Add dirty version of qubits and multi-qubit states
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@@ -724,7 +724,7 @@ $\ket{e_n},~n\in \mathbb{N}$.
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We can see that the determinate states are the \emph{eigenstates} of
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the observable operator $\hat{Q}$ and that the measurement values are
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the corresponding \emph{eigenvalues} $\lambda_n$
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\cite[Postulate~3]{griffiths_introduction_1995}.
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\cite[Sec.~3.3]{griffiths_introduction_1995}.
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% Determinate states as a basis
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@@ -788,7 +788,7 @@ the determinate states.
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Further measurements should then yield the same value.
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More general methods of modelling measurements exist, e.g., describing
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destructive measurements \cite[Box~2.5]{nielsen_quantum_2010}, but
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they are not relevant to us here.
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they are not relevant to this work.
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% Projection operators
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@@ -829,7 +829,7 @@ only has the eigenvalues $0$ or $1$
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\cite[Prob.~3.57a)]{griffiths_introduction_1995}.
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% tex-fmt: on
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The eigenvalues can again be interpreted as possible measurement results.
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We can thus use the $\hat{P}$ as an observable and treat
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We can thus use $\hat{P}$ as an observable and treat
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the eigenvalue as an indicator of the state having a component along
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the related basis vector.
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@@ -837,112 +837,159 @@ the related basis vector.
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\subsection{Qubits and Multi-Qubit States}
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\label{subsec:Qubits and Multi-Qubit States}
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% The qubit
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% TODO: Make sure `quantum gate` is proper terminology
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A central concept for quantum computing is that of the \emph{qubit}.
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We employ it analogously to the classical \emph{bit}.
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For classical computers, we alter bits' states using \emph{gates}.
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We can chain multiple of these gates together to build up more complex logic,
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such as half-adders or eventually a full processor.
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In principle, quantum computers work in a similar fashion, only that
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instead of bits we use qubits and instead of, e.g. {AND}, OR, and XOR
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operations we use \emph{quantum gates} \cite[Sec.~1.3]{nielsen_quantum_2010}.
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We define a qubit to be a component with determinate
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states $\ket{0}$ and $\ket{1}$.
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The general description of the state $\ket{\psi}$ of a qubit is thus
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\begin{align*}
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\ket{\psi} = \alpha\ket{0} + \beta\ket{1}, \hspace{5mm} \alpha,
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\beta \in \mathbb{C}
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.%
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\end{align*}
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% The tensor product and multi-qubit states
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The overall state of a composite quantum system is described using
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the \emph{tensor product}, denoted as $\otimes$
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\cite[Sec.~2.2.8]{nielsen_quantum_2010}.
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Take for example the two qubits
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\begin{align*}
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\ket{\psi_1} = \alpha_1 \ket{0} + \beta_1 \ket{1},\hspace*{10mm}
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\ket{\psi_2} = \alpha_2 \ket{0} + \beta_2 \ket{1}
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.%
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\end{align*}
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% TODO: Fix the fact that \psi is used above for the single-qubit
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% case and below for the multi-qubit case
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We denote the state of the composite system as $\ket{\psi}$.
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Assuming the qubits are independent, this is a \emph{product state}
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$\ket{\psi} = \ket{\psi_1}\otimes\ket{\psi_2}$.
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When not ambiguous, we may omit the tensor product symbol or even write
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the entire product state as a single ket
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\cite[Sec.~6.2]{griffiths_consistent_2001}.
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We have
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\begin{align}
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\label{eq:product_state}
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\begin{split}
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\ket{\psi} = \ket{\psi_1} \ket{\psi_2}
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&= \left( \alpha_1 \ket{0} + \beta_1 \ket{1} \right)
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\left( \alpha_2 \ket{0} + \beta_2 \ket{1} \right) \\
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&= \alpha_1\alpha_2\ket{00}
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+ \alpha_1\beta_2\ket{01}
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+ \beta_1\alpha_2\ket{10}
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+ \beta_1\beta_2\ket{11}
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.%
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\end{split}
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\end{align}
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We call $\ket{x_0, \ldots, x_n}~, x_i \in \{0,1\}$ the
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\emph{computational basis states}.
