26 Def 8: Detectors
A detector is a structure \((D, f, c)\) where:
\(D \subseteq M\) is a finite set of measurement labels (a \(\operatorname {Finset} M\)),
\(f : M \to \mathbb {Z}/2\mathbb {Z}\) assigns to each measurement its ideal outcome (\(0 \leftrightarrow +1\), \(1 \leftrightarrow -1\)),
\(c\) is a proof of the detector constraint: \(\sum _{m \in D} f(m) = 0\) in \(\mathbb {Z}/2\mathbb {Z}\).
In the \(\{ +1, -1\} \) encoding, the constraint states that the product of ideal outcomes over the detector’s measurements equals \(+1\).
Given a detector \(D\) and a set of time-faults \(F\), the observed parity is defined as
In \(\{ +1,-1\} \) encoding, \(0\) means the product of observed outcomes is \(+1\), and \(1\) means it is \(-1\).
A detector \(D\) is violated by a set of time-faults \(F\) if the observed parity equals \(1\):
The flip parity of a detector \(D\) with respect to faults \(F\) is the sum in \(\mathbb {Z}/2\mathbb {Z}\) of the indicator of faulted measurements in \(D\):
In the absence of faults, the observed parity of any detector equals \(0\):
Unfolding the definitions of observed parity and observed outcome, for every \(m \in D.\mathrm{measurements}\) we have \(\langle m \rangle \notin \emptyset \), so the observed outcome reduces to the ideal outcome \(D.\mathrm{idealOutcome}(m)\). Rewriting the sum using this simplification, the result is \(\sum _{m \in D.\mathrm{measurements}} D.\mathrm{idealOutcome}(m)\), which equals \(0\) by the detector constraint.
In the absence of faults, no detector is violated:
Unfolding the definition of violation, we need \(\operatorname {observedParity}(D, \emptyset ) \neq 1\). By the no-fault observed parity theorem, the observed parity is \(0\), and \(0 \neq 1\).
The observed parity equals the flip parity:
That is, the observed parity reduces to counting (mod 2) how many of the detector’s measurements are faulted, because the ideal outcomes cancel by the detector constraint.
Unfolding the definitions of observed parity, flip parity, and observed outcome, the sum \(\sum _{m \in D.\mathrm{measurements}} (D.\mathrm{idealOutcome}(m) + \mathbf{1}_{\langle m \rangle \in F})\) splits by the distributivity of summation as
By the detector constraint, the first sum is \(0\), so the result equals the flip parity.
A detector is violated if and only if its flip parity equals \(1\):
Unfolding the definition of violation, we rewrite the observed parity as the flip parity using the equality \(\operatorname {observedParity} = \operatorname {flipParity}\).
The flip parity with no faults is \(0\):
Unfolding the definition of flip parity, since no measurement label belongs to the empty fault set, every summand is \(0\), and the result follows by simplification.
The violation of a detector depends only on which of the detector’s measurements appear in the fault set. If two fault sets \(F_1, F_2\) satisfy \(\langle m \rangle \in F_1 \iff \langle m \rangle \in F_2\) for all \(m \in D.\mathrm{measurements}\), then
By the characterization of violation in terms of flip parity, it suffices to show the flip parities are equal. Both directions follow by converting the goal and applying the sum congruence: for each \(m \in D.\mathrm{measurements}\), the indicator \(\mathbf{1}_{\langle m \rangle \in F_1}\) equals \(\mathbf{1}_{\langle m \rangle \in F_2}\) by hypothesis \(h\).
Violation is invariant if we add faults outside the detector. If no measurement of \(D\) appears in \(\mathrm{extra}\), then
We apply the intersection-dependence theorem. For each \(m \in D.\mathrm{measurements}\), we show \(\langle m \rangle \in F \cup \mathrm{extra} \iff \langle m \rangle \in F\). The forward direction: if \(\langle m \rangle \in F \cup \mathrm{extra}\), then either \(\langle m \rangle \in F\) (done) or \(\langle m \rangle \in \mathrm{extra}\), which contradicts the disjointness hypothesis. The backward direction: if \(\langle m \rangle \in F\), then \(\langle m \rangle \in F \cup \mathrm{extra}\) by left inclusion.
A repeated measurement detector is formed from two consecutive measurements \(m_1 \neq m_2\) of the same stabilizer check with common ideal outcome \(o \in \mathbb {Z}/2\mathbb {Z}\). The detector has:
measurements \(= \{ m_1, m_2\} \),
ideal outcome \(f(m) = o\) if \(m = m_1\) or \(m = m_2\), and \(0\) otherwise.
