54 lines
2.6 KiB
TeX
54 lines
2.6 KiB
TeX
% vim: ft=tex
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\section{The naive Simulator}
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A quite big part of the simulations interesting for students and researchers is not
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covered by stabilizer states and stabilizer circuits. In particular the
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phase estimation algorithm is essential for many applications. Being able to simulate
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such an algorithm is essential for education.
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\subsection{Simulator Core}
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Recalling \ref{ref:nqbitsystems} an arbitrary $N$ qbit state $\ket{\psi}$ can be written as such:
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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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Which yields $2^N$ coefficients $c_i = \braket{\psi}{i}$. These coefficients are used to
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store the quantum mechanical state of the simulator and are stored in NumPy arrays \cite{numpy_array}.
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They can be modified and viewed without overhead
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using python and allow fast modification using so-called NumPy ufuncs\cite{numpy_ufunc}.
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A simulator state also contains a classical state which is a length $N$ integer array containing
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the measured classical bits with a default value of $-1$.
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The gates are implemented as NumPy ufuncs which allows an efficient
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manipulation of NumPy arrays using C code. Every gate maps a length $2^N$
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\lstinline[basicstyle=\ttfamily, language=Python]{numpy.cdouble} and a length
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$N$ \lstinline[basicstyle=\ttfamily, language=Python]{numpy.int8} array to
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a length $2^N$ \lstinline[basicstyle=\ttfamily,
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language=Python]{numpy.cdouble}, a length $N$ \lstinline[basicstyle=\ttfamily,
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language=Python]{numpy.int8} and a \lstinline[basicstyle=\ttfamily,
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language=Python]{numpy.uint64} scalar. The \lstinline[basicstyle=\ttfamily,
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language=Python]{numpy.cdouble} arrays are the quantum mechanical state before
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and after the gate; the \lstinline[basicstyle=\ttfamily,
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language=Python]{numpy.int8} arrays are the respective classical states and the
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\lstinline[basicstyle=\ttfamily, language=Python]{numpy.uint64} scalar has
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a logical one at every bit that has been measured.
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\subsection{Implemented Gates}
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As stated in \ref{ref:singleqbitsystems} and \ref{ref:nqbitsystems} a just a small set of gates is required to
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approximate an arbitrary unitary matrix arbitrarily good. In principle just the $X$, $R_\phi$, $H$ and $CNOT$ gate
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would be sufficient for a simulator. As however both $X$ and $Z$ are often used in practice FIXME: CITATION NEEDED
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$X$, $Z$, $R_\phi$, $H$ and $CNOT$ are implemented. They can be accessed using
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\lstinline[basicstyle=\ttfamily]{pyqcs.X}, \lstinline[basicstyle=\ttfamily]{pyqcs.Z}, \lstinline[basicstyle=\ttfamily]{pyqcs.R},
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\lstinline[basicstyle=\ttfamily]{pyqcs.H} and \lstinline[basicstyle=\ttfamily]{pyqcs.C}.
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\subsection{Usage}
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FIXME
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\subsection{Performance}
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FIXME
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