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\begin{document}

\title{\vspace{-10mm}Introduction to Quantum Computation, UPB\\Summer 2021, Assignment 2\\{\large To be completed by: Friday, April 30}}
\date{}
\maketitle

\section{Exercises}
\begin{enumerate}
    \item %($4$ marks)
    Use the spectral decompositions of $X$ and $Z$ to prove that $HXH^\dagger=Z$. (Do not simply write out the matrices and multiply!) Why does this immediately also yield that $HZH^\dagger=X$?
    \item %($6$ marks)
    \begin{enumerate}
        \item %($2$ marks)
        Write out the 4-dimensional vector for $(\alpha\ket{0}+\beta\ket{1})\otimes(\gamma\ket{0}+\delta\ket{1})$.
        \item %($4$ marks)
        Let $\mathcal{B}_1=\set{\ket{\psi_1},\ket{\psi_2}}$ and $\mathcal{B}_2=\set{\ket{\phi_1},\ket{\phi_2}}$ be two orthonormal bases for $\complex^2$. Prove that
            \[
            \mathcal{B}_3 = \set{\ket{\psi_1}\otimes\ket{\phi_1},\ket{\psi_1}\otimes\ket{\phi_2},\ket{\psi_2}\otimes\ket{\phi_1},\ket{\psi_2}\otimes\ket{\phi_2}}
            \]
            is an orthonormal basis for $\complex^4$. In other words, show that for each $\ket{v}\in\mathcal{B}_3$, $\enorm{\ket{v}}=1$, and for all pairs of distinct $\ket{v},\ket{w}\in\mathcal{B}_3$, $\braket{v}{w}=0$.
    \end{enumerate}
    \item %($10$ marks)
            \begin{enumerate}
                %\item %($2$ marks)
                %Write out the $4\times 4$ matrix representing $Y\otimes Y$.
                \item %($2$ marks)
                Prove that $(Z\otimes Y)^\dagger=Z\otimes Y$. Do not write out any matrices explicitly; rather, you must use the properties of the tensor product, dagger, and $Y$.
                \item %($6$ marks)
                In class, we saw a quantum circuit which, given starting state $\ket{0}\otimes\ket{0}$, prepared the Bell state $\ket{\Phi^+}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$. In fact, that circuit is a change of basis matrix, mapping the standard basis $\ket{00},\ket{01},\ket{10},\ket{11}$ to the Bell basis $\ket{\Phi^+},\ket{\Psi^+},\ket{\Phi^-},\ket{\Psi^-}$.
                \begin{enumerate}
                    \item %($3$ marks)
                    Write down a quantum circuit which maps $\ket{\Phi^+}$ to $\ket{0}\otimes\ket{0}$. (Hint: Write out the circuit from class as a sequence of matrix operations. Then, recalling that the inverse of any unitary operation $U$ is given by $U^\dagger$, take the inverse of this entire sequence by taking the dagger.)
                    \item %($3$ marks)
                    Your circuit from 3(b)(i) is actually a change of basis which maps the Bell basis \emph{back} to the standard basis. To verify this, run your circuit from 3(b)(i) on input $\ket{\Psi^-}$ and check that the output is $\ket{1}\otimes\ket{1}$.
                \end{enumerate}
            \end{enumerate}
\end{enumerate}

\end{document}