Certification Resources and the Value of Retained Quantum-Reader Records

An information lower boundary for terminal-risk certification

Jeremy RodgersSection 4 of 10Original PDF ↗Publication DOI ↗

An information lower boundary for terminal-risk certification

A direct decision-risk gate has a weaker output contract than identifying an entire continuous detector response. We can nevertheless bound its best possible resource cost in a precise black-box interface, and compare an attainable sequential gate with that boundary. This distinguishes genuine task-relative economy from a claim of universal physical-calibration savings.

The declared validation experiment

A frozen reader uses $B$ fresh source copies and returns a sign at its fixed deadline. A trusted evaluator prepares one of $H$ labels, keeps its identity unavailable to the reader, and scores the terminal error $Y ∈{0, 1}$ . The calibration algorithm may select labels predictably, randomize, stop or reject, but its allocation and acceptance use only these label/error observations. Reader settings, nulls and root seeds remain append-only and unread by the validator; its own allocation coins are independent of the reader seed law. Querying additional trajectory information changes this lower-bound model. Conditional on a fixed device and label $h$ , successive calibration experiments have independent Bernoulli error probability $ph$ . This is an explicit fresh-stock, preparation and stability promise, not a conclusion from reset. The target is $ph ≤ε$ for every label. An acceptance event $A$ must satisfy

sup
p: maxh ph>ε Pr
p (A) ≤α. — Source page 6.

For a good boundary device $ph = p0 < ε$ for every label, require Pr $p0(A) ≥1 −β$ , with 0 $≤β < 1 −α.$ Let $Nh$ be the total number of validation experiments queried at label $h$ , possibly random and sequential. Each such experiment consumes $B$ source preparations; trusted-label setting, controller, references, resets, archives and time are additional resources.

Theorem 3.1 (Sequential terminal-validation lower boundary). Every validator in the declared interface satisfies

Ep0Nh ≥kl(1 −β∥α)
kl(p0∥ε)
for each h, — Source page 6.

and consequently expected source consumption is at least $BH$ times the right-hand side. Here kl is binary relative entropy in natural logarithms. For power one ( $β = 0$ ), the numerator is log $(1/α)$ . The bound allows adaptive label allocation and acceptance-only procedures that need not terminate under a bad device.

Proof. Fix a label $h$ . Compare the good vector to an alternative changing only that label’s error to $q > ε$ . At a finite global query budget $T$ , the likelihood chain rule gives transcript relative entropy

D(P T
p0∥P T
q,h) = Ep0Nh(T) kl(p0∥q), — Source page 6.

where stopped algorithms are extended by a no-query action. Adaptive label choices and ready coins have the same conditional controller law in the two worlds and add no direct relative entropy. The event $AT = {$ accepted by query $T}$ is measurable in this finite transcript. Data processing gives the corresponding binary event-relative-entropy bound. The sequential change-of-measure principle is established in Kaufmann et al. [4], Lemma 1 and Appendix A.1; the finite-prefix form here also covers a bad world in which acceptance never occurs. Since Pr $q,h(AT ) ≤α$ and Pr $p0(AT ) ↑$ Pr $p0(A) ≥1 −β > α$ , monotonicity and continuity of binary relative entropy imply, as $T →∞,$

Ep0Nh kl(p0∥q) ≥kl(1 −β∥α). — Source page 6.

Infinite expected consumption makes the inequality automatic. Otherwise monotone convergence justifies the finite limit. Let $q ↓ε$ to respect the strict bad-device condition. Sum the resulting bounds over all labels and charge $B$ preparations per experiment. No Bonferroni factor is introduced into the lower bound; every one-bad-label alternative must separately be controlled by the same global acceptance rule.

This is a lower bound for an unrestricted terminal-Bernoulli validation model. Its one-bad-label alternatives need not all be realizable within a specified low-dimensional Pauli detector family. Therefore it does not lower-bound every structured detector-calibration method, a protocol using full quantum records, or a method with extra known preparation information. It also does not assert that all physical apparatus resources equal sample count. Those boundaries are necessary to interpret the comparison fairly.

A nearly attained expected-resource boundary

Section 2 constructs the likelihood process for each label,

Eh,n = (p0/ε)Sh,n[(1 −p0)/(1 −ε)]n−Sh,n, — Source page 7.

and, under its serial exhaustive allocation, accepts only after every label has crossed 1 $/α$ at some time. Its intersection–union guarantee uses $α$ per label without dividing by $H$ , while controlling the global false-certificate event at $α$ . Under every good device with $ph ≤p0$ , the same frozen rule accepts almost surely. Put

A = log(1/α),
b = log 1 −p0
1 −ε ,
D = kl(p0∥ε). — Source page 7.

Only a non-error observation increases the log likelihood and its increment is $b$ . Thus its stopped log value never exceeds $A + b$ . Positive drift at least $D$ , applied first at a truncated stopping time, proves $ENh ≤(A + b)/D$ and then almost-sure acceptance. Combined with Theorem 3.1 at $β$ = 0, the matched frontier is

BH A
D ≤Copt ≤BH A + b
D — Source page 7.

for expected source consumption at the good boundary, in this terminal-only interface. The attained upper rule has a stronger uniform good-device promise $ph ≤p0$ ; the lower uses its included boundary vector. The multiplicative gap is at most 1 + $b/A$ . There is no finite worst-case stopping budget in this expectation theorem; a finite-budget alternative is supplied separately in Section 2. For $H = 4, B = 14, ε = .$ 01 and $α = 2−20$ , the declared nominal planning cases are:

Good-risk promise Expected source lower Attainable expected upper

$p0 = .$ 00887 (bisector planning) $11579968.82 . . .$ $11580921.73 . . .$ $p0 = .$ 00875 (adaptive planning) $9422661.68 . . .$ $9423519.35 . . .$

The exact upper figures and outward enclosures are supplied by the calibration verifier. These are mathematical planning bounds, not measured validation costs. The nominal design certificates justify the two chosen risk margins under their calibrated model; a real deployment still needs stable trusted validation. Different $p0$ values are explicitly different good-device planning promises. No uniform superiority across all noisy apparatus families follows merely from this table. The smaller error of the frozen fourteen-copy DAG therefore has a concrete possible benefit despite an unchanged copy upper threshold: it increases the margin to one percent and lowers the cost of terminal-risk certification. Controller/feedback time and actual reference/calibration setup must still be charged. Appendix A supplies the corresponding comparison conditions rather than inferring time from preparations alone.

Exact logarithmic enclosure and verification

The supplied research11/calibration/calibration_gate.py and research11/calibration/ver ify_calibration.py, relative to supplement/readers/, evaluate outward rational enclosures of all

displayed information quantities. To enclose log $x$ , reduce $x$ by powers of two to $u ∈[1,$ 2) and use

m−1
∑︂
z2j+1
u + 1,
0 ≤Rm ≤
2z2m+1
2j + 1 + Rm,
z = u −1
log u = 2
(2m + 1)(1 −z2).
j=0 — Source page 8.

The same series encloses log 2; signed integer multiples are rounded in the proper direction. The exported rational cost intervals are then rounded outward to twelve decimal places for readability. Floating displays never choose a certified inequality. Independent rational range reduction and high-precision checks supplement the analytic proof.