QC framework

content.tex

@@ -1,4 +1,6 @@
 
+\todo[inline]{Mention that transition functions / matrices need to be effectively computable by a polynomial algorithm. Otherwise, the hard computations can be outsourced to the preparation of the circuit. Consider a decision problem. The decision could be computed in advance for each input and the transition matrix just writes a designated bit: $1$ for \texttt{ACCEPT} or $1$ for \text{REJECT}}
+
 \section{A Simple Computational Model}
 What are Qubits? That's usually the first question getting addressed in any introduction to quantum computing, for a good reason. If we want to construct a new computational model, we first need to define the most basic building block: a single \emph{bit} of information. In classical computer science, the decision on how to define this smallest building block of information seems quite straight forward. We just take the most basic logical fact: either something is \emph{true} or \emph{false}, either 1 or 0. We have a name for an object holding this information: a \textbf{Bit}. Let's envision a computational model based on logical gates. Such a gate has one or more inputs and an output, with each either being \emph{true} or \emph{false}. Now consider a bit $b$ and a gate $f : \{0, 1\} \to \{0, 1\}$. We have a \emph{bit} of information $b$ and can get another \emph{bit} of information $b' \coloneqq f(b)$. In a final third step, we introduce a timescale, which means that now our \emph{bit} of information is time dependent. It can have different values at different times. To make it easier, we choose a discrete timescale. Our Bit $b$ has a distinct value on each point on the timescale. A value of a bit can only be changed in between time steps, by applying a logical gate to it:
 $$
@@ -268,13 +270,14 @@ The set of operations mapping $\mathscr{B}_{\R}^n$ to $\mathscr{B}_{\R}^n$ is ex
 \begin{definition}  
   A matrix $A \in \R^n$ is orthogonal iff
   \begin{equation*}
-    A^{-1} = A^t \Leftrightarrow AA^t = A^tA = \idmat
+    A^{-1} = A^t \quad \Leftrightarrow \quad AA^t = A^tA = \idmat
   \end{equation*}
 \end{definition}
 \begin{remark}
   It is important that orthogonal matrices from a group, because this means the composition of two orthogonal operators is orthogonal again. So, orthogonal computation can be composed or decomposed by or into other orthogonal computations. This is extremely useful for describing and developing algorithms in this model.
 \end{remark}
-\begin{definition}{Orthogonal Computations}
+\begin{definition}[Orthogonal Computation]
+  \label{def:orthogonal_computation}
   A computation on the state space $\mathcal{B}_{\R}^n$ is defined by an orthogonal matrix $A \in \R^N$ with $N = 2^n$.
 \end{definition}
 What does it mean if a matrix is orthogonal? Let $A = (a_{ij}) = (\mathbf{a}_i,\dots,\mathbf{a}_n) \in \R^{(n,n)}$ be orthogonal with. Then, it follows directly from $AA^t = (b_{ij}) = \idmat$ that $b_{ij} = \mathbf{a}_i^t \mathbf{a}_j = \delta_{ij}$. Hence, the columns (and rows) of $A$ form an orthonormal basis of $\R^n$. It is also easy to check that $A$ preserves the dot product making it angle and length preserving. Another direct consequence of $AA^t = \idmat$ is the reversibility of orthogonal computations.
@@ -283,6 +286,7 @@ What does it mean if a matrix is orthogonal? Let $A = (a_{ij}) = (\mathbf{a}_i,\
 The measurement operators of \cref{sec:measurements_probabilistic} obviously also need to be adjusted to the new state space and also suffered from some shortcomings. The dot product operators of \cref{def:measurment_operator_probabilistic} completely leave the state space when applied to a state vector. But the most important reason for why to redesign measurements are the newly used probably amplitudes. Just extracting an amplitude and squaring it would of course be a possible, but alien solution to the linear framework developed so far. This is because squaring is not linear. The desired goal is to design a linear operator nicely fitting in the framework at hand.
  
