diff --git a/src/thesis/acronyms.tex b/src/thesis/acronyms.tex index 5f6b7fb..856661b 100644 --- a/src/thesis/acronyms.tex +++ b/src/thesis/acronyms.tex @@ -72,3 +72,8 @@ short=AWGN, long=additive white Gaussian noise } + +\DeclareAcronym{pdf}{ + short=PDF, + long=probability density function +} diff --git a/src/thesis/chapters/2_fundamentals.tex b/src/thesis/chapters/2_fundamentals.tex index 82c3198..e30d2e3 100644 --- a/src/thesis/chapters/2_fundamentals.tex +++ b/src/thesis/chapters/2_fundamentals.tex @@ -592,15 +592,30 @@ decoding of subsequent blocks \cite[Sec.~III.~C.]{hassan_fully_2016}. \section{Quantum Mechanics and Quantum Information Science} \label{sec:Quantum Mechanics and Quantum Information Science} -% TODO: Should the brief intro to QC be made later on or here? +Designing codes and decoders for \ac{qec} is generally performed on a +layer of abstraction far removed from the quantum mechanical +processes underlying the actual qubits. +Nevertheless, having a fundamental understanding of the related +quantum mechanical concepts is useful to understand the unique constraints +of this field. +The purpose of this section is to convey these concepts to the reader. + %%%%%%%%%%%%%%%% \subsection{Core Concepts and Notation} \label{subsec:Notation} -\ldots can be very elegantly expressed using the language of -linear algebra. -\todo{Mention that we model the state of a quantum mechanical system -as a vector} +% Wave functions + +In quantum mechanics, the evolution of a state of a particle over tme +and space is described by a \emph{wave function} $\psi(x,t)$. +The connection between this function and the world that we can observe +is the fact that $\lvert \psi (x,t) \rvert^2$ is the \ac{pdf} of +finding a praticle in that particular state. + +% Dirac notation + +A lot of the related mathematics can be very elegantly expressed +using the language of linear algebra. The so called Bra-ket or Dirac notation is especially appropriate, having been proposed by Paul Dirac in 1939 for the express purpose of simplifying quantum mechanical notation \cite{dirac_new_1939}. @@ -611,9 +626,170 @@ For example, two vectors specified by the labels $a$ and $b$ respectively are written as $\ket{a}$ and $\ket{b}$. Their inner product is $\braket{a\vert b}$. +% Expressing wave functions using linear algebra + +We can model a wave function $\psi(x,t)$ as a linear combination of different +\emph{basis functions} $e_n(x,t),~n\in \mathbb{N}$ as% +% +\begin{align*} + \psi(x,t) = \sum_{n=1}^{\infty} c_n \cdot e_n(x,t) + .% +\end{align*} +To express this relation using linear algebra, we represent +$\psi(x,t)$ and $e_n(x,t)$ as vectors $\ket{\psi}$ and $\ket{e_n}$. +We write% +% +\begin{align*} + \ket{\psi} = \sum_{n=1}^{\infty} c_n \ket{e_n} + .% +\end{align*} + +% Operators + +Another important notion is that of an \emph{operator}, a component +that takes a function as an input and returns another function as an output. +Operators are useful to describe the relations between different +quantities relating to a particle. +An example of this is the differential operator $\partial x$. + +%%%%%%%%%%%%%%%% +\subsection{Observables} +\label{subsec:Observables} + +% Observable quantities + +An \emph{observable quantity} $Q$ is \ldots . +Due to the probabilistic nature of quantum mechanics, the result of a +measurement is not deterministic. +Thus, it is useful to consider the \emph{expected value} $\braket{Q}$ +of an observable quantity in addition to individual measurement results. + +% General expression for expected value of observable quantity + +If we know the wave function of a particle, we should be able to +compute $\braket{Q}$ for any observable quantity we wish. +It can be shown that for any $Q$, we can compute a +corresponding operator $\hat{Q}$ such that% +% +\begin{align} + \label{eq:gen_expr_Q_exp} + \braket{Q} = \int_{-\infty}^{\infty} \psi^*(x,t) \hat{Q} \psi(x,t) dx + .% +\end{align}% +% +While the derivation of this relationship is out of the scope of this +work, we can at least look at an example to illustrate it. +Considering the position $Q = x$ of a particle and setting the observable +operator to $\hat{Q} = x$, we can write% +% +\begin{align*} + \braket{x} = \int_{-\infty}^{\infty} \psi^*(x,t) \cdot x \cdot \psi(x,t) dx + = \int_{-\infty}^{\infty} x \lvert \psi(x,t) \rvert ^2 dx + .