by such hermitian matrices. How computations and
%measurements would work using this method is not clear at the moment as many
%basic properties of the stabilizers are lost. One important property is that the
%stabilization: The simulated state is the $+1$ eigenstate of the stabilizers.
%This is another property that will have to be fulfilled by the hermitians as it
%is a key property used in \ref{ref:dynamics_stabilizer}. To ensure that the
%state is well defined one will have to demand that the eigenvalues fulfill
%$\lambda_1 = 1$ and $\lambda_2 < 1$.
%
%One should also note that the sabilizer states do not form a Hilbert (sub)space.
%Linear combinations of stabilizer states are (in general) no stabilizer states.
%The superposition principle is quite essential to many quantum algorithms and
%quantum physics which limits the use of the stabilizer formalism drastically.
%}}}
Writing $h_j = \bigotimes\limits_{i=1}^{n} h_{j,i}$,
$A := \left(\bigotimes\limits_{l<j} I\right)$
and $B := \left(\bigotimes\limits_{l>i} I\right)$ this can be seen easily
by transforming a general $h_k$ with $CX_{i,j}$, $i = j+1$:
The stabilizer formalism as introduced in \ref{ref:stab_states} has since been
generalized to normalizers of a finite abelian group over the Hilbert space
\cite{bermejovega_lin_vdnest2015}\cite{bermejovega_vdnest2018}\cite{vandennest2019}\cite{vandennest2018}.
This allows to simulate more classes of circuits efficiently on classical computers
including the Quantum Fourier Transforms which is often believed to be
responsible for exponential speedups. One must note that in the definition of
the QFTs as in \cite{vandennest2018} the QFT depends on the Abelian group it
acts on. In particular the QFT on the group that decomposes the Hilbert space
as seen in this paper ($Z_2^n$) is just the tensor product of the $H$ gates.
The QFT as used in \ref{ref:quantum_algorithms} for the phase estimation
however normalizes the group $Z_{2^n}$ \cite{vandennest2018}. This yields two
interesting results: It is not the tensor product property of the multilocal
Pauli group that makes computations efficient but the normalization property of
the Clifford group \cite{vandennest2018}.
\begin{equation}
\begin{aligned}
CX_{i,j} h_k CX_{i,j}^\dagger &= \left( A\otimes |1\rangle\langle 1| \otimes X \otimes B
+ A \otimes |0\rangle\langle 0| \otimes I \otimes B\right)\\
&h_k\\
&\left(A \otimes |1\rangle\langle 1| \otimes X \otimes B
+ A \otimes |0\rangle\langle 0| \otimes I \otimes B\right) \\
&= h_{k,A} \otimes h_{k,j,11} |1\rangle\langle 1| \otimes Xh_{k,i}X \otimes h_{k,B}\\
&+ h_{k,A} \otimes h_{k,j,00}|0\rangle\langle 0| \otimes Ih_{k,i}I \otimes h_{k,B}\\
&+ h_{k,A} \otimes h_{k,j,01}|0\rangle\langle 1| \otimes Ih_{k,i}X\otimes h_{k,B}\\
&+ h_{k,A} \otimes h_{k,j,10}|1\rangle\langle 0| \otimes Xh_{k,i}I\otimes h_{k,B}\\
\end{aligned}
\end{equation}
Searching for hermitians $h_1, h_2$ that fulfill
\begin{equation}
CX_{1,2} (h_1 \otimes h_2) CX_{1,2} = h_1' \otimes h_2'
\end{equation}
and
\begin{equation}
CX_{2,1} (h_1 \otimes h_2) CX_{2,1} = h_1'' \otimes h_2''
\end{equation}
might be a good step to find new classes of states that can be simulated
efficiently using this method. This property has to be fulfilled by all
elements of a group generated by such hermitian matrices. How computations and
measurements would work using this method is not clear at the moment as many
basic properties of the stabilizers are lost. One important property is that the
stabilization: The simulated state is the $+1$ eigenstate of the stabilizers.
This is another property that will have to be fulfilled by the hermitians as it
is a key property used in \ref{ref:dynamics_stabilizer}. To ensure that the
state is well defined one will have to demand that the eigenvalues fulfill
$\lambda_1 = 1$ and $\lambda_2 < 1$.
One should also note that the sabilizer states do not form a Hilbert (sub)space.
Linear combinations of stabilizer states are (in general) no stabilizer states.
The superposition principle is quite essential to many quantum algorithms and
quantum physics which limits the use of the stabilizer formalism drastically.
The exponential speedup of quantum computing is often attributed to
entanglement, superposition and interference effects
\cite{uwaterloo}\cite{21732}\cite{vandennest2018}. Stabilizer states
however show both entanglement, superposition and interference effects;
as do computations done using general normalizers \cite{vandennest2018}.

