From 6ea151ffebb33c3dfe3111025422b42ffc5d0ef5 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Fri, 24 Apr 2026 09:25:55 +0200 Subject: [PATCH] Add backlog problem explanation --- src/thesis/chapters/2_fundamentals.tex | 32 +++++++++++++++++++------- 1 file changed, 24 insertions(+), 8 deletions(-) diff --git a/src/thesis/chapters/2_fundamentals.tex b/src/thesis/chapters/2_fundamentals.tex index 7cc994d..527c6b3 100644 --- a/src/thesis/chapters/2_fundamentals.tex +++ b/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}