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Daniel Knüttel 2020-02-21 17:59:00 +01:00
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66
thesis/chapters/quantum_computing.tex
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@ -7,8 +7,8 @@
\begin{definition}
A qbit is a two level quantum mechanical system, i.e. it has the eigenbasis
$ \{\ket{\uparrow} \equiv \ket{1}, \ket{\downarrow} \equiv \ket{0}\} $
with $\braket{\uparrow}{\downarrow} = 0$. In the folling this basis will be called
the $Z$ basis in analogy to the conventions used in spin systems. For some computations
with $\braket{\uparrow}{\downarrow} = 0$. In the following this basis will be called
the $Z$ basis in analogy to the conventions used in spin systems ($\sigma_Z$). For some computations
it can be useful to have component vectors, $\ket{\uparrow} \equiv \left(\begin{array}{c} 0 \\ 1\end{array} \right)$ and
$\ket{\downarrow} \equiv \left(\begin{array}{c} 1 \\ 0 \end{array} \right)$
are used in these cases.
@ -16,7 +16,7 @@
A gate acting on a qbit is a unitary operator $G \in SU(2)$. One can show that
$\forall G \in SU(2)$ $G$ can be approximated arbitrarily good as a product of unitary generator matrices
\cite[Chapter 4.3]{kaye_ea2007}\cite[Chapter 2]{marquezino_ea_2019},
\cite[Chapter 4.3]{kaye_ea2007}\cite[Chapter 2]{marquezino_ea_2019};
common choices for the generators are $ X, H, R_{\phi}$ and $Z, H, R_{\phi}$ with
\label{ref:singleqbitgates}
@ -31,18 +31,18 @@ common choices for the generators are $ X, H, R_{\phi}$ and $Z, H, R_{\phi}$ wit
\end{equation}
Note that $X = HZH$ and $Z = R_{\pi}$, so the set of $H, R_\phi$ is sufficient.
Further note that the basis vectors are chosen s.t. $Z\ket{0} = +\ket{0}$ and $Z\ket{1} = -\ket{1}$,
Further note that the basis vectors are chosen s.t. $Z\ket{0} = +\ket{0}$ and $Z\ket{1} = -\ket{1}$;
transforming to the other Pauli eigenstates is done using $H$ and $SH$:
\begin{equation}
S = R_{\frac{\pi}{2}} = \left(\begin{array}{cc} 1 & 0 \\ 0 & i\end{array}\right)
S := R_{\frac{\pi}{2}} = \left(\begin{array}{cc} 1 & 0 \\ 0 & i\end{array}\right)
\end{equation}
\begin{equation}
S H Z H^\dagger S^\dagger = S X S^\dagger = Y
\end{equation}
The following states are the $\pm 1$ eigenstates of the $X$, $Y$, $Z$ operators
and will be used in some calculations later.
The following states are the $\pm 1$ eigenstates of the $X,Y,Z$ operators
and will be used in some calculations later:
\begin{equation}
\begin{aligned}
@ -55,16 +55,16 @@ and will be used in some calculations later.
\end{aligned}
\end{equation}
\subsubsection{Many Qbits}
\subsubsection{Many-Qbit Systems}
\label{ref:many_qbits}
\begin{postulate}
A $N$ qbit quantum mechanical state is the tensor product\cite[Definition 14.3]{wuest1995} of the $N$ single qbit
A $n$-qbit quantum mechanical state is the tensor product\cite[Definition 14.3]{wuest1995} of the $n$ one-qbit
states.
\end{postulate}
Let $\ket{0}_s := \left(\begin{array}{c} 1 \\ 0 \end{array} \right)$ and $\ket{1}_s := \left(\begin{array}{c} 0 \\ 1 \end{array} \right)$
be the basis of the one-qbit systems. Then two-qbit basis states are
be the basis of the one-qbit systems. Then two-qbit basis states are:
\begin{equation}
\ket{0} := \ket{0b00} := \ket{0}_s \otimes \ket{0}_s := \left(\begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right)
@ -79,15 +79,17 @@ be the basis of the one-qbit systems. Then two-qbit basis states are
\ket{3} := \ket{0b11} := \ket{1}_s \otimes \ket{1}_s := \left(\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right)
\end{equation}
The $N$ qbit basis states can then be constructed in a similar manner.