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% Entanglement
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States that are not able to be decomposed into such a product
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are called \emph{entangled} \cite[Sec.~2.2.8]{nielsen_quantum_2010}.
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An example of such states are the \emph{Bell states}
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\begin{align*}
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\begin{split}
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\ket{\psi_{00}} &= \frac{\ket{00} + \ket{11}}{\sqrt{2}} \hspace{15mm}
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\ket{\psi_{01}} = \frac{\ket{01} - \ket{10}}{\sqrt{2}} \\
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\ket{\psi_{10}} &= \frac{\ket{00} + \ket{11}}{\sqrt{2}} \hspace{15mm}
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\ket{\psi_{11}} = \frac{\ket{01} - \ket{10}}{\sqrt{2}}
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\end{split}
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\hspace{4mm}.%
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\end{align*}
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Quantum entanglement plays a major role in the way information
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is encoded on quantum systems compared to classical ones.
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Instead of employing only the individual qubit states, the
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information is stored in the correlations between the qubits
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\cite[Sec.~2]{preskill_quantum_2018}.
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% The size of the vector spaced
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As we can see in \autoref{eq:product_state}, the number of
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computational basis states needed to express the full composite state
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is $2^n$ \cite[Sec.~4.6]{nielsen_quantum_2010}.
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This is in contrast to classical systems, where the dimensionality of
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the state space only grows linearly with $n$.
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This exponential growth of the state space is what makes it difficult
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to simulate quantum systems on classical hardware.
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It is also what motivated the research into performing computations
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using quantum hardware in the first place
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\cite[Sec.~3]{feynman_simulating_1982}.
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% Basic types of gates: The X,Y,Z operators, Bloch sphere
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After examining the modelling of single- and multi-qubit systems,
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we now shift our focus to describing the evolution of their states.
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\red{[Bloch sphere]}
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We do this using operators, also called \emph{gates}.
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The \emph{Pauli operators} $I$,
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$X$, $Z$ and $Y$ \cite[Appendix~2]{roffe_quantum_2019} are the most
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fundamental ones:
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\begin{align*}
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\begin{array}{c}
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I\text{ Operator} \\
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\hline\\
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\ket{0} \mapsto \ket{0} \\
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\ket{1} \mapsto \ket{1} \\
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\end{array}%
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\hspace{10mm}%
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\begin{array}{c}
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X\text{ Operator} \\
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\hline\\
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\ket{0} \mapsto \ket{1} \\
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\ket{1} \mapsto \ket{0} \\
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\end{array}%
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\hspace{10mm}%
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\begin{array}{c}
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Z\text{ Operator} \\
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\hline\\
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\ket{0} \mapsto -\ket{0} \\
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\ket{1} \mapsto -\ket{1} \\
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\end{array}%
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\hspace{10mm}%
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\begin{array}{c}
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Y\text{ Operator} \\
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\hline\\
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\ket{0} \mapsto -j\ket{1} \\
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\ket{1} \mapsto -j\ket{0} \\
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\end{array}
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\end{align*}
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$I$ is the identity operator and $X$ and $Z$ are referred to as
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\emph{bit-flips} and \emph{phase-flips} respectively.
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% % TODO: Move this further down to the digitization of errors?
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% $Y$ can be represented as a combination of $X$ and $Z$ as $Y = jXZ$.
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% Operators over multiple qubits
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We can also perform operations on multi-qubit states.
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% TODO: Maybe the Hadamard operator and X <-> Z?