% +\end{align*} +% +Note that $\lvert \psi(x,t) \rvert $ represents the \ac{pdf} of +finding a particle in a specific state. We immediately see that the +formula simplifies to the direct calculation of the expected value. + +% Determinate states and eigenvalues + +% TODO: Introduce determinate states above +% TODO: Nicer phrasing +% TODO: Use different symbol for determinate states (not psi) +% TODO: Fix equation +Let us now examine how the observable operator $\hat{Q}$ relates to +the determinate states that make up the overall superposition state +of the particle. +We begin by translating \autoref{eq:gen_expr_Q_exp} into linear alebra as% +% +\begin{align} + \label{eq:gen_expr_Q_exp_lin} + \braket{Q} = \braket{\psi \vert \hat{Q}\psi} + .% +\end{align} +% +\autoref{eq:gen_expr_Q_exp_lin} expresses an inherently probabilistic +relationhip. +The determinate states are inherently deterministic. +To relate the two, we look at those states $\ket{\psi}$, where the +variance of the measurements of $Q$ is zero. These are exactly the +determinate states.% +% +\begin{align*} + 0 &\overset{!}{=} \braket{(Q - \braket{Q})^2} + = \braket{\psi \vert (\hat{Q} - \braket{Q})^2 \psi} \\ + &= \braket{(Q - \braket{Q})\psi \vert (\hat{Q} - \braket{Q}) \psi} \\ + &= \lVert (Q - \braket{Q}) \psi \rVert^2 \\[3mm] + &\hspace{-8mm}\Leftrightarrow (\hat{Q} - \braket{Q}) \ket{\psi} = 0 \\ + \label{eq:observable_eigenrelation} + &\hspace{-8mm}\Leftrightarrow \hat{Q}\ket{\psi} = \braket{Q} \ket{\psi} + .% +\end{align*}% +% +Because we have assumed the variance to be zero, $\braket{Q}$ is now +the deterministic measurement value corresponding to the determinate +state $\ket{\psi}$. +We can see that the determinate states are the \emph{eigenstates} of +the observable operator $\hat{Q}$ and that the corresponding +(deterministic) measurement values are the corresponding \emph{eigenvalues}. + +% Recap + +To summarize, we can mathematically express any observable quantity +$Q$ using a corresponding operator $\hat{Q}$. +This operator allows us to both compute the expected value of the +observable using \autoref{eq:gen_expr_Q_exp_lin}, and describe the +individual determinate states and corresponding measurement values +using \autoref{eq:observable_eigenrelation}. + +%%%%%%%%%%%%%%%% +\subsection{Projective Measurements} +\label{subsec:Projective Measurements} + +%%%%%%%%%%%%%%%% +\subsection{Qubits and Multi-Qubit States} +\label{subsec:Qubits and Multi-Qubit States} + +\red{ + \begin{itemize} + \item Projective measurements + \begin{itemize} + \item Using the determinate states as a basis for the + superposition state + \item The effect of using the obserrvable operator on + the superposition state + \item Modelling the observable operator as a series + of projections + \item The projection operator and its eigenvalues + \end{itemize} + \item Qubits and multi-qubit states + \begin{itemize} + \item The qubit + \begin{itemize} + \item Similar structure to classical + computing: bits are modified with gates + -> quantum bits are modified with quantum gates + \end{itemize} + \item The tensor product + \item Information is not stored in the individual bit + states but in the correlations / entanglement between them + \item -> The size of the vector space + \item The X,Z and Y operators + \item (?) Notation of operators on multi-qubit states + \end{itemize} + \end{itemize} +} + +\red{ + \begin{itemize} + \item Representing wave functions as vectors (psi as label, + building a vector space using basis functions) + \end{itemize} +} + \red{\textbf{Tensor product}} \red{\ldots - \todo{Introduce determinate state or use a different word?} Take for example two systems with the determinate states $\ket{0}$ and $\ket{1}$. In general, the state of each can be written as the superposition% @@ -668,11 +844,6 @@ Introducing a new notation for entangled states, we can write% \end{align*} % -\subsection{Projective Measurements} -\label{subsec:Projective Measurements} - -% TODO: Write - %%%%%%%%%%%%%%%% \subsection{Quantum Gates} \label{subsec:Quantum Gates}