A general $N$ qbit state can then be written as a superposition of the
The $n$-qbit basis states can be constructed in a similar manner.
A general $n$-qbit state can then be written as a superposition of the
integer states:
\begin{equation}
\ket{\psi} = \sum\limits_{i = 0}^{2^N - 1} c_i \ket{i}
\ket{\psi} = \sum\limits_{i = 0}^{2^n - 1} c_i \ket{i}
\end{equation}
With the normation condition:
\begin{equation}
\sum\limits_{i = 0}^{2^N - 1} |c_i|^2 = 1
\sum\limits_{i = 0}^{2^n - 1} |c_i|^2 = 1
\end{equation}
The states $\ket{i}$ for $i = 0, ..., 2^{n}-1$ are called integer states. Note
@ -95,16 +97,19 @@ that integer states are eigenstates of the $Z$ operators. The computational basi
is $\{\ket{i_0} \otimes ... \otimes \ket{i_{n-1}} | i_0, ..., i_{n-1} = 0, 1\}$.
\begin{definition}
For a single qbit gate $U$ and a qbit $j = 0, 1, ..., n - 1$
For a single-qbit gate $U$ and a qbit $j = 0, 1, ..., n - 1$
\begin{equation}
U_j := \left(\bigotimes\limits_{i = 0}^{j - 1} I\right)
\otimes U
\otimes \left(\bigotimes\limits_{i = j + 1}^{n - 1} I \right)
\end{equation}
is the $U$ gate acting on qbit $j$.
is acting on qbit $j$.
\end{definition}
% XXX
\newpage
\begin{definition}\label{def:CU}
For two qbits $i,j = 0, 1, ..., n - 1$, $i \neq j$ and a gate $U_i$ acting on $i$
the controlled version of $U$ is defined by
@ -141,8 +146,8 @@ In Definition \ref{def:CU} $i$ is called the act-qbit and $j$ the control-qbit.
$CU$ applies the gate $U$ to the act-qbit if the control-qbit is in its $\ket{1}$ state.
One can show that the gates in \ref{ref:singleqbitgates} together with an entanglement gate, such as $CX$ or $CZ$ are enough
to generate an arbitrary $N$ qbit gate\cite[Chapter 4.3]{kaye_ea2007}\cite{barenco_ea_1995}.
The matrix representation of $CX$ and $CZ$ for two qbits is given by
to generate an arbitrary $n$-qbit gate\cite[Chapter 4.3]{kaye_ea2007}\cite{barenco_ea_1995}.
The matrix representation of $CX$ and $CZ$ for two qbits is given by:
\begin{equation}
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)
@ -156,7 +161,7 @@ The matrix representation of $CX$ and $CZ$ for two qbits is given by
\begin{postulate}
Let
$$\ket{\psi} = \alpha\ket{\phi_1} \otimes \ket{1}_j + \beta\ket{\phi_0} \otimes \ket{0}_j$$
be a $n$ qbit state
be a $n$-qbit state
where $\ket{1}_j, \ket{0}_j$ denote the $j$-th qbit state and $|\alpha|^2 + |\beta|^2 = 1$.
Then the measurement of the $j$-th qbit will yield
$$\ket{\phi_1} \otimes \ket{1}_j$$
@ -167,8 +172,7 @@ The matrix representation of $CX$ and $CZ$ for two qbits is given by
Measuring a qbit will also yield a classical result $0$ or $1$ with the respective probabilities.
Measurements are always performed in the computational basis, i.e. for a qbit
$i$ $Z_i$ is measured. With the eigenstate of the $+1$ eigenvalue being the $\ket{0}$
and $\ket{1}$ for the $-1$ eigenvalue of $Z$.
