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Rephrase some sentences

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Andreas Tsouchlos 2025-11-19 09:49:27 +01:00
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src/intro/main.tex
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@ -39,7 +39,7 @@
Much like in classical error correction, in \ac{qec} information Much like in classical error correction, in \ac{qec} information
is protected by mapping it onto codewords in an expanded is protected by mapping it onto codewords in an expanded
Hilbert space, thereby introducing redundancy. Take for example, the Hilbert space, thereby introducing redundancy. Take, for example, the
two-qubit code \cite{roffe_quantum_2019}, where we map% two-qubit code \cite{roffe_quantum_2019}, where we map%
% %
\begin{align*} \begin{align*}
@ -59,7 +59,7 @@ To determine if an error occurred, we want to know
whether a state belongs% whether a state belongs%
\footnote{ \footnote{
It is possible for a state to not completely lie in either subspace. It is possible for a state to not completely lie in either subspace.
In this case, we can interpret the state as being in In this case, we can interpret it as being in
$\mathcal{C}$ or $\mathcal{F}$ with a certain probability. $\mathcal{C}$ or $\mathcal{F}$ with a certain probability.
} }
to $\mathcal{C}$ or $\mathcal{F}$. to $\mathcal{C}$ or $\mathcal{F}$.
@ -69,8 +69,8 @@ described using operators \cite[Section
1.5]{griffiths_introduction_1995}. Because of the way these operators are 1.5]{griffiths_introduction_1995}. Because of the way these operators are
defined, their eigenvalues correspond to the possible outcomes of defined, their eigenvalues correspond to the possible outcomes of
measuring that observable, and the corresponding measuring that observable, and the corresponding
eigenstates are the states that yield those values as measurements \cite[Section eigenstates are the determinate states that yield those values as
3.3]{griffiths_introduction_1995}. measurements \cite[Section 3.3]{griffiths_introduction_1995}.
In our case, we need an operator with two eigenvalues, and the corresponding In our case, we need an operator with two eigenvalues, and the corresponding
eigenspaces should be $\mathcal{C}$ and $\mathcal{F}$ respectively. eigenspaces should be $\mathcal{C}$ and $\mathcal{F}$ respectively.
For the two-qubit code, $Z_1Z_2$ is such an operator:% For the two-qubit code, $Z_1Z_2$ is such an operator:%
@ -95,7 +95,7 @@ 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
$\ket{\psi}_\text{L}$ in such a way that determining that instead $\ket{\psi}_\text{L}$ in such a way that determining that instead
indicates the eigenvalue. More specifically, using a syndrome indicates the eigenvalue. More specifically, using a syndrome
extraction circuit as shown in fig. \ref{fig:syndrome extraction}, we extraction circuit as shown in Figure \ref{fig:syndrome extraction}, we
transform the state of the three-qubit system as% transform the state of the three-qubit system as%
% %
@ -114,8 +114,8 @@ transform the state of the three-qubit system as%
If $E \ket{\psi}_\text{L} \in \mathcal{C}$, the second term will If $E \ket{\psi}_\text{L} \in \mathcal{C}$, the second term will
cancel and we will deterministically measure $\ket{0}_\text{A}$ for cancel and we will deterministically measure $\ket{0}_\text{A}$ for
the ancilla qubit. Similarly, if $E \ket{\psi}_\text{L} \in 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, 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