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mhjensenmhjensen
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update week 4
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‎doc/pub/week4/html/week4-bs.html‎

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‎doc/pub/week4/html/week4-reveal.html‎

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‎doc/pub/week4/html/week4-solarized.html‎

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‎doc/pub/week4/html/week4.html‎

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‎doc/pub/week4/ipynb/ipynb-week4-src.tar.gz‎

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‎doc/pub/week4/ipynb/week4.ipynb‎

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‎doc/pub/week4/pdf/week4.pdf‎

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‎doc/src/week4/Backup/week4_2025.do.txt‎

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‎doc/src/week4/_minted/09B7072857FCA14FDCC597BB5DE67221.highlight.minted‎

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\begin{MintedVerbatim}[commandchars=\\\{\},codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8\relax}]
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\PYG{c+c1}{\PYGZsh{} Now rewrite it in terms of the identity matrix and the Pauli matrix X and Z}
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\PYG{n}{X} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{array}\PYG{p}{(}\PYG{p}{[}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{,}\PYG{l+m+mi}{1}\PYG{p}{]}\PYG{p}{,}\PYG{p}{[}\PYG{l+m+mi}{1}\PYG{p}{,}\PYG{l+m+mi}{0}\PYG{p}{]}\PYG{p}{]}\PYG{p}{)}
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\PYG{n}{Y} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{array}\PYG{p}{(}\PYG{p}{[}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{,}\PYG{o}{\PYGZhy{}}\PYG{l+m+mi}{1}\PYG{n}{j}\PYG{p}{]}\PYG{p}{,}\PYG{p}{[}\PYG{l+m+mi}{1}\PYG{n}{j}\PYG{p}{,}\PYG{l+m+mi}{0}\PYG{p}{]}\PYG{p}{]}\PYG{p}{)}
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\PYG{n}{Z} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{array}\PYG{p}{(}\PYG{p}{[}\PYG{p}{[}\PYG{l+m+mi}{1}\PYG{p}{,}\PYG{l+m+mi}{0}\PYG{p}{]}\PYG{p}{,}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{,}\PYG{o}{\PYGZhy{}}\PYG{l+m+mi}{1}\PYG{p}{]}\PYG{p}{]}\PYG{p}{)}
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\PYG{c+c1}{\PYGZsh{} identity matrix}
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\PYG{n}{I} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{array}\PYG{p}{(}\PYG{p}{[}\PYG{p}{[}\PYG{l+m+mi}{1}\PYG{p}{,}\PYG{l+m+mi}{0}\PYG{p}{]}\PYG{p}{,}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{,}\PYG{l+m+mi}{1}\PYG{p}{]}\PYG{p}{]}\PYG{p}{)}
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\PYG{n}{epsilon} \PYG{o}{=} \PYG{p}{(}\PYG{n}{e0}\PYG{o}{+}\PYG{n}{e1}\PYG{p}{)}\PYG{o}{*}\PYG{l+m+mf}{0.5}\PYG{p}{;} \PYG{n}{omega} \PYG{o}{=} \PYG{p}{(}\PYG{n}{e0}\PYG{o}{\PYGZhy{}}\PYG{n}{e1}\PYG{p}{)}\PYG{o}{*}\PYG{l+m+mf}{0.5}
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\PYG{n}{c} \PYG{o}{=} \PYG{l+m+mf}{0.0}\PYG{p}{;} \PYG{n}{omega\PYGZus{}z}\PYG{o}{=}\PYG{n}{Xdiag}\PYG{p}{;} \PYG{n}{omega\PYGZus{}x} \PYG{o}{=} \PYG{n}{Xnondiag}
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\PYG{n}{Hamiltonian} \PYG{o}{=} \PYG{p}{(}\PYG{n}{epsilon}\PYG{o}{+}\PYG{n}{c}\PYG{p}{)}\PYG{o}{*}\PYG{n}{I}\PYG{o}{+}\PYG{p}{(}\PYG{n}{omega\PYGZus{}z}\PYG{o}{+}\PYG{n}{omega}\PYG{p}{)}\PYG{o}{*}\PYG{n}{Z}\PYG{o}{+}\PYG{n}{omega\PYGZus{}x}\PYG{o}{*}\PYG{n}{X}
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\PYG{n}{EigValues}\PYG{p}{,} \PYG{n}{EigVectors} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{linalg}\PYG{o}{.}\PYG{n}{eig}\PYG{p}{(}\PYG{n}{Hamiltonian}\PYG{p}{)}
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\PYG{n}{permute} \PYG{o}{=} \PYG{n}{EigValues}\PYG{o}{.}\PYG{n}{argsort}\PYG{p}{(}\PYG{p}{)}
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\PYG{n}{EigValues} \PYG{o}{=} \PYG{n}{EigValues}\PYG{p}{[}\PYG{n}{permute}\PYG{p}{]}
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\PYG{c+c1}{\PYGZsh{} print only the lowest eigenvalue}
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\PYG{n+nb}{print}\PYG{p}{(}\PYG{n}{EigValues}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{]}\PYG{p}{)}
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\end{MintedVerbatim}