$i$ $Z_i$ is measured. The $+1$ eigenvalue of $Z_i$ is $\ket{0}_i$, $\ket{1}_i$ for $-1$.
\begin{corrolary}
In general the measurement of a qbit is not invertible, in particular it cannot be represented as a
@ -182,21 +186,21 @@ and $\ket{1}$ for the $-1$ eigenvalue of $Z$.
Any unitary matrix $U$ has the inverse $U^\dagger \equiv U^{-1}$.
\end{proof}
As a measurement is not unitary it is not a gate as in the definition above.
Because a measurement is not unitary it is not a gate in the sense the definition above.
In the following discussion the term \textit{measurement gate} will be used from time
to time as a measurement can be treated similarely while doing numerics.
\subsection{Quantum Circuits}
As mentioned in \ref{ref:many_qbits} one can approximate an arbitrary $n$
qbit gate $U$ as a product of some single qbit gates and either $CX$ or $CZ$.
As mentioned in \ref{ref:many_qbits} one can approximate an arbitrary $n$-qbit
gate $U$ as a product of some single-qbit gates and either $CX$ or $CZ$.
Writing (possibly huge) products of matrices is quite unpractical and very much
unreadable. To address this problem quantum circuits have been introduced.
These represent the qbits as a horizontal line, a gate acting on a qbit is
These represent the qbits as a horizontal line and a gate acting on a qbit is
a box with a name on the respective line. Quantum circuits are read from
left to right. This means that a gate $U_i = Z_i X_i H_i$ has the
circuit representation
circuit representation:
\[
\Qcircuit @C=1em @R=.7em {
@ -204,8 +208,8 @@ circuit representation
}
\]
The controlled gates (such as $CX$ and $CZ$) have a vertical line from the control qbit to
the gate, for instance the circuit for $CZ_{2, 1}CX_{2,0}$ is
The controlled gates (such as $CX$ and $CZ$) have a vertical line from the control-qbit to
the gate, for instance the circuit for $CZ_{2, 1}CX_{2,0}$ is:
\[
\Qcircuit @C=1em @R=.7em {
@ -230,7 +234,7 @@ The great hope behind quantum computing is that it will speed up some computatio
exponentially using algorithms that utilize the laws of quantum mechanics. Current
algorithms are based upon quantum search and quantum fourier transform \cite{nielsen_chuang_2010}.
The latter is particular interesting for physical problems as it is used in the phase estimation
algorithm that can be used to analyze the spectrum of the transfer matrix
algorithm that can be used to analyze the spectrum of the transfer matrix:
\begin{equation}
T = \exp(itH)
@ -246,6 +250,6 @@ The eigenvalue of $T$ can now be estimated by using the phase estimation circuit
\]
Where $T\ket{\varphi} = \exp(2\pi i\varphi) \ket{\varphi}$ and the measurement result $\tilde{\varphi} = \frac{x}{2^n}$ is an
estimate for $\varphi$. If a success rate of $1-\epsilon$ and an accuracy of $| \varphi - \tilde{\varphi} | < 2^{-m}$
is wanted $N = m + \log_2(2 + \frac{1}{2\epsilon})$ qbits are required\cite{nielsen_chuang_2010}\cite{lehner2019}.
estimation for $\varphi$. If a success rate of $1-\epsilon$ and an accuracy of $| \varphi - \tilde{\varphi} | < 2^{-m}$
is wanted, then $N = m + \log_2(2 + \frac{1}{2\epsilon})$ qbits are required\cite{nielsen_chuang_2010}\cite{lehner2019}.

32
thesis/chapters/stabilizer.tex
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@ -2,7 +2,7 @@
\section{The Stabilizer Formalism}
The stabilizer formalism was originally introduced by Gottesman\cite{gottesman1997}
The stabilizer formalism was originally introduced by Gottesman \cite{gottesman1997}
for quantum error correction and is a useful tool to encode quantum information
such that it is protected against noise. The prominent Shor code \cite{shor1995}
is an example of a stabilizer code (although it was discovered before the stabilizer
@ -37,8 +37,8 @@ the elements of $P$ either commute or anticommute.
is called the multilocal Pauli group on $n$ qbits.
