From 8d6df8a79d7375e4090df16e6372d19de39aea5a Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Mon, 4 May 2026 20:06:18 +0200 Subject: [PATCH] Final readthrough corrections for fault tolerance chapter --- src/thesis/chapters/3_fault_tolerant_qec.tex | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/src/thesis/chapters/3_fault_tolerant_qec.tex b/src/thesis/chapters/3_fault_tolerant_qec.tex index eeed27f..0a3457a 100644 --- a/src/thesis/chapters/3_fault_tolerant_qec.tex +++ b/src/thesis/chapters/3_fault_tolerant_qec.tex @@ -16,19 +16,19 @@ using qubits. While the use of error correcting codes may facilitate this, it also introduces two new challenges \cite[Sec.~4]{gottesman_introduction_2009}: \begin{itemize} - \item For realizing a quantum algorithm, we must be able to + \item To realize a quantum algorithm, we must be able to perform operations on the encoded state in such a way that we do not lose the protection against errors. \item \ac{qec} systems, in particular the syndrome extraction circuit, are themselves partially implemented in quantum hardware. In addition to the errors we have originally introduced them - for, these systems must be able to account for the fact they - are implemented on noisy hardware themselves. + for, these systems must therefore be able to account for the + fact they are implemented on noisy hardware themselves. \end{itemize} In the literature, both of these points are viewed under the umbrella of \emph{fault-tolerant} quantum computing. -In this thesis, we focus only on the second aspect. +In this thesis, we focus on the second aspect. It was recognized early on as a challenge of \ac{qec} that the correction machinery itself may introduce new faults \cite[Sec.~III]{shor_scheme_1995}. @@ -938,10 +938,10 @@ triggered the measurements in the syndrome extraction round immediately afterwards, but all subsequent ones as well. To only see the effect of errors in the syndrome measurement round immediately following them, we consider our newly defined detectors -instead of the measurements, that effectively compute the difference -between the measurements. +instead of the measurements. +These effectively compute the difference between the measurements. -Hereby, each error can only trigger syndrome bits that follow it. +Each error can only trigger syndrome bits that follow it. This is reflected in the triangular structure of $\bm{\Omega}$ in \Cref{eq:syndrome_matrix_ex}. Combining the measurements into detectors according to @@ -1121,7 +1121,7 @@ In fact, it was in this tool that the concept of the \ac{dem} was first introduced. One capability of stim, and \acp{dem} in general, that we did not -explain in detail about in this chapter, is the merging of error mechanisms. +explain in detail in this chapter, is the merging of error mechanisms. Since \acp{dem} differentiate errors based on their effect on the measurements and not on their Pauli type and location \cite[Sec.~1.4.3]{higgott_practical_2024}, it is natural to group