The detector constraint holds because \(o + o = 0\) in \(\mathbb {Z}/2\mathbb {Z}\).
An initialization-measurement detector is formed from an initialization event \(m_{\mathrm{init}}\) and a later measurement \(m_{\mathrm{meas}}\) with \(m_{\mathrm{init}} \neq m_{\mathrm{meas}}\), both having the same ideal outcome \(o\). Per Definition 7, initializations are treated as measurements, so this forms a valid detector with measurements \(\{ m_{\mathrm{init}}, m_{\mathrm{meas}}\} \) and ideal outcome \(o\) on both, satisfying the constraint \(o + o = 0\) in \(\mathbb {Z}/2\mathbb {Z}\).
A repeated measurement detector on \(m_1, m_2\) with \(m_1 \neq m_2\) is violated by faults \(F\) if and only if exactly one of the two measurements is faulted:
We rewrite using the flip parity characterization and unfold the definitions. The flip parity over \(\{ m_1, m_2\} \) equals \(\mathbf{1}_{\langle m_1 \rangle \in F} + \mathbf{1}_{\langle m_2 \rangle \in F}\) in \(\mathbb {Z}/2\mathbb {Z}\). For the forward direction, assuming this sum equals \(1\), we case-split on whether \(\langle m_1 \rangle \in F\) and \(\langle m_2 \rangle \in F\); in each case we simplify to conclude exactly one holds, giving exclusive or. For the backward direction, given the exclusive or, we similarly case-split and simplify to show the sum equals \(1\).
The syndrome of a set of time-faults \(F\) with respect to a family of detectors \((D_i)_{i \in I}\) is the set of detector indices whose detectors are violated:
A detector index \(i\) is in the syndrome if and only if the detector \(D_i\) is violated:
This follows by simplification, unfolding the definitions of syndrome and violation.
The syndrome is empty when there are no faults:
We show the filter is empty by checking that for every \(i\), the observed parity \(\operatorname {observedParity}(D_i, \emptyset ) = 0 \neq 1\), using the no-fault observed parity theorem.
Given two detectors \(D_1, D_2\) with disjoint measurement sets and identical ideal outcomes, their disjoint union is the detector with:
measurements \(= D_1.\mathrm{measurements} \cup D_2.\mathrm{measurements}\),
ideal outcome \(= D_1.\mathrm{idealOutcome}\).
The detector constraint holds because the sum over the union splits into \(\sum _{D_1} + \sum _{D_2}\), both of which are \(0\).
The empty detector has no measurements (\(D.\mathrm{measurements} = \emptyset \)). The detector constraint is trivially satisfied since the empty sum is \(0\).
The empty detector is never violated:
for any fault set \(F\).
Unfolding the definitions of violation, observed parity, and empty detector, the sum over the empty measurement set is \(0\), and \(0 \neq 1\).
A single-measurement detector for measurement \(m\) has measurements \(= \{ m\} \) and ideal outcome constantly \(0\). The constraint \(\sum _{m' \in \{ m\} } 0 = 0\) holds trivially. This detector fires when measurement \(m\) is faulted.
A single-measurement detector on \(m\) is violated if and only if the measurement \(m\) is faulted:
We rewrite using the flip parity characterization and unfold the definitions. The flip parity over the singleton \(\{ m\} \) is \(\mathbf{1}_{\langle m \rangle \in F}\). For the forward direction, if this equals \(1\) and \(\langle m \rangle \notin F\), we obtain a contradiction since the indicator would be \(0\). For the backward direction, if \(\langle m \rangle \in F\) then the indicator is \(1\).
The violation status of a detector under a spacetime fault \(F\) is determined by the time-fault component. Space-faults change the quantum state but the detector constraint is purely about measurement outcome flips:
This holds by reflexivity, as it is the definition of \(\operatorname {isViolated}\).
The weight of a detector \(D\) is the number of measurements it contains:
The repeated measurement detector has weight \(2\):
Unfolding the definitions of detector weight and repeated measurement detector, the measurement set is \(\{ m_1, m_2\} \) with \(m_1 \neq m_2\), so its cardinality is \(2\).
The initialization-measurement detector has weight \(2\):
Unfolding the definitions, the measurement set is \(\{ m_{\mathrm{init}}, m_{\mathrm{meas}}\} \) with \(m_{\mathrm{init}} \neq m_{\mathrm{meas}}\), so its cardinality is \(2\).
The empty detector has weight \(0\):
Unfolding the definitions, the measurement set is \(\emptyset \), which has cardinality \(0\).
The single-measurement detector has weight \(1\):
Unfolding the definitions, the measurement set is \(\{ m\} \), which has cardinality \(1\).