PaperlayerView exact revision on arXiv

Source-linked AI summary

The Sample Complexity of Quantum Entanglement Allocation

Nathan Roll

arXiv:2609.10141v1quant-phcs.LG

TL;DR

The paper asks how many past requests are needed to learn where entanglement should be allocated when different queries favor different qubit groups. It characterizes attainable contrasts, derives learning laws for paths and regions, and studies calibration under preparation noise, finding that sample demand depends on query-created choices rather than memory size alone.

  • Problem

    The paper asks how many past requests are needed to choose entanglement allocations when different requests favor different groups of qubits.

  • Method

    The paper characterizes attainable prediction contrasts for independent commuting X- and Z-type Pauli queries and analyzes allocation learning with preparation calibration.

  • Results

    Sample demand depends on allocation choices: paths add choices as they grow, whereas boundedly connected biclique regions can grow without increasing the sample requirement.

  • Takeaways & Limitations

    Memory size alone does not determine allocation-learning cost; workload structure and preparation costs also affect practical gains.

  • Takeaways & Limitations

    The analysis does not yet characterize general measurements or test the learning laws across independent hardware jobs.

Abstract

from arXiv · show

How many past requests are needed to decide which qubits should share entanglement? We show that the answer depends on the allocation choices created by the queries: a larger memory can require no more data. The memory stores a classical bit and answers requests through a fixed detector that preserves coherence within each measured sector. For independent commuting $X$- and $Z$-type Pauli queries, we characterize the full attainable prediction-contrast region and construct encodings that preserve the bit at every nonzero vertex. With sharp reports, a $d$-qubit path and groups of at most $k$ qubits have minimax excess error after $m$ requests proportional to $k^{-1}\min\{1,\sqrt{d\log(k+1)/m}\}$, uniformly for $2\leq k<d$. Connected biclique regions can grow without increasing sample demand when depth, region count and connections per region stay bounded. Preparation noise introduces a separate calibration requirement. We derive an exact tradeoff with extra fresh detector calls and transfer the learning law to structured transaction co-location. Population-risk experiments test the statistical predictions. We also compare encodings on a native 15-qubit device and learned partitions on public purchase baskets. The full chain wins on the device; frequency grouping outperforms basket search in the largest-capacity retail setting.

1 INTRODUCTION

The paper asks how many past requests are needed to choose where a quantum memory should place limited entanglement. It shows that query frequencies and geometry determine the allocation choices, while the detector is held fixed.

  • 1 INTRODUCTION: A three-qubit example reduces optimal prediction error from 25% to 12.5% by entangling the pair favored by the request distribution, without losing the stored bit.The better pair depends on request frequencies, so the encoder learns how to arrange existing information rather than acquire new information.
  • 1 INTRODUCTION: The detector preserves arbitrary coherence within each measured parity sector, so the analysis varies stored-state entanglement while keeping the measurement fixed.Measuring individual qubits and classically combining outcomes would reveal parity but destroy this coherence.
  • 1 INTRODUCTION: For commuting X- and Z-type Pauli queries, the paper characterizes attainable prediction contrasts and shows that path and biclique geometries impose different learning costs.Paths create more allocation choices as they grow, whereas boundedly connected biclique regions can grow without increasing the sample requirement.
  • 1 INTRODUCTION: Preparation noise requires calibration, and extra detector calls can substitute for entangled preparation; the learning law also transfers to classical transaction co-location.The paper studies request and calibration budgets together.
  • 1 INTRODUCTION: A native 15-qubit commissioning comparison favors the full chain over fixed even, odd, and product allocations.The figure reports fixed allocations and 1,920 shots over all 15 queries and both stored bits.

2 THE MEMORY AND ITS DETECTOR

The memory encodes a noisy classical bit into a quantum state before an independently sampled request arrives. A fixed parity-preserving detector restricts the available report to postprocessed parity information, after which the decoder predicts the bit.