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% CNOT gates
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\red{[CNOT gates]}
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\red{
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\begin{itemize}
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\item Qubits and multi-qubit states
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\begin{itemize}
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\item The qubit
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\begin{itemize}
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\item Similar structure to classical
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computing: bits are modified with gates
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-> quantum bits are modified with quantum gates
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\end{itemize}
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\item The tensor product
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\item Information is not stored in the individual bit
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states but in the correlations / entanglement between them
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\item -> The size of the vector space
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\item The X,Z and Y operators
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\item (?) Notation of operators on multi-qubit states
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\end{itemize}
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\end{itemize}
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}
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\red{
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\begin{itemize}
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\item Representing wave functions as vectors (psi as label,
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building a vector space using basis functions)
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\end{itemize}
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}
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\red{\textbf{Tensor product}}
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\red{\ldots
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Take for example two systems with the determinate states $\ket{0}$
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and $\ket{1}$. In general, the state of each can be written as the
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superposition%
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%
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\begin{align*}
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\alpha \ket{0} + \beta \ket{1}
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.%
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\end{align*}
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%
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Combining these two sytems into one, the overall state becomes%
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%
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\begin{align*}
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&\mleft( \alpha_1 \ket{0} + \beta_1 \ket{1} \mright) \otimes
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\mleft( \alpha_2 \ket{0} + \beta_2 \ket{1} \mright) \\
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= &\alpha_1 \alpha_2 \ket{0} \ket{0}
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+ \alpha_1 \alpha_2 \ket{0} \ket{1}
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+ \beta_1 \alpha_2 \ket{1} \ket{0}
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+ \beta_1 \beta_2 \ket{1} \ket{1}
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% =: &\alpha_{00} \ket{00}
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% + \alpha_{01} \ket{01}
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% + \alpha_{10} \ket{10}
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% + \alpha_{11} \ket{11}
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.%
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\end{align*}%
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%
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\ldots When not ambiguous in the context, the tensor product
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symbol may be omitted, e.g.,
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\begin{align*}
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\ket{0} \otimes \ket{0} = \ket{0}\ket{0}
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.%
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\end{align*}
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}
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As we will see, the core concept that gives quantum computing its
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power is entanglement. When two quantum mechanical systems are
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entangled, measuring the state of one will collapse that of the other.
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Take for example two subsystems with the overall state
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%
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\begin{align*}
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\ket{\psi} = \frac{1}{\sqrt{2}} \mleft( \ket{0}\ket{0} +
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\ket{1}\ket{1} \mright)
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.%
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\end{align*}
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%
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If we measure the first subsystem as being in $\ket{0}$, we can
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be certain that a measurement of the second subsystem will also yield $\ket{0}$.
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Introducing a new notation for entangled states, we can write%
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%
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\begin{align*}
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\ket{\psi} = \frac{1}{\sqrt{2}} \left( \ket{00} + \ket{11} \right)
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.%
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\end{align*}
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%
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%%%%%%%%%%%%%%%%
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\subsection{Quantum Gates}
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\label{subsec:Quantum Gates}
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\red{
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\textbf{Content:}
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\begin{itemize}
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\item Bra-ket notation
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\item The tensor product
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\item Projective measurements (the related operators,
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eigenvalues/eigenspaces, etc.)
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\begin{itemize}
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\item First explain what an operator is
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\end{itemize}
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\item Abstract intro to QC: Use gates to process qubit
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states, similar to classical case
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\item X, Z, Y operators/gates
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\item Hadamard gate (+ X and Z are the same thing in differt bases)
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\item Notation of operators on multi-qubit states
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\item The Pauli, Clifford and Magic groups
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\end{itemize}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Quantum Error Correction}
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\label{sec:Quantum Error Correction}
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@@ -950,6 +997,8 @@ Introducing a new notation for entangled states, we can write%
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\red{
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\textbf{Content:}
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\begin{itemize}
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\item Why we need commutativity of the stabilizers [Journal,
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p.~51], [Got97, p.~6]
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\item General context
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\begin{itemize}
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\item Why we want QC
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@@ -968,6 +1017,10 @@ Introducing a new notation for entangled states, we can write%
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% TODO: Actually binary linear codes or just linear codes?
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equivalent of binary linear codes (e.g.,
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expressible via check matrix)
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\item Similar to parity checks, quantum states can be
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more conveniently described using stabilizers
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rather than working with the states directly
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\cite[Sec.~10.5.1]{nielsen_quantum_2010}
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\end{itemize}
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\item Digitization of errors
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\item CSS codes
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