\documentclass{article}
\usepackage{amsthm,amsmath,amsfonts,fullpage}
\newcounter{probpart}
\newenvironment{probparts}{\begin{list}{\alph{probpart})}%
{\usecounter{probpart}}}{\end{list}}
\newcommand{\problem}[2]{\noindent {\bf Problem \#{#1}. {#2}}}
\newcommand{\ket}[1]{|{#1}\rangle}
\newcommand{\bra}[1]{\langle{#1}|}
\newcommand{\enczero}{\ket{\overline{0}}}
\newcommand{\encone}{\ket{\overline{1}}}
\newcommand{\Hilbert}{\mathcal{H}} % a generic Hilbert space
\newcommand{\Hilbertk}[1]{\mathcal{H}_{{#1}}} % a Hilbert space of dimension k
\newcommand{\channel}[1]{\mathcal{#1}} % font choice for a quantum channel
\newcommand{\solution}{\paragraph{Solution:}}
\begin{document}
\title{Problem Set \#4}
\author{Quantum Error Correction\\Instructors: Daniel Gottesman}
\date{Due Tuesday, Apr.~9, 2024}
\maketitle
\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}