Add backlog problem explanation

src/thesis/chapters/2_fundamentals.tex

@@ -1082,7 +1082,7 @@ An example of this is the CNOT gate introduced in
 
 One of the major barriers on the road to building a functioning
 quantum computer is the inevitability of errors during quantum
-computation due to the difficulty in sufficiently isolating the
+computation. These arise due to the difficulty in sufficiently isolating the
 qubits from external noise \cite[Intro.]{roffe_quantum_2019}.
 This isolation is critical for quantum systems, as the constant interactions
 with the environment act as small measurements, leading to the
@@ -1094,8 +1094,8 @@ correction.
 
 % The unique challenges of QEC
 
-The problem setting of \ac{qec} differs slightly from the classical case, as
-three main restrictions apply \cite[Sec.~2.4]{roffe_quantum_2019}:
+The problem setting of \ac{qec} differs slightly from the classical case.
+Three main restrictions apply \cite[Sec.~2.4]{roffe_quantum_2019}:
 \begin{itemize}
     \item The no-cloning theorem states that it is
         impossible to exactly copy the state of one qubit into another.
@@ -1116,12 +1116,27 @@ To this end, $k \in \mathbb{N}$ \emph{logical qubits} are mapped onto
 $n \in \mathbb{N}$ \emph{physical qubits}, $n>k$.
 We circumvent the no-cloning restriction by not copying the state of any of
 the $k$ logical qubits, instead spreading the total state out over all $n$
-physical ones \cite[Intro.]{calderbank_good_1996}.
+physical qubits \cite[Intro.]{calderbank_good_1996}.
 To differentiate quantum codes from classical ones, we denote a
 code with parameters $k,n$ and minimum distance $d_\text{min}$ using
 double brackets, as $\llbracket n,k,d_\text{min} \rrbracket$
 \cite[Sec.~4]{roffe_quantum_2019}.
 
+% The backlog problem
+
+Another difference between quantum and classical error correction
+lies in the resource constraints.
+For \ac{qec}, the most important property is low latency, not, e.g.,
+low overall computational complexity.
+This is due to the \emph{backlog problem}
+\cite[Sec.~II.G.3.]{terhal_quantum_2015}: There are certain gates
+at which the effect of existing errors on single qubits may be
+exacerbated by transforming them to mutli-qubit errors.
+We wish to correct the errors before passing qubits through such gates.
+If the \ac{qec} system is not fast enough, there will be an increasing
+backlog of information at this point in the circuit, leading to an
+exponential slowdown in computation.
+
 %%%%%%%%%%%%%%%%
 \subsection{Stabilizer Measurements}
 \label{subsec:Stabilizer Measurements}
@@ -1306,15 +1321,16 @@ We call codes constructed this way \emph{stabilizer codes}.
 Similar to the classical case, we can use a syndrome vector to
 describe which local codes are violated.
 To obtain the syndrome, we simply measure the corresponding
-operators, each using a circuit as explained in
+operators $P_i$, each using a circuit as explained in
 \autoref{subsec:Stabilizer Measurements}.
 A full \emph{syndrome extraction circuit} is depicted in \autoref{fig:sec}.
 
 % TODO: Move this further up to the commutativity of operators?
 \indent\red{[Fixing the error after finding it
-\cite[Sec.~10.5.5]{nielsen_quantum_2010}]} \\
+        \cite[Sec.~10.5.5]{nielsen_quantum_2010} -> This may require
+introducing the gates as unitary]} \\
 \indent\red{[Logical operators \cite[Sec.~4.2]{roffe_quantum_2019}]} \\
-\indent\red{[Measuring logical operators gives yields the outcomes of
+\indent\red{[Measuring logical operators yields the outcomes of
 the encoded computations \cite[Sec.~2.6]{derks_designing_2025}]} \\
 \indent\red{[X and Z measurements can be performed with only CNOT and
 Hadamard gates \cite[Sec.~10.5.8]{nielsen_quantum_2010}]} \\
@@ -1353,7 +1369,7 @@ $Z$ operators and some with only $X$ operators.
 
 \indent\red{[Z-type operators for X type errors and vice versa ]} \\
 \indent\red{[Construction from two binary linear codes
-\cite[p.~452,469]{nielsen_quantum_2010}]}
+\cite[p.~452,469]{nielsen_quantum_2010}]} \\
 
 \subsection{Quantum Low-Density Parity-Check Codes}