handouts/Advanced/Introduction to Quantum/week 1.tex

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% use the [nosolutions] flag to hide solutions,
% use the [solutions] flag to show solutions.
\documentclass[
	solutions,
	singlenumbering
]{../../resources/ormc_handout}
\usepackage{../../resources/macros}

\def\ket#1{\left|#1\right\rangle}
\def\bra#1{\left\langle#1\right|}

\usepackage{units}
\input{tikzset}


\uptitlel{Advanced 2}
\uptitler{Winter 2022}
\title{Intro to Quantum Computing I}
\subtitle{Prepared by \githref{Mark} on \today{}}


\begin{document}

	\maketitle

	\input{parts/01 bits}
	\input{parts/02 qubit}
	\input{parts/03 two qubits}
	\input{parts/04 logic gates}
	\input{parts/05 quantum gates}

	\section{Bonus Problems (Putnam)}

	\problem{}
	Suppose $A$ is a real, square matrix that satisfies $A^3 = A + I$. \par
	Show that $\text{det}(A)$ is positive.

	\vfill

	\problem{}
	Suppose $A, B$ are $2 \times 2$ complex matrices satisfying $AB = BA$, \par
	and assume $A$ is not of the form $aI$ for some complex $a$. \par
	Show that $B = xA + yI$ for complex $x$ and $y$.

	\vfill

	\problem{}
	Is there an infinite sequence of real numbers $a_1, a_2, ...$ so that \par
	$a_1^m + a_2^m + ... = m$ for every positive integer $m$?

	\vfill
\end{document}