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Change paragraphs; add TODO; change margins

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Andreas Tsouchlos 2025-11-19 10:43:49 +01:00
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src/intro/main.tex
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@ -1,6 +1,6 @@
\documentclass[dvipsnames]{article} \documentclass[dvipsnames]{article}
\usepackage[a4paper,left=3cm,right=2cm,top=2.5cm,bottom=2.5cm]{geometry} \usepackage[a4paper,left=3cm,right=3cm,top=2.5cm,bottom=2.5cm]{geometry}
\usepackage{float} \usepackage{float}
\usepackage{amsmath} \usepackage{amsmath}
\usepackage{amsfonts} \usepackage{amsfonts}
@ -17,6 +17,7 @@
style=ieee, style=ieee,
sorting=nty, sorting=nty,
]{biblatex} ]{biblatex}
\usepackage{todonotes}
\usetikzlibrary{calc, positioning} \usetikzlibrary{calc, positioning}
@ -89,7 +90,6 @@ $E$ is an operator describing a possible error and $E
\ket{\psi}_\text{L}$ is the resulting state after that error. \ket{\psi}_\text{L}$ is the resulting state after that error.
By measuring the corresponding eigenvalue, we can determine if By measuring the corresponding eigenvalue, we can determine if
$E\ket{\psi}_\text{L}$ lies in $\mathcal{C}$ or $\mathcal{F}$. $E\ket{\psi}_\text{L}$ lies in $\mathcal{C}$ or $\mathcal{F}$.
To do this without directly measuring (and thus disturbing) the To do this without directly measuring (and thus disturbing) the
logical state $\ket{\psi}_\text{L}$, we prepare an ancilla logical state $\ket{\psi}_\text{L}$, we prepare an ancilla
qubit with state $\ket{0}_\text{A}$ and we entangle it with qubit with state $\ket{0}_\text{A}$ and we entangle it with
@ -117,18 +117,21 @@ the ancilla qubit. Similarly, if $E \ket{\psi}_\text{L} \in
\mathcal{F}$, we will deterministically measure $\ket{1}_\text{A}$. \mathcal{F}$, we will deterministically measure $\ket{1}_\text{A}$.
In general, however, the resulting state of the three-qubit system will be a In general, however, the resulting state of the three-qubit system will be a
superposition of the two cases. superposition of the two cases.
Note that the expressions $P_\mathcal{C}$ and $P_\mathcal{F}$ above Note that the expressions $P_\mathcal{C}$ and $P_\mathcal{F}$ above
essentially constitute projection operators onto $\mathcal{C}$ and essentially constitute projection operators onto $\mathcal{C}$ and
$\mathcal{F}$. E.g., $P_\mathcal{C}$ will eliminate all $\mathcal{F}$. E.g., $P_\mathcal{C}$ will eliminate all
components of a superposition state $E \ket{\psi}_\text{L}$ that lie components of a superposition state $E \ket{\psi}_\text{L}$ that lie
in $\mathcal{F}$. in $\mathcal{F}$.
By measuring the ancilla qubit, we collapse the overall state By measuring the ancilla qubit, we collapse the overall state
into one of two configurations. In each of those configurations, $E into one of two configurations. In each of those configurations, $E
\ket{\psi}_\text{L}$ is projected back onto either $\mathcal{C}$ or \ket{\psi}_\text{L}$ is projected back onto either $\mathcal{C}$ or
$\mathcal{F}$. At the same time, because $Z_1Z_2 \ket{\psi}_\text{L} $\mathcal{F}$. At the same time, because $Z_1Z_2 \ket{\psi}_\text{L}
= \ket{\psi}_\text{L}$, the projections leave the logical state = \ket{\psi}_\text{L}$, the projections leave the logical state
$\ket{\psi}_\text{L}$ untouched. We have thus managed to determine $\ket{\psi}_\text{L}$ untouched.
\todo{Explain that the collapse of the error superposition enables
the digitization of arbitrary error operations \cite{nielsen_quantum_2010}}
We have thus managed to determine
whether an error occurred without disturbing the encoded whether an error occurred without disturbing the encoded
quantum information. quantum information.