bachelor_thesis/thesis/chapters/graph_simulator.tex

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\section{The Graph Simulator}
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\subsection{Mathematical Prerequisites}
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The following definitions and lemmata are required to understand both how the
graph formalism works and how the simulator handles gates.
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\begin{definition}
\begin{equation}
p \in P_n \Rightarrow p = \bigotimes\limits_{i=0}^n p_i \\
\forall i: p_i \in P := \{\pm 1, \pm i\} \cdot \{I, X, Y, Z\}
\end{equation}
Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$,
$Y = \left(\begin{array}{cc} 0 & i \\ -i & 0\end{array}\right)$ and
$Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)$ are the Pauli matrices and
$I$ is the identity.
\end{definition}
\begin{definition}
Let $p_i \in P_n \forall i = 1, ..., n$, $[p_i, p_j] = 0 \forall i,j$ be commuting multi-local Pauli operators.
Then a $n$ qbit state $\ket{\psi}$ is called a stabilizer state iff
\begin{equation}
\forall i: p_i\ket{\psi} = +1\ket{\psi}
\end{equation}
\end{definition}
%A $n$ qbit graph or stabilizer state is a $+1$ eigenstate of some $ p \in P_n$ where $P_n$ is the Pauli group\cite{andersbriegel2005}.
\begin{definition}
\begin{equation}
C_n := \{U \in SU(2) | UpU^\dagger \in P_n \forall p \in P_n\}
\end{equation}
is called the Clifford group on $n$ qbits.
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$C_1$ is called the local Clifford group.
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\end{definition}
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One can show that the Clifford group $C_n$ can be generated using the elements of $C_1$ acting on all qbits and
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the controlled phase gate $CZ$ between all qbits\cite{andersbriegel2005}. It is worth noting that the $CX$ gate can be
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generated using $CZ$ and $C_1$ gates.
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\begin{lemma}
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Let $a \in C_1$ then $\forall \phi \in [0, 2\pi)$ also $\exp(i\phi)a \in C_1$.
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\textbf{Note}: This is also true for $C_n \forall n >= 1$.
\end{lemma}
\begin{proof}
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Let $a' := \exp(i\phi)a$. $a' \in C_1$ iff $a'pa^{\prime\dagger} \in P \forall p \in P$.
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\begin{equation}
\begin{aligned}
a'pa^{\prime\dagger} &= (\exp(i\phi)a)p(\exp(i\phi)a)^\dagger \\
& = \exp(i\phi)ap\exp(-i\phi)a^\dagger \\
& = \exp(i\phi)\exp(-i\phi) apa^\dagger \\
& = apa^\dagger \in P
\end{aligned}
\end{equation}
\end{proof}
\begin{lemma}
One cannot measure phases by projecting states.
\end{lemma}
\begin{proof}
Let $\ket{\psi}$ be a state, $\ket{\varphi}\bra{\varphi}$ a projector. $\ket{\psi'} := \exp(i\phi)\ket{\psi}$ for some $\phi \in [0, 2\pi)$.
\begin{equation}
\begin{aligned}
\bra{\psi'}\ket{\varphi}\bra{\varphi}\ket{\psi'} &= \exp(-i\phi)\bra{\psi}\ket{\varphi}\bra{\varphi}\exp(i\phi)\ket{\psi} \\
& = \exp(-i\phi)\exp(i\phi)\bra{\psi}\ket{\varphi}\bra{\varphi}\ket{\psi} \\
& = \bra{\psi}\ket{\varphi}\bra{\varphi}\ket{\psi}
\end{aligned}
\end{equation}
\end{proof}
\begin{definition}
A phase $\phi \in [0, 2\pi)$ is called qbit-global, if for some qbit states $\ket{\psi}, \ket{\varphi}$ $\ket{\psi} = \exp(i\phi)\ket{\varphi}$.
\end{definition}
\begin{lemma}
When entangling qbits via projections one can disregard qbit-global phases.
