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\begin{document}

\title{Problem Set \#4}
\author{Quantum Error Correction\\Instructors:  Daniel Gottesman}
\date{Due Tuesday, Apr.~14, 2026}

\maketitle

General instructions: No use of LLMs.  If you collaborate or use any outside resources, remember to cite them in your solutions.

\problem{1}{Repetition of syndrome measurement in Shor error correction (40 points)}

For Shor error correction and a distance 3 code, consider the following method of repeated syndrome measurement: Measure the error syndrome twice.  If both syndrome measurements are the same, use that value.  If the syndrome measurements differ but the first syndrome measured is $0$ (corresponding to no error), deduce the trivial error.  If the syndrome measurements differ but the first syndrome is non-zero, use the second syndrome to deduce the error.

\begin{probparts}
\item (15 points) Show that this method of repeating the syndrome and deducing the error satisfies the ECCP for a code correcting $1$ error.

\item (15 points) For the $7$-qubit code, show that the ECRP is not satisfied by giving a combination of an error on the input state to the EC gadget and a fault during the gadget that cause the ECRP to fail.

\item (10 points) For the $5$-qubit code, show that the ECRP is satisfied.

\end{probparts}

\medskip

\problem{2}{Teleportation of the controlled-phase gate (20 points)}

Let $U$ be the gate
\begin{equation}
U = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & i 
\end{pmatrix}.
\end{equation}

\begin{probparts}
\item (5 points) Find $UPU^\dagger$ for $P = X \otimes I$, $I \otimes X$, $Z \otimes I$, and $I \otimes Z$.

\item (15 points) Find a \emph{two-qubit} ancilla state and a circuit involving two data qubits and the ancilla consisting of Clifford group gates, Pauli basis measurements, and classical feed-forward, such that the output of the circuit is $U \ket{\psi}$.  ($\ket{\psi}$ is the input state of the two data qubits.)

\end{probparts}





\end{document}