 \begin{definition}[(Real) Measurement Operators]
+  \label{def:measurment_operator_orthogonal_state_space}
   Let $\Omega$ be the set of possible outcomes of an experiment. Then the corresponding measurement is described by a set of linear operators $\parensc{M_m}_{m \in \Omega} \subset \R^{(N,N)}$ with:
   \begin{itemize}
     \item $P(m) = \ev{M_m^t M_m}{b}$
@@ -443,6 +447,7 @@ The best theoretical model is of no value if it relies on some kind of magic tha
 \end{remark}
   
 \begin{definition}[Computational Basis]
+  \label{def:computational_basis}
   The standard bais of quantum computing is the computational basis. The basis $2^n$ values of a $n$-bit register are mapped to an orthonormal basis $\mathcal{B} = \parensc*{\ket{i}}_{i=0}^{2^n - 1}$, with $i$ being the decimal value of the $n$-bit register. In coordinate space $\ket{i}$ is represented by $e_i \in \C^{2^n}$:
   \begin{equation}
     \ket{i} \cong e_i = \parens{\underbrace{0,\dots,0}_{i - 1},1,0,\dots,0}^t
@@ -450,7 +455,33 @@ The best theoretical model is of no value if it relies on some kind of magic tha
 \end{definition}
 
 \begin{definition}[Quantum State Space]
+  \label{def:quantum_state_space}
+  Spanning a complex vector space over an arbitrary set of $2^n$ many orthonormal base vectors we get the state space of a $n$-qubit system. The standard base of choice is the computational basis (\cref{def:computational_basis}), giving the standard $n$-qubit state space $\mathcal{B}$ 
 \begin{equation*}
   \mathcal{B}^n = \parensc*{\ket{\psi} = \sum_{i=0}^{2^n -1} \alpha_i \ket{i} \:\middle|\: \alpha_i \in \C ,\: \braket{\psi} = 1}
 \end{equation*}
 \end{definition}
+
+\begin{definition}[Unitary Operator]
+  \label{def:unitary_operator}
+  Unitary operators and matrices are the complex extensions to orthogonal operators and matrices. An operator $\Phi$ is unitary if its matrix representation $U_\Phi$ is a unitary matrix, which is the case iff
+  \begin{equation*}
+    U_\Phi^{-1} = U_\Phi^\dagger \quad \Leftrightarrow \quad U_\Phi U_\Phi^\dagger = U_\Phi^\dagger U_\Phi = \idmat
+  \end{equation*}
+  with $U_\Phi^\dagger$ being the transposed complex conjugate of $U_\Phi$, defined as
+  \begin{equation*}
+    \forall n \in \N,\: A = (a_{ij}) \in \C^n \::\quad A^\dagger = (a_{ji}^*)
+  \end{equation*}
+\end{definition}
+
+\begin{definition}[Unitary Computation]
+  \label{def:unitary_computation}
+  In accordance with \cref{def:orthogonal_computation}, a computation on the state space $\mathcal{B}^n$ is defined by a unitary matrix $U \in \C^{2^n}$.
+\end{definition}
+
+\begin{definition}[Measurement Operators]
+  \label{def:measurment_quantum}
+  The definition of measurements on quantum state spaces follow the definition of measurements on orthogonal state spaces (\cref{def:measurment_operator_orthogonal_state_space}), but with states $\ket{\psi} \in \mathcal{B}^n$ and complex operators $\parensc*{M_m}_{m \in \Omega} \subset \C^{(N,N)}$.
+\end{definition}
+
+This wraps up the framework of quantum computing. Fortunately, \cref{def:quantum_state_space,def:unitary_operator,def:measurment_quantum} correspond to the fundermental postulates of quantum mechanics, meaning that quantum computing with all its seemingly strange properties is in fact a physically realizable computational model!

main.tex

@@ -32,6 +32,7 @@
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+\DeclareMathOperator{\N}{\mathbb{N}}
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