  • 2 THE MEMORY AND ITS DETECTOR: The encoder receives a noisy bit, prepares a d-qubit state, and discards its classical record and environment before an independent request is sampled.The decoder receives only the request and detector report, with no additional copy or later unrestricted measurement.
  • 2 THE MEMORY AND ITS DETECTOR: Entanglement depth constrains the stored codewords through mixtures of states product across allowed qubit partitions, while the same collective detector remains available to every encoder.The depth bound applies to the stored state, not to subsequent detector operations.
  • 2 THE MEMORY AND ITS DETECTOR: Orthogonal codeword supports preserve the noisy bit and attain unrestricted error h, although the optimization also permits information-losing encodings.The paper shows that an optimum can preserve the bit.
  • 2 THE MEMORY AND ITS DETECTOR: The detector’s nonselective channel preserves coherence within each parity sector, whereas refining individual outcomes can preserve parity while violating that channel.This distinction fixes the operational meaning of the detector used in the analysis.
  • 2 THE MEMORY AND ITS DETECTOR: Every compatible instrument is a stochastic postprocessing of the parity outcome, so enlarging the classical report alphabet does not add information beyond parity.The all-input requirement ensures that the detector preserves unknown coherent inputs, not merely the particular stored bit.
  • 2 THE MEMORY AND ITS DETECTOR: With symmetric report noise, contrast is attenuated by known visibility vt, and small violations of channel preservation admit dimension-independent continuity bounds.Under exact preservation, the remaining problem is determining which contrast vectors fit the entanglement budget.

3 WHICH QUERIES CAN AN ENCODING SUPPORT?

For independent commuting CSS Pauli queries, the paper converts entanglement allocation into a graph-independent-set problem. It exactly characterizes feasible contrasts and identifies the workload-optimal encoding.

  • 3 WHICH QUERIES CAN AN ENCODING SUPPORT?: Binary codeword contrasts have the same attainable region as single-state absolute correlations because a tensor product of single-qubit Paulis flips all independent generators while preserving partition classes.This reduction supports the graph formulation for pairs of codewords.
  • 3 WHICH QUERIES CAN AN ENCODING SUPPORT?: Theorem 1 identifies the exact attainable contrast region by representing query conflicts as a bipartite graph for each allowed partition.Edges connect queries whose restrictions anticommute on a shared block.
  • 3 WHICH QUERIES CAN AN ENCODING SUPPORT?: Every nonempty independent-set vertex has an explicit allowed encoding with orthogonal supports, so it preserves the stored bit while attaining that contrast.Mixtures fill the contrast region, although mixtures of bit-preserving pairs need not themselves preserve the bit.
  • 3 WHICH QUERIES CAN AN ENCODING SUPPORT?: The optimal workload risk is 1/2 − (D/2) times the maximum weighted independent-set value over allowed partitions.Because the objective is linear, a best vertex is optimal and its encoding preserves the bit; mixtures may enlarge feasibility without improving this optimum.
  • 3 WHICH QUERIES CAN AN ENCODING SUPPORT?: For graph-state queries, allocation deletes a minimum-weight set of query vertices so every surviving connected component has at most k vertices, then entangles within those components.Deleted vertices represent queries sacrificed according to workload weight.
  • 3 WHICH QUERIES CAN AN ENCODING SUPPORT?: Sharp-report error h is attainable exactly when every positive-weight connected component fits the entanglement budget, while fixed excess error can use constant-size groups independent of path length.Exact recovery through every query requires depth d, and local outcome errors contribute at most three visibility factors per path query.

4 LEARNING WHERE ENTANGLEMENT BELONGS

The learner observes only past request identities and chooses an entanglement allocation before future requests arrive. Sample complexity is governed by the number and structure of competing feasible allocations, not simply by memory size.

