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Rework detector matrix and detector error matrix sections

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Andreas Tsouchlos
2026-04-29 13:26:35 +02:00
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src/thesis/chapters/3_fault_tolerant_qec.tex
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@@ -312,7 +312,7 @@ the decoding process.
The core idea behind detector error models is to do this by defining
a new \emph{circuit code} that describes the circuit.
Each \ac{vn} of this new code corresponds to an error location in the
circuit and each \ac{cn} corresponds to a to a syndrome measurement.
circuit and each \ac{cn} corresponds to a syndrome measurement.
% This circuit code, combined with the prior probabilities of error
% given by the noise model, incorporates all information necessary for decoding.
@@ -508,7 +508,8 @@ arriving at the circuit depicted in
\autoref{fig:rep_code_multiple_rounds_phenomenological}.
For each additional error location, we extend $\bm{\Omega}$ by
appending the corresponding syndrome vector as a column.
\begin{gather*}
\begin{gather}
\label{eq:syndrome_matrix_ex}
\bm{\Omega} =
\left(
\begin{array}{ccccccccccccccc}
@@ -534,11 +535,21 @@ appending the corresponding syndrome vector as a column.
\end{array}
}
}_\text{Original matrix}
\end{gather*}
\end{gather}
Notice that the first three columns correspond to the original
measurement syndrome matrix, as these columns correspond to the error
locations on the data qubits.
In this example, all measurements we considered were syndrome measurements.
Assuming no errors, the results of those measurements were
deterministic, irrespective of the actual logical state
$\ket{\psi}_\text{L}$, as they only depend on whether
$\ket{\psi}_\text{L} \in \mathcal{C}$, not on the concrete state.
It is, in general, possible to also consider non-deterministic measurements.
As an example, it is usual to consider a round of noiseless
measurements of the actual data qubit states after the last syndrome
extraction round.
\begin{figure}[t]
\centering
@@ -745,22 +756,21 @@ locations on the data qubits.
\end{figure}
%%%%%%%%%%%%%%%%
\subsection{Detector Error Matrix}
\label{subsec:Detector Error Matrix}
\subsection{Detector Matrix}
\label{subsec:Detector Matrix}
% Core idea
% TODO: Make this a proper definition?
Instead of using the measurements as parity indicators directly, we
may wish to combine them in some way.
We call such combinations \emph{detectors}.
Formally, a detector is a parity constraint on a set of measurement
outcomes \cite[Def.~2.1]{derks_designing_2025}.
Changing the perspective in this way does not alter the theoretical
error correcting capabilities of the circuit, but it may change the
decoding performance when using a practical decoder.
\red{[Possibly a few more words on this (maybe a mathematical
proof/intuition?)]}
Instead of using stabilizer measurement results directly, we
generalize the notion of what constitutes a parity check slightly.
We formally define a \emph{detector} as a deterministic parity constraint on
a set of measurement outcomes \cite[Def.~2.1]{derks_designing_2025}.
In the most straight forward case, we may simply use the stabilizer
measurements as detectors.
We immediately recognize that we will have as many linearly
independent detectors as there are separate deterministic measurements.
We generally aim to utilize the maximum number of linearly
independent detectors \cite[Sec.~2.2]{derks_designing_2025}.
% The detector matrix
@@ -768,21 +778,36 @@ proof/intuition?)]}
We describe the relationship between measurements and detectors using
the \emph{detector matrix} $\bm{D} \in \mathbb{F}_2^{d\times m}$
\cite[Def.~2.2]{derks_designing_2025}.
Similar to the way a \ac{pcm} connects bits with parity checks, the
Similar to the way a \ac{pcm} associates bits with parity checks, the
detector matrix links measurements and detectors.
Each column corresponds to a measurement, while the rows correspond
to the detectors.
Each column corresponds to a measurement, while each rows corresponds
to a detector.
We should note at this point that the combination of measurements
into detectors has no bearing on the actual construction of the
syndrome extraction circuitry.