Two qbits are entangled via projection, if for some single qbit gates $M,N$
and two orthonormal states $\ket{a}, \ket{b}$
\begin{equation}
C^{M,N}(i,j) = \ket{a}\bra{a}_j \otimes M_i + \ket{b}\bra{b}_j \otimes N_i
\end{equation}
\textbf{Remark.}
In particular when entangling states using $CX$ and $CZ$ one can disregard qbit-global phases.
This is immideatly clear when recalling \eqref{eq:CX_pr} and \eqref{eq:CZ_pr}.
\end{lemma}
\begin{proof}
Let $\alpha, \beta \in [0, 2\pi)$ be some phases, $\ket{\psi}, \ket{\varphi}, \ket{\psi'} := \exp(i\alpha)\ket{\psi}, \ket{\varphi'} := \exp(i\beta)\ket{\varphi}$ some single qbit states,
$M, N, \ket{a}, \ket{b}, C^{M,N}(i,j)$ as defined above.
\begin{equation}
\begin{aligned}
C^{M,N}(1, 0) (\ket{\psi'}\otimes\ket{\varphi'}) & = \ket{a}\braket{a}{\varphi'}\otimes M\ket{\psi'} + \ket{b}\braket{b}{\varphi'} \otimes N\ket{\psi'} \\
& = \exp(i\beta)\ket{a}\braket{a}{\varphi}\otimes\exp(i\alpha)M\ket{\phi} + \exp(i\beta)\ket{b}\braket{b}{\varphi}\otimes\exp(i\alpha)N\ket{\phi}\\
& = \exp(i(\beta + \alpha))(\ket{a}\braket{a}{\varphi}\otimes M\ket{\psi} + \ket{b}\braket{b}{\varphi} \otimes N\ket{\psi})\\
& = \exp(i(\beta + \alpha))C^{M,N}(1, 0) (\ket{\psi}\otimes\ket{\varphi})
\end{aligned}
\end{equation}
Where $\exp(i(\beta + \alpha))$ is a multi-qbit-global phase which can be (following the above argumentation) disregarded.
\end{proof}
\begin{corrolary}
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One can disregard global phases of elements of the $C_1$ group.
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\end{corrolary}
\begin{proof}
As it has been shown above a quantum computer cannot measure global phases. Also
the entanglement gates $CX, CZ$ map qbit-global phases to multi-qbit-global phases which cannot
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be measured. It has been shown above that one can choose the $C_1$ operators such that they do not yield
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a phase.
\end{proof}
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\begin{definition}
\begin{equation}
C_L := \{a \in C_1 | \nexists \phi \in [0, 2\pi), b \in C_L : a = \exp(i\phi)b\}
\end{equation}
Is called the non-trivial local Clifford group.
\end{definition}
\textbf{Remark.} When computing the elements of $C_L$ and their products one will realize that $C_L$ is not a group.
If one however disregards a global phase the product of two $C_L$ elements will be in $C_L$ again. Because the global phases
can be disregarded as discussed above $C_L$ will be used from now on instead of $C_1$.
\begin{theorem}
\begin{equation}
| C_L | = 24
\end{equation}
\end{theorem}
\begin{proof}
It is clear that $\forall a \in C_L$ a is a group isomorphism $P \circlearrowleft$: $apa^\dagger a p' a^\dagger = a pp'a^\dagger$.
Therefore $a$ will preserve the (anti-)commutator relations of $P$. Also $P$ is generated by $X,Z$ when disregarding a phase wich
does not matter for anticommutator relations.
This means that $X$ can be mapped to any $p \in P$ which are six elements disregarding
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FIXME
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\end{proof}
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\subsection{Introduction to the Graph Formalism}
The first step towards the simulation in the graph formalism has been
the discovery of the stabilizer states and stabilizer circuits \cite{gottesman2009}\cite{gottesman1997}.
They led to the faster simulation using stabilizer tableaux\cite{gottesman_aaronson2008} and later
to the graph formalism\cite{schlingenmann2001}\cite{andersbriegel2005}\cite{vandennest_ea2004}.