  • 4 LEARNING WHERE ENTANGLEMENT BELONGS: The learner maximizes empirical workload weight over inclusion-maximal feasible query sets, yielding finite-class excess O(D sqrt((q + log(1/δ))/m)) under exact empirical optimization.Each maximal feasible set specifies an explicit lossless code.
  • 4 LEARNING WHERE ENTANGLEMENT BELONGS: Paths create more allocation choices as depth grows, whereas complete bipartite regions reduce the oracle to comparing color totals with the k largest weights.This structural difference explains why memory size alone does not determine learning cost.
  • 4 LEARNING WHERE ENTANGLEMENT BELONGS: Connected biclique regions can grow without increasing sample demand when depth, region count, and connections per region remain bounded.The corresponding constants are uniform in d, k, and m, and the second law is uniform in region count, depth, and region sizes.
  • 4 LEARNING WHERE ENTANGLEMENT BELONGS: A chain of connected regions has VC dimension controlled by interior choices and connection vertices, with exact empirical optimization in O(rk^2) operations after sorting.The bound separates region count and size from the number of connection vertices.
  • 4 LEARNING WHERE ENTANGLEMENT BELONGS: The minimax laws describe worst-case workloads; clear gaps can permit faster learning, while regional reductions in sample demand may impose physical costs through larger query supports.General graphs additionally require access to an optimization oracle.

5 LEARNING ALLOCATION UNDER PREPARATION ERRORS

Preparation errors make allocation learning depend on both request history and device calibration. The paper identifies lossless noisy encodings, optimizes them within a circuit family, and establishes a sharp two-resource tradeoff.

  • Noisy allocation model: Preparation errors make neighboring query support interact through edge reliability, so allocation value cannot be reduced to fixed workload weights.Retaining an edge changes neighboring queries’ visibility; the noisy risk therefore depends on the selected structure.
  • Lossless noisy encodings: An odd-sized retained component preserves the encoded bit under ZZ preparation errors, whereas an isolated two-site run can fail.The stabilizer product of an odd component remains in orthogonal sectors, but a two-site ZZ error flips the encoded bit.
  • Optimization: The noisy optimization tracks selected-run length and odd-run closure with O(dk) dynamic-program transitions, while an independent O(dk^2) implementation checks the answers.The optimization covers a lossless circuit family, whereas the ideal theorem covers the full physical class.
  • Two-resource learning: Learning requires m past requests and B calibration trials per parameter, with calibration adding B(2d−1) observations beyond the request log.The comparator is the best allocation in the same noisy circuit family.
  • Two-resource learning: The minimax excess rate is sharp on depth-three paths because four-site choices force the request term while near-indistinguishable three-site channels force the calibration term.The theorem applies to the same circuit family for d ≥ 4 under correctly specified noise.

6 EXPERIMENTS

Experiments test whether request and calibration budgets produce the predicted learning behavior, then evaluate the resulting allocation choices in simulation and on hardware. The results show distinct request and calibration bottlenecks, with the full chain outperforming simpler fixed encodings on the device.

  • 6.1 WHAT CHANGES WHEN THE MEMORY GROWS?: On the hard family, empirical and data-free allocations follow the square-root slope, while on fixed workloads only the empirical learner improves.The experiment varies d from 128 to 1,024 and k from 2 to 32, scaling excess by k and requests by d log(k + 1).
  • 6.1 WHAT CHANGES WHEN THE MEMORY GROWS?: Connected-region curves remain similar when region size changes at fixed m/(rk), as predicted by Theorem 2.Selective connection repair plus local improvement is within 0.145 error points of exact optimization at the primary condition.
  • 6.2 REQUESTS AND CALIBRATION LIMIT DIFFERENT DECISIONS: At B = 4, increasing requests from 16 to 4,032 reduces excess from 6.48 to 1.33 points, but exact workload knowledge still leaves 1.30 points.At B = 256, the same request increase reduces excess from 5.26 to 0.085 points; with m = 16, 1,024 calibration trials still leave 5.14 points.
  • 6.2 REQUESTS AND CALIBRATION LIMIT DIFFERENT DECISIONS: In the 63-qubit heterogeneous-noise comparison, Joint reduces error from 9.79% for Task only to 8.58%, a 1.21-point gain.This uses 1,008 requests and 256 calibration trials per parameter; the paired 95% interval is [0.95, 1.50].
  • 6.3 TESTING THE ENCODINGS ON A QUANTUM PROCESSOR: Full chain reaches 5.57% uniform-request error, versus 14.27% for fixed depth-three and 25.05% or 30.42% for product encodings.The comparison uses 7,680 commissioning shots; learning and calibration jobs were not measured successfully, and a coherence witness still needs validation.

