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\subsection*{PHYS 570/MA 595: Unit 3 Exercises, Due October 9, 2025}
Exercises will be added to this list but no new problems will be added after Thursday October 2.\\

\textbf{Tuesday September 23 and Thursday September 25}

\begin{enumerate}
\item In Kitaev's quantum double model for a finite group $G$, the vertex gauge transformation operators $A_g(s)$ and the operators $B_h(s)$ that project onto sites with flux $h$ do not commute for all $g,h \in G$. \\
\textbf{Explain} why they satisfy $A_g(s)B_h(s)=B_{ghg^{-1}}(s)A_g(s).$ \hfill (5 points)\\
This completes the calculation that shows that the local operators at a site $s$ form a representation of the quantum double algebra $DG$. 

\item  Recall that $A_g(s)$ only depends on the vertex $v$ at the site $s$, and so we can write $A_g(v)=A_g(s)$. \\\textbf{Verify} that the quantum double vertex operator $A(v)= \frac{1}{|G|} \sum_{g \in G} A_g(v)$ is a projector, i.e.~ $A(v)^2=A(v)$. \hfill (5 points) 

\item Consider the quantum double model on a square lattice as in class. When you plug in $G = \mathds{Z}_2$, \textbf{describe} the relationship of the vertex and plaquettes operators $A(v)$ and $B(p)$ to the vertex and plaquette operators $A_v$ and $B_p$ that appeared in Kitaev's toric code Hamiltonian from Lectures 4 and 5.  \hfill (5 points) \\
\end{enumerate}

\textbf{Tuesday September 30}
\begin{enumerate}
\item[4.] Consider a periodic 1d lattice with $N$ sites with a qubit on each site. Show that the operator $S_{i,j}$ that swaps the $i$th and $j$th qubits $S_{i,j}= \frac{1}{2}\left ( X_iX_j +Y_iY_j + Z_iZ_j + 1 \right )$ acts on the Pauli $X$ operator on site $i$ to the Pauli $X$ operator on site $j$, i.e.~ \textbf{show} that $$S_{i,j} X_j (S_{i,j})^{-1}=X_i. \hspace{100pt} \text{(6 points)}$$\\
\noindent It will be helpful to remember the relations in the single-qubit Pauli group from the Unit 1 Exercises, which can be summarized with the notation $\sigma _{i}\sigma _{j}=\delta _{ij}I+{\text{i}}\epsilon _{ijk}\sigma _{k}$ where $\sigma_1 = X$, $\sigma_2 =Y$, and $\sigma_3=Z$ and $\epsilon_{ijk}$ is the \hl{Levi-Civita symbol}. 

\item[5.] Let $H = -g\sum_{j=1}^N X_j - \sum_{j=1}^N Z_jZ_{j+1}$ be the transverse-field Ising Hamiltonian, and put $T_{j,j+1}=S_{j,j+1}$ where $S$ is the operator defined in the previous problem. \textbf{Argue} that the $\mathds{Z}_N$ lattice translation symmetry operator 
$$T=\prod_{j=1}^{N-1} T_{j,j+1}$$
commutes with $H$. \hfill (4 points)
\end{enumerate}


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