‎doc/src/week4/_minted/6351E1F6AEE389ABB9D6CD536FF6252D.highlight.minted‎

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\begin{MintedVerbatim}[commandchars=\\\{\},codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8\relax}]
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\PYG{c+c1}{\PYGZsh{} define the rotation matrices}
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\PYG{c+c1}{\PYGZsh{} Define angles theta and phi}
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\PYG{n}{theta} \PYG{o}{=} \PYG{l+m+mf}{0.5}\PYG{o}{*}\PYG{n}{np}\PYG{o}{.}\PYG{n}{pi}\PYG{p}{;} \PYG{n}{phi} \PYG{o}{=} \PYG{l+m+mf}{0.2}\PYG{o}{*}\PYG{n}{np}\PYG{o}{.}\PYG{n}{pi}
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\PYG{n}{Rx} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{cos}\PYG{p}{(}\PYG{n}{theta}\PYG{o}{*}\PYG{l+m+mf}{0.5}\PYG{p}{)}\PYG{o}{*}\PYG{n}{I}\PYG{o}{\PYGZhy{}}\PYG{l+m+mi}{1}\PYG{n}{j}\PYG{o}{*}\PYG{n}{np}\PYG{o}{.}\PYG{n}{sin}\PYG{p}{(}\PYG{n}{theta}\PYG{o}{*}\PYG{l+m+mf}{0.5}\PYG{p}{)}\PYG{o}{*}\PYG{n}{X}
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\PYG{n}{Ry} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{cos}\PYG{p}{(}\PYG{n}{phi}\PYG{o}{*}\PYG{l+m+mf}{0.5}\PYG{p}{)}\PYG{o}{*}\PYG{n}{I}\PYG{o}{\PYGZhy{}}\PYG{l+m+mi}{1}\PYG{n}{j}\PYG{o}{*}\PYG{n}{np}\PYG{o}{.}\PYG{n}{sin}\PYG{p}{(}\PYG{n}{phi}\PYG{o}{*}\PYG{l+m+mf}{0.5}\PYG{p}{)}\PYG{o}{*}\PYG{n}{Y}
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\PYG{c+c1}{\PYGZsh{}define basis states}
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\PYG{n}{basis0} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{array}\PYG{p}{(}\PYG{p}{[}\PYG{l+m+mi}{1}\PYG{p}{,}\PYG{l+m+mi}{0}\PYG{p}{]}\PYG{p}{)}
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\PYG{n}{basis1} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{array}\PYG{p}{(}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{,}\PYG{l+m+mi}{1}\PYG{p}{]}\PYG{p}{)}
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\PYG{n}{NewBasis} \PYG{o}{=} \PYG{n}{Ry} \PYG{o}{@} \PYG{n}{Rx} \PYG{o}{@} \PYG{n}{basis0}
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\PYG{n+nb}{print}\PYG{p}{(}\PYG{n}{NewBasis}\PYG{p}{)}
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\PYG{c+c1}{\PYGZsh{} Compute the expectation value}
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\PYG{c+c1}{\PYGZsh{}Note hermitian conjugation}
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\PYG{n}{Energy} \PYG{o}{=} \PYG{n}{NewBasis}\PYG{o}{.}\PYG{n}{conj}\PYG{p}{(}\PYG{p}{)}\PYG{o}{.}\PYG{n}{T} \PYG{o}{@} \PYG{n}{Hamiltonian} \PYG{o}{@} \PYG{n}{NewBasis}
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\PYG{n+nb}{print}\PYG{p}{(}\PYG{n}{Energy}\PYG{p}{)}
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\end{MintedVerbatim}

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