7 CONSEQUENCES OF THE ALLOCATION LAW

The allocation law extends beyond request learning to detector-call tradeoffs and classical transaction placement. These consequences show when extra calls or entanglement are useful, while practical gains remain dependent on costs and workload structure.

  • Detector-call tradeoffs: If fresh calls are dominated by E_p, reproducing an E_q experiment requires at least Nq/p expected calls, and a stopped policy attains equality.This equality concerns expected call budgets for equivalent experiments; fixed-horizon power comparisons can differ.
  • Detector-call tradeoffs: For the three-query menu, product and Bell coverages are .6 and .8, but a product-initialized adaptive policy matches Bell at 1.2 expected calls.The fresh-call ratio is 4/3; repeated unsupported queries can diagnose noise without recovering the stored sign.
  • Cost boundary: Entanglement lowers lifetime cost only when its extra startup cost is smaller than accumulated preparation and detection savings.Prior GHZ diagnosis results cannot provide a matched-cost hardware comparison with the adaptive policy because their calibrations differ.
  • Classical transaction co-location: Theorem 2 transfers to structured row/column transactions, yielding minimax locality regret Θ(min{1,√(rk/m)}) while larger tables need not require more requests.The result applies under the stated capacity and table-size conditions, and 15,360 placement checks reproduce the predicted masks.
  • Classical transaction co-location: At capacity 512 and 512 training baskets, frequency grouping reaches 11.37% locality, while basket-based search reaches 10.90% and fixed key-range placement 10.35%.The full retail study evaluates 3,992 held-out invoices without projection; the simpler frequency control wins this comparison.

8 RELATED WORK AND IMPLICATIONS

The paper situates its allocation framework among stabilizer-polytope, graph, information, data-hiding, and channel-discrimination work. It concludes that learning cost is governed by allocation-choice structure rather than memory size alone, while broader measurement classes and independent hardware validation remain open.

  • Related work: The constructive region for independent CSS families builds on prior stabilizer-polytope, cut-inequality, and graph-based correlation results.The cited approaches include weighted independent-set optimization and linear graph-generator bounds.
  • Related work: The framework extends predictor-dependent information ideas to logarithmic loss with physically restricted observations and unrestricted classical decoders.Related work also separates storage from access through data hiding and studies entanglement advantages in channel discrimination.
  • Implications: Memory size alone does not determine allocation-learning cost: paths create more choices as they grow, whereas boundedly connected biclique regions need not.The same distinction transfers to classical co-location.
  • Implications: Practical gains depend on preparation costs and workload, which can favor simpler allocations, and the paper has not characterized general measurements or tested independent hardware jobs.These are stated scope boundaries for the conclusions.

REPRODUCIBILITY STATEMENT

The manuscript provides code, data, seeds, encoders, resource counts, replay code, and an installable package for reproducing key analyses.

  • The ancillary materials include raw observations, seeds, encoders, resource counts, replay code, and a package reproducing the path optimizer and online sampler.The README identifies entry points for hardware, classical, and interface evidence and distinguishes recorded observations from unacquired experiments.

AI USE STATEMENT

Generative AI assisted across the paper’s theoretical, experimental, and writing workflows, while the author retained responsibility for the final manuscript and artifacts.

  • Generative AI assisted with modeling, proofs, writing, hypothesis refinement, experiments, implementation, data processing, interpretation, figures, literature review, and editing.
  • The author takes responsibility for the final manuscript and artifacts.

ETHICS STATEMENT

The paper frames its results within explicit access, detector, calibration, and data-use constraints. It reports aggregate purchase locality without identifying customers and does not claim a general quantum learning speedup.