\end{definition}
The group property of $P_n$ follows directly from its definition
via the tensor product as do the (anti-)commutator relationships.
The group property of $P_n$ and the (anti-)commutator relationships follow directly from its definition
via the tensor product.
%Further are $p \in P_n$ hermitian and have the eigenvalues $\pm 1$ for
%$p \neq \pm I$, $+1$ for $p = I$ and $-1$ for $p = -I$.
@ -48,14 +48,15 @@ via the tensor product as do the (anti-)commutator relationships.
\label{def:stabilizer}
An abelian subgroup $S = \{S^{(0)}, ..., S^{(N)}\}$ of $P_n$ is called a set of stabilizers iff
\begin{enumerate}
\item{$\forall i,j = 1, ..., N$ $[S^{(i)}, S^{(j)}] = 0$ $S^{(i)}$ and $S^{(j)}$ commute}
\item{$\forall i,j = 1, ..., N$ $[S^{(i)}, S^{(j)}] = 0$%: $S^{(i)}$ and $S^{(j)}$ commute
}
\item{$-I \notin S$}
\end{enumerate}
\end{definition}
\begin{lemma}
If $S$ is a set of stabilizers, the following statements are follow
directly
If $S$ is a set of stabilizers, the following statements follow
directly:
\begin{enumerate}
\item{$\pm iI \notin S$}
@ -78,9 +79,9 @@ via the tensor product as do the (anti-)commutator relationships.
\end{proof}
As considering all elements of a group can be unpractical for some calculations
the generators of a group are introduced. It is usually enough to discuss the generator's
properties to understand the properties of the group.
Considering all the elements of a group can be impractical for some calculations,
the generators of a group are introduced. Often it is enough to discuss the generator's
properties in order to understand the properties of the group.
\begin{definition}
For a finite group $G$ and some $m \in \mathbb{N}$ one denotes the generators
@ -93,16 +94,16 @@ properties to understand the properties of the group.
\end{definition}
In the following discussions $\langle S^{(i)} \rangle_{i=0, ..., n-1}$ will be used as
the properties of a set of stabilizers that are used in the discussions can be studied using only its
the required properties of a set of stabilizers that can be studied on its
generators.
\subsubsection{Stabilizer States}
One important basic property of quantum mechanics is that hermitian operators have real eigenvalues
and eigenspaces associated with these eigenvalues. Finding these eigenvalues and eigenvectors
and eigenspaces which are associated with these eigenvalues. Finding these eigenvalues and eigenvectors
is what one calls solving a quantum mechanical system. One of the most fundamental insights of
quantum mechanics is that operators that commute have a common set of eigenvectors, i.e. they
can be diagonalized simultaneously. This motivates and justifies the following definition
quantum mechanics is that commuting operators have a common set of eigenvectors, i.e. they
can be diagonalized simultaneously. This motivates and justifies the following definition.
\begin{definition}
For a set of stabilizers $S$ the vector space
@ -115,8 +116,9 @@ can be diagonalized simultaneously. This motivates and justifies the following d
$\ket{\psi}$ is stabilized by $S$.
\end{definition}
It is clear that it is sufficient to show the stabilization property for the generators of
$S$, as all the generators forming an element in $S$ can be absorbed into $\ket{\psi}$.
It is clear that to show the stabilization property of
$S$ the proof for the generators is sufficient,
as all the generators forming an element in $S$ can be absorbed into $\ket{\psi}$.
The dimension of $V_S$ is not immediately clear. One can however show that
for a set of stabilizers $\langle S^{(i)} \rangle_{i=1, ..., n-m}$ the dimension
$dim V_S = 2^m$ \cite[Chapter 10.5]{nielsen_chuang_2010}. This yields the following important