189 lines
7.4 KiB
TeX
189 lines
7.4 KiB
TeX
% vim: ft=tex
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\section{Quantum Computing}
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\subsection{Qbits and Gates}
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\subsubsection{Single Qbits}
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\begin{definition}
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A qbit is a two level quantum mechanical system, i.e. it has the eigenbasis
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$ \{\ket{\uparrow} \equiv \ket{1}, \ket{\downarrow} \equiv \ket{0}\} $
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with $\braket{\uparrow}{\downarrow} = 0$. In the folling this basis will be called
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the $Z$ basis in analogy to the conventions used in spin systems. For some computations
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it can be useful to have component vectors, $\ket{\uparrow} \equiv \left(\begin{array}{c} 0 \\ 1\end{array} \right)$ and
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$\ket{\downarrow} \equiv \left(\begin{array}{c} 1 \\ 0 \end{array} \right)$
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are used in these cases.
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\end{definition}
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A gate acting on a qbit is a unitary operator $G \in U(2)$. One can show that
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$\forall G \in U(2)$ $G$ can be approximated arbitrarily good as a product of unitary generator matrices
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\cite[Chapter 4.3]{kaye_ea2007}\cite[Chapter 2]{marquezino_ea_2019},
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common choices for the generators are $ X, H, R_{\phi}$ and $Z, H, R_{\phi}$ with
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\label{ref:singleqbitgates}
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\begin{equation}
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X := \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)
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\end{equation}
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\begin{equation}
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Z := \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)
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\end{equation}
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\begin{equation}
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H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right)
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\end{equation}
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\begin{equation}
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R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)
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\end{equation}
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\begin{equation}
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I := \left(\begin{array}{cc} 1 & 0 \\ 0 & 1\end{array}\right)
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\end{equation}
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Note that $X = HZH$ and $Z = R_{\pi}$, so the set of $H, R_\phi$ is sufficient.
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Further note that the basis vectors are chosen s.t. $Z\ket{0} = +\ket{0}$ and $Z\ket{1} = -\ket{1}$,
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transforming to the other Pauli eigenstates is done using $H$ and $SH$:
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\begin{equation}
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S = R_{\frac{\pi}{2}} = \left(\begin{array}{cc} 1 & 0 \\ 0 & i\end{array}\right)
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\end{equation}
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\begin{equation}
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S H Z H^\dagger S^\dagger = S X S^\dagger = Y
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\end{equation}
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\subsubsection{Many Qbits}
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\begin{postulate}
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A $N$ qbit quantum mechanical state is the tensor product\cite[Definition 14.3]{wuest1995} of the $N$ single qbit
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states.
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\end{postulate}
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Let $\ket{0}_s := \left(\begin{array}{c} 1 \\ 0 \end{array} \right)$ and $\ket{1}_s := \left(\begin{array}{c} 0 \\ 1 \end{array} \right)$
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be the basis of the one-qbit systems. Then two-qbit basis states are
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\begin{equation}
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\ket{0} := \ket{0b00} := \ket{0}_s \otimes \ket{0}_s := \left(\begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right)
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\end{equation}
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\begin{equation}
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\ket{1} := \ket{0b01} := \ket{0}_s \otimes \ket{1}_s := \left(\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array} \right)
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\end{equation}
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\begin{equation}
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\ket{2} := \ket{0b10} := \ket{1}_s \otimes \ket{0}_s := \left(\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array} \right)
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\end{equation}
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\begin{equation}
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\ket{3} := \ket{0b11} := \ket{1}_s \otimes \ket{1}_s := \left(\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right)
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\end{equation}
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The $N$ qbit basis states can then be constructed in a similar manner.
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A general $N$ qbit state can then be written as a superposition of the
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integer states:
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\begin{equation}
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\ket{\psi} = \sum\limits_{i = 0}^{2^N - 1} c_i \ket{i}
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\end{equation}
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\begin{equation}
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\sum\limits_{i = 0}^{2^N - 1} |c_i|^2 = 1
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\end{equation}
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The states $\ket{i}$ for $i = 0, ..., 2^{n}-1$ are called integer states. Note
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that integer states are eigenstates of the $Z$ operators. The computational basis
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is $\{\ket{i_0} \otimes ... \otimes \ket{i_{n-1}} | i_0, ..., i_{n-1} = 0, 1\}$.