It is something that happens ``virtually'' after the fact and only
affects the decoder.
Note that we can use the detector matrix $\bm{D}$ to describe the set
of possible measurement outcomes under the absence of noise.
The same way we use a \ac{pcm} to describe the code space as
\begin{align*}
\mathcal{C}
= \{ \bm{x} \in \mathbb{F}_2^{n} : \bm{H}\bm{x}^\text{T} = \bm{0} \}
,%
\end{align*}
the set of possible measurement outcomes is simply $\text{kern}\{\bm{D}\}$
\cite[Sec.~2.2]{derks_designing_2025}.
%%%%%%%%%%%%%%%%
\subsection{Detector Error Matrix}
\label{subsec:Detector Error Matrix}
% The detector error matrix
We now know how the errors at different locations in the circuit
affect the measurements ($\bm{\Omega}$), and we know how the
measurements relate to the detectors ($\bm{D}$).
affect the measurements (through $\bm{\Omega}$), and we know how the
measurements relate to the detectors (through $\bm{D}$).
For decoding, we are interested in the effect of the errors on the
detectors directly.
We thus construct the \emph{detector error matrix} $\bm{H} \in
@@ -791,25 +816,58 @@ We thus construct the \emph{detector error matrix} $\bm{H} \in
\bm{H} := \bm{D}\bm{\Omega}
.%
\end{align*}
Note that, in particular when $d=m$, this is equivalent to performing row
additions on the matrix $\bm{\Omega}$.
% There are multiple ways of choosing the detectors
There is a degree of freedom in how we choose the detectors, which is
reflected in the fact that we can construct multiple different
detector matrices $\bm{D}$ from the same circuit.
For two detector matrices $\bm{D}_1$ and $\bm{D}_2$, as long as
\begin{gather}
\label{eq:kern_condition}
\text{kern}\{\bm{D}_1\} = \text{kern}\{\bm{D}_2\}
\end{gather}
they describe the same set of possible measurement outcomes (under
the absence of noise) and thus the same circuit.
In fact, as long as \autoref{eq:kern_condition} holds, the detector
error matrices we construct from them can distinguish between the
same pairs of error sets \cite[Lemma~6]{derks_designing_2025}.
To see this, we note that we can distinguish between two circuit
error vectors $\bm{e}_1$ and $\bm{e}_2$ as long as they do not
violate the same set of detectors, i.e.,
\begin{align*}
\hspace{-15mm}
% tex-fmt: off
&& \bm{H} \bm{e}_1^\text{T} & \neq \bm{H} \bm{e}_2^\text{T} \\
\iff \hspace{-33mm} && \bm{H} \left( \bm{e}_1 - \bm{e}_2 \right)^\text{T} & \neq 0 \\
\iff \hspace{-33mm} && \bm{D} \bm{\Omega} \left( \bm{e}_1 - \bm{e}_2 \right)^\text{T} & \neq 0 \\
\iff \hspace{-33mm} && \bm{\Omega} \left( \bm{e}_1 - \bm{e}_2 \right)^\text{T} & \notin \text{kern} \{\bm{D}\}
% tex-fmt: on
.%
\end{align*}
We conclude that altering our perspective by choosing a different
detector matrix
does not modify the error correcting capabilities of the code.
It may, however, change the decoding performance when using a practical decoder.
% How to choose the detectors
% TODO: Fix notation (n used for both number of measurements and
% measurements themselves)
% TODO: Properly define the ranges i and r belong to
We still have a degree of freedom in how we choose the detectors.
\red{[No way to automate this, NP-complete problem to find detectors
which yield highest sparity]}
What constitutes a good set of detectors is difficult to assess
without performing explicit decoding simulations, since it ultimately
depends on the decoder employed.
For iterative decoders, high sparsity is generally beneficial, but
finding detectors that maximize sparsity is an NP-complete problem
\cite[Sec.~2.6]{derks_designing_2025}.
There is, however, one way of defining the detectors that will prove useful
at a later stage.
