an.tsouchlos/ma-thesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases 12 Wiki Activity

Final readthrough corrections for fault tolerance chapter

Browse Source
This commit is contained in:
Andreas Tsouchlos
2026-05-04 20:06:18 +02:00
parent c41ac9f61f
commit 8d6df8a79d
1 changed files with 8 additions and 8 deletions
Download Patch File Download Diff File Expand all files Collapse all files

16
src/thesis/chapters/3_fault_tolerant_qec.tex
View File

@@ -16,19 +16,19 @@ using qubits.
While the use of error correcting codes may facilitate this, it also While the use of error correcting codes may facilitate this, it also
introduces two new challenges \cite[Sec.~4]{gottesman_introduction_2009}: introduces two new challenges \cite[Sec.~4]{gottesman_introduction_2009}:
\begin{itemize} \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 perform operations on the encoded state in such a way that we
do not lose the protection against errors. do not lose the protection against errors.
\item \ac{qec} systems, in particular the syndrome extraction \item \ac{qec} systems, in particular the syndrome extraction
circuit, are themselves partially implemented in circuit, are themselves partially implemented in
quantum hardware. quantum hardware.
In addition to the errors we have originally introduced them In addition to the errors we have originally introduced them
for, these systems must be able to account for the fact they for, these systems must therefore be able to account for the
are implemented on noisy hardware themselves. fact they are implemented on noisy hardware themselves.
\end{itemize} \end{itemize}
In the literature, both of these points are viewed under the umbrella In the literature, both of these points are viewed under the umbrella
of \emph{fault-tolerant} quantum computing. 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 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}. 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. afterwards, but all subsequent ones as well.
To only see the effect of errors in the syndrome measurement round To only see the effect of errors in the syndrome measurement round
immediately following them, we consider our newly defined detectors immediately following them, we consider our newly defined detectors
instead of the measurements, that effectively compute the difference instead of the measurements.
between 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 This is reflected in the triangular structure of $\bm{\Omega}$ in
\Cref{eq:syndrome_matrix_ex}. \Cref{eq:syndrome_matrix_ex}.
Combining the measurements into detectors according to 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. first introduced.
One capability of stim, and \acp{dem} in general, that we did not 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 Since \acp{dem} differentiate errors based on their effect on the
measurements and not on their Pauli type and location measurements and not on their Pauli type and location
\cite[Sec.~1.4.3]{higgott_practical_2024}, it is natural to group \cite[Sec.~1.4.3]{higgott_practical_2024}, it is natural to group