The following discussion eludicates the graph formalism and explains how the graph simulator works.
Some parts will be kept short as they can be looked up in \cite{andersbriegel2005}.
A naive state is just a vector containing the coefficients $c_i$ as defined in \ref{ref:nqbitsystems}.
It is a quite straight forward approach and gates are applied by updating the coefficients according
to the gate's matrix representation. A naive state has the time and space complexity $\mathcal{O}(2^n)$ which limits the number
of qbits drastically. \\
The stabilizer tableaux represent the state by its stabilizers i.e. by those Pauli operators of which the
state is a $+1$ eigenstate. This has a space complexity of $\mathcal{O}(n^2)$ while updating the tableaux
has a time complexity of $\mathcal{O}(n)$ for unitary gates and $mathcal{O}(n^2)$ for measurements.
A graph state now represents the state by the gates that have been applied to it starting from the $\ket{+}$ state:
\begin{equation}
\ket{+} := \bigotimes\limits_{i=0}^{n-1} H_i \ket{0}
\end{equation}
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\begin{definition}
\label{def:graph_state}
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A graph state $\ket{G}$ is a 3-tuple $(V, E, O)$ where $(V = \{0, ..., n-1\}, E)$ is a graph with the vertices $V$, edges $E$
and vertex operators $O = \{o_i | i = 0, ..., n-1; o_i \in C_L \forall i\}$. The vertex operators and edges are defined
by the following relation:
\begin{equation}
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\label{eq:g_state}
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\ket{G} = \left(\bigotimes\limits_{i=0}^{n-1} o_i \right)\left(\bigotimes\limits_{\{i, j\} \in E} CZ_{i,j} \right) \ket{+}
\end{equation}
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\end{definition}
One can show that any stabilizer state can be realized as a graph state (for instance in \cite{schlingenmann2001}).
\subsection{Operations on the Graph State}
\subsubsection{Single Qbit Gates}
Recalling \eqref{eq:g_state}
Makes it clear that for any single qbit gate $o \in C_L$ with $o^{(k)}$ being the gate
acting on qbit $k$ the state changes according to
\begin{equation}
\begin{aligned}
o^{(k)} \ket{G} &= o^{(k)} \left(\bigotimes\limits_{i=0}^{n-1} o_{i} \right)\left(\bigotimes\limits_{\{i, j\} \in E} CZ_{i,j} \right) \ket{+} \\
&= \left(\bigotimes\limits_{i=0}^{n-1} o^{\delta_{i,k}}o_{i} \right)\left(\bigotimes\limits_{\{i, j\} \in E} CZ_{i,j} \right)\ket{+}
\end{aligned}
\end{equation}
meaning that the graph state $(V, E, O)$ changes to $(V, E, \{o_0, ..., o_{k-1}, oo_k, o_{k+1}, ..., o_{n-1}\})$
as $C_L$ is almost a group the element $oo_k \in C_L$ up to a global phase that is disregarded. All the results
of $C_L \times C_L \rightarrow C_L, a,b \mapsto ab$ have been precomputed in a lookup table and the vertex operators
are updated according to that lookup table.
\subsubsection{Controlled Phase Gate}
Recalling \eqref{eq:g_state}
it is clear that some $CZ$ application is less trivial
than a single qbit gate.
%\begin{struktogramm}(100, 50)
% \ifthenelse[10]{1, 4}
% {Both Vertex operators Commute with CZ}{\sTrue}{\sFalse}
% \change
% \ifend
%\end{struktogramm}
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\subsection{Graph Storage}
One of the gread advantages of simulating in the graph formalism is a great increase
in simulation performance and a lower memory requirement. The simulation of
at least $10^6$ qbits on a common desktop computer should be possible\cite{andersbriegel2005}.
Therefore one has to take care when choosing a representation of the graph state.
The following
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\subsection{Usage}
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\subsection{Performance}
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