To the measurement results from each syndrome extraction round, we
To the measurement results from each syndrome extraction round we
can add the results from the previous round, as illustrated in
\autoref{fig:detectors_from_measurements_general}.
Concretely, we denote the outcome of the
$i$-th syndrome measurement in round $r$ by $m_i^{(r)} \in \mathbb{F}_2$
and define
$i$-th measurement in round $r$ by $m_i^{(r)} \in \mathbb{F}_2$ and define
\begin{gather*}
\bm{m}^{(r)} :=
\begin{pmatrix}
@@ -832,50 +890,8 @@ $d_i^{(r)} \in \mathbb{F}_2$ and define
:= \bm{m}^{(r)} + \bm{m}^{(r-1)}
,%
\end{gather}
where $\bm{m}^{(0)} = \bm{0}$.
with $\bm{m}^{(0)} = \bm{0}$.
We again turn our attention to the three-qubit repetition code.
In \autoref{fig:rep_code_multiple_rounds_phenomenological} we can see
that $E_6$ has occurred and has subsequently tripped the last four measurements.
We now take those measurements and combine them according to
\autoref{eq:measurement_combination}.
We can see this process graphically in
\autoref{fig:detectors_from_measurements_rep_code}.
To understand why this way of defining the detectors is useful, we
note that the error $E_6$ in
\autoref{fig:rep_code_multiple_rounds_phenomenological} has not only
tripped the measurements in the syndrome extraction round immediately
afterwards, but all subsequent ones as well.
To only see errors in the rounds immediately following them, we
consider our newly defined detectors instead of the measurements,
that effectively compute the difference between the measurements.
Each error can only trip syndrome bits that follow it.
We can see this in the triangular structure of $\bm{\Omega}$ in
\autoref{fig:rep_code_multiple_rounds_phenomenological}.
Combining the measurements into detectors according to
\autoref{eq:measurement_combination}, we are performing row additions
in such a way as to clear the bottom left of the matrix.
This yields a block-diagonal structure for the detector error matrix
$\bm{H}$, as in the example in
\autoref{fig:detectors_from_measurements_rep_code}.
\begin{align*}
\bm{H} =
\left(
\begin{array}{ccccccccccccccc}
1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1
\end{array}
\right)
\end{align*}
Note that we exploit the fact that each syndrome measurement round is
identical to obtain this structure.
% TODO: Change notation (\bm{D})
\begin{figure}[t]
\centering
@@ -901,6 +917,46 @@ identical to obtain this structure.
\label{fig:detectors_from_measurements_general}
\end{figure}
We again turn our attention to the three-qubit repetition code.
In \autoref{fig:rep_code_multiple_rounds_phenomenological} we can see
that $E_6$ has occurred and has subsequently tripped the last four measurements.
We now take those measurements and combine them according to
\autoref{eq:measurement_combination}.
We can see this process graphically in
\autoref{fig:detectors_from_measurements_rep_code}.
To understand why this way of defining the detectors is useful, we
note that the error $E_6$ in
\autoref{fig:rep_code_multiple_rounds_phenomenological} has not only
tripped the measurements in the syndrome extraction round immediately
afterwards, but all subsequent ones as well.
To only see errors in the rounds immediately following them, we
consider our newly defined detectors instead of the measurements,
that effectively compute the difference between the measurements.
Each error can only trip syndrome bits that follow it.
This is reflected in the triangular structure of $\bm{\Omega}$ in
\autoref{eq:syndrome_matrix_ex}.
Combining the measurements into detectors according to
\autoref{eq:measurement_combination}, we are effectively performing
row additions in such a way as to clear the bottom left of the matrix.
The detector error matrix
\begin{align*}
\bm{H} =
\left(
\begin{array}{ccccccccccccccc}
1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1
\end{array}
\right)
\end{align*}
we obtain this way has a block-diagonal structure.
Note that we exploit the fact that each syndrome measurement round is
identical to obtain this structure.
\begin{figure}[t]
\centering