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\begin{definition}
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For a single qbit gate $U$ and a qbit $j = 0, 1, ..., n - 1$
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\begin{equation}
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U_j := \left(\bigotimes\limits_{i = 0}^{j - 1} I\right)
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\otimes U
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\otimes \left(\bigotimes\limits_{i = j + 1}^{n - 1} I \right)
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\end{equation}
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is the $U$ gate acting on qbit $j$.
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\end{definition}
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\begin{definition}\label{def:CU}
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For two qbits $i,j = 0, 1, ..., n - 1$, $i \neq j$ and a gate $U_i$ acting on $i$
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the controlled version of $U$ is defined by
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\begin{equation}\label{eq:CU}
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CU_{i, j} = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes U_i
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\end{equation}
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In particular for $X, Z$:
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\begin{equation}\label{eq:CX_pr}
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CX(i, j) = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes X_i
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\end{equation}
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\begin{equation}\label{eq:CZ_pr}
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CZ(i, j) = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes Z_i
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\end{equation}
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\end{definition}
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In the definition \ref{def:CU} $i$ is called the act-qbit and $j$ the control-qbit. In words
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$CU$ applies the gate $U$ to the act-qbit if the control-qbit is in its $\ket{1}$ state.
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One can show that the gates in \ref{ref:singleqbitgates} together with an entanglement gate, such as $CX$ or $CZ$ are enough
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to generate an arbitrary $N$ qbit gate\cite[Chapter 4.3]{kaye_ea2007}\cite{barenco_ea_1995}.
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The matrix representation of $CX$ and $CZ$ for two qbits is given by (this is quickly verified by applying the
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matrices to the basis states)
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\begin{equation}
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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)
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\end{equation}
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\begin{equation}
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CZ_{1, 0} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1 \end{array}\right)
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\end{equation}
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\subsubsection{Measurements}
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\textbf{FIXME} I don't like this at all.
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\begin{postulate}
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Let
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$$\ket{\psi} = \alpha\ket{\phi_1} \otimes \ket{1}_j + \beta\ket{\phi_0} \otimes \ket{0}_j$$
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be a $n$ qbit state
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where $\ket{1}_j, \ket{0}_j$ denote the $j$-th qbit state and $|\alpha|^2 + |\beta|^2 = 1$.
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Then the measurement of the $j$-th qbit will yield
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$$\ket{\phi_1} \otimes \ket{1}_n$$
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with probability $|\alpha|^2$ and
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$$\ket{\phi_0} \otimes \ket{0}_n$$
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with probability $|\beta|^2$. This is called collapse of the wave function.
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\end{postulate}
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Measuring a qbit will also yield a classical result $0$ or $1$ with the respective probabilities.
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\begin{corrolary}
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In general the measurement of a qbit is not invertible, in particular it cannot be represented as a
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unitary operator.
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\end{corrolary}
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\begin{proof}
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The measuerment in not injective: Measuring both
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$\ket{0}$ and $\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ (can) map to $\ket{0}$.
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Any unitary matrix $U$ has the inverse $U^\dagger \equiv U^{-1}$.
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\end{proof}
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As a measurement is not unitary it is not a gate as in the definition above.
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In the following discussion the term \textit{measurement gate} will be used from time
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to time as a measurement can be treated similarely while doing numerics.
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Measurements are always performed in the computational basis, i.e. for a qbit
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$i$ $Z_i$ is measured. Let the state to be measured be
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\begin{equation}
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\ket{\psi} = \alpha\ket{0}\otimes\ket{\psi_0} + \beta\ket{1}\otimes\ket{\psi_1}
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\end{equation}
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then a result $0$ is measured with probability $|\alpha|^2$ and $1$ with probability
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$|\beta|^2 = 1 - |\alpha|^2$. The wave function is then collapsed to
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\begin{equation}
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\begin{aligned}
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\ket{\psi'} = \left\{\begin{array}{c}\ket{0}\otimes\ket{\psi_0}, \mbox{ for a result 0 } \\
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\ket{1}\otimes\ket{\psi_1}, \mbox{ for a result 1 }
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\end{array}\right\}
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\end{aligned}
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\end{equation}
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\subsection{Quantum Circuits}
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Quantum circuits are a simple and well-readable way to express the application
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of several gates on a state.
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\textbf{TODO}
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