24
thesis/chapters/implementation.tex
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@ -296,7 +296,7 @@ interpreted as qbit indices and are measured in the order they appear.
If the keyword argument \lstinline{keep_states} is \lstinline{True} the
sampling function will include the resulting states in the result. At the
moment this works for dense vectors only. Checking for equality on graphical
states has yet to be implemented but can be done in polynomial time
states has yet to be implemented but has $NP$ computational hardness
\cite{dahlberg_ea2019}.
Writing circuits out by hand can be rather painful. The function\\
@ -439,6 +439,13 @@ regimes:
higher than in the low-linear regime.}
\end{description}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{../performance/regimes/scaling_circuits_measurements_linear.png}
\caption[Circuit Execution Timer for Scaling Circuit Length with Random Measurements]{Circuit Execution Timer for Scaling Circuit Length with Random Measurements}
\label{fig:scaling_circuits_measurements_linear}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{../performance/regimes/graph_low_linear_regime.png}
@ -469,6 +476,21 @@ Figure \ref{fig:graph_high_linear_regime}. The latter is hardly visible; this is
to the great amount of edges in this regime. Further the regimes are not clearly
visibe for $n>30$ qbits so choosing smaller graphs is not possible.
The Figure \ref{fig:scaling_circuits_measurements_linear} brings more substance
to this interpretation. In this simulation the Pauli $X$ gate has been replaced
by the measurement gate $M$, .i.e. in every gate drawn from the probability
space a qbit is measured with probability $\frac{1}{4}$. As described in
\cite{hein_eisert_briegel2008} the Schmidt measure for entropy is bounded from
above by Pauli persistency, i.e. the minimal amount of Pauli measurements
required to disentangle a state. This Pauli persistency is closely related to
the amount (and structure of) vertices in the graph
\cite{hein_eisert_briegel2008}. In particular Pauli measurements decrease the
entanglement (and the amount of edges) in a state
\cite{hein_eisert_briegel2008}\cite{li_chen_fisher2019}. The frequent
measurements in the simulation therefore keeps the amount of edges low thus
preventing a transition from the low linear regime to the intermediate regime.
Because states with more qbits reach the intermediate regime at higher circuit
lengths it is important to account for this virtual performance boost when
comparing with other simulation methods. This explains why the circuit length

1
thesis/chapters/quantum_computing.tex
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@ -232,6 +232,7 @@ Several qbits can be abbreviated by writing a slash on the qbit line:
\subsection{Quantum Algorithms}
\label{ref:quantum_algorithms}
The great hope behind quantum computing is that it will speed up some
computations exponentially using algorithms that utilize the laws of quantum

54
thesis/main.bib
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@ -194,7 +194,7 @@
}
@online{
intelqc,
url={https://newsroom.intel.com/press-kits/quantum-computing/#gs.2s0dux},
url={https://newsroom.intel.com/press-kits/quantum-computing/\#gs.2s0dux},
urldate={19.09.2019},
title={Intel Press Kit: Quantum Computing},
author={Intel},
@ -217,7 +217,7 @@
}
@online{
lrzqc,
url={https://www.lrz.de/wir/newsletter/2019-08/#LRZ_bereit_fuer_bayerische_Quantechnologie},
url={https://www.lrz.de/wir/newsletter/2019-08/\#LRZ_bereit_fuer_bayerische_Quantechnologie},
urldate={19.09.2019},
title={LRZ-Newsletter Nr. 08/2019 vom 01.08.2019},
author={S. Vieser},
@ -231,3 +231,53 @@
author={Arne Grävemeyer},
year=2018
}
@article{
li_chen_fisher2019,
year=2019,
author={Yaodong Li, Xiao Chen, Matthew P. A. Fisher},
title={Measurement-driven entanglement transition in hybrid quantum circuits},
note={https://arxiv.org/abs/1901.08092}
}
@article{
bermejovega_lin_vdnest2015,
year=2015,
author={Juan Bermejo-Vega, Cedric Yen-Yu Lin, Maarten Van den Nest},
title={Normalizer circuits and a Gottesman-Knill theoremfor infinite-dimensional systems},
note={https://arxiv.org/pdf/1409.3208.pdf}
}
@article{
vandennest2019,
year=2019,
author={Maarten Van den Nest},
title={Classical simulation of quantum computation,the Gottesman-Knill theorem, and slightly beyond},
note={https://arxiv.org/pdf/0811.0898v1.pdf}
}
@article{
vandennest2018,
year=2018,
author={Maarten Van den Nest},
title={Efficient classical simulations of quantum Fourier transforms and normalizer circuits over Abelian groups},
note={https://arxiv.org/abs/1201.4867v1}
}
@article{
bermejovega_vdnest2018,
year=2018,
author={Juan Bermejo-Vega, Maarten Van den Nest},
title={Classical simulations of Abelian-group normalizer circuits with intermediate measurements},
note={https://arxiv.org/abs/1210.3637v2}
}
@MISC {21732,
TITLE = {Why and how is a quantum computer faster than a regular computer?},
AUTHOR = {babou (https://cs.stackexchange.com/users/8321/babou)},
HOWPUBLISHED = {Computer Science Stack Exchange},
NOTE = {URL:https://cs.stackexchange.com/q/21732 (version: 2014-02-17)},
EPRINT = {https://cs.stackexchange.com/q/21732},
URL = {https://cs.stackexchange.com/q/21732}
}
@MISC {uwaterloo,
TITLE = {Quantum computing 101},
AUTHOR = {University of Waterloo: Institute for Quantum Computing},
NOTE = {https://uwaterloo.ca/institute-for-quantum-computing/quantum-computing-101\#Quantum-effects-matter},
URL = {https://uwaterloo.ca/institute-for-quantum-computing/quantum-computing-101\#Quantum-effects-matter}
}