  • ETHICS STATEMENT: The transaction study uses stock codes, invoice identifiers, times, quantities, and prices, but no customer identifiers or person-linked purchases.It reports aggregate locality, while hardware results compare fixed implementations.
  • ETHICS STATEMENT: The hardware interpretation is limited to the stated measurement and control assumptions and does not establish a general quantum learning speedup.
  • ETHICS STATEMENT: The detector must preserve coherence within each parity sector for all input states, not merely preserve parity values on the stored codewords.The compatible reports are stochastic postprocessings of sharp parity, whereas refinements inside eigenspaces may destroy coherence.
  • ETHICS STATEMENT: A device comparison requires an informative predictor below the comparator bound because channel calibration alone does not guarantee a useful report.The one-outcome instrument can have zero channel error while retaining risk 1/2 for every encoder.
  • ETHICS STATEMENT: For the three-vertex gadget family, the depth-two oracle beats the relaxed separable class whenever δ < (1 + θ)/8, with the sufficient bound δ < 1/64 independent of gadget count.

C.3 ALLOCATION GEOMETRY CHANGES THE LEARNING RATE

Allocation geometry determines statistical difficulty: different query families with the same memory size can have different sample rates. Bounded-connection biclique regions can scale while retaining rates controlled by region count and depth rather than region size.

  • C.3 ALLOCATION GEOMETRY CHANGES THE LEARNING RATE: The maximal complete-bipartite allocation class has VC dimension k, while the full downward-closed feasible class has dimension max(a, b).Counting dominated feasible indicators can therefore overstate statistical difficulty.
  • C.3 ALLOCATION GEOMETRY CHANGES THE LEARNING RATE: At depth two, complete bipartite graphs have rate Θ(D min{1, m^-1/2}), whereas connected paths have Θ(D min{1, sqrt(q/m)}).This comparison concerns statistical allocation geometry, not universal learning difficulty or equal detector cost.
  • C.3 ALLOCATION GEOMETRY CHANGES THE LEARNING RATE: For connected bipartite families with fixed maximum degree, the depth-two rate is Θ_Δ(D min{1, sqrt(d/m)}), with bounded query support.The constants are not claimed sharp in Δ, and dimension-free rates require leaving the bounded-degree regime.
  • C.6 CONNECTED REGIONS AND THE EFFECTIVE ALLOCATION DIMENSION: With r biclique regions, at most p connection vertices per region, and k nonconnection vertices per color, the rate scales as Θ(D min{1, sqrt(rk/m)}) independently of region sizes.The matching lower bound permits physical groups crossing region boundaries, while the upper bound uses maximal feasible sets.
  • C.6 CONNECTED REGIONS AND THE EFFECTIVE ALLOCATION DIMENSION: Increasing region size need not increase statistical cost, but increasing the number of regions does; increasing connections can restore size-dependent rates.A chain has rate Θ(D min{1, sqrt(rk/m)}), whereas growing matching connections yields an Ω(D min{1, sqrt(s/m)}) lower bound.

D.3 INFORMATION EQUALITY AFTER PREPARATION ERRORS

Preparation errors preserve the classical experiment when codewords remain in opposite stabilizer sectors, but contrast alone does not determine logarithmic loss for every code. The resulting learning guarantees extend to noisy preparation, with separate request and calibration limits and efficient information-preserving encodings.

  • Information preservation after errors: Opposite eigenspaces of a retained odd-component operator preserve the exact classical experiment under correlated or coherent ZZ preparation errors.The support argument allows correlated mixtures or coherent operations, while the contrast product assumes independent errors.
  • Information preservation after errors: At p = .2, a noisy codeword pair has trace overlap .32, showing that preparation noise can destroy information preservation.The overlap counterexample is retained alongside the odd-code control.
  • Path allocation: For γ ≤2/3, singleton density 1/2 is optimal, while for γ > 2/3 the best capped-run density is max{1/2, [2γ + (k −2)γ2]/(k + 1)}.The result includes separator costs and vanishing boundary terms for long paths.
  • Logarithmic loss: The optimal logarithmic-loss value is determined by βt = 1 −H2((1 −Dvt)/2), with information-preserving projector codes attaining it.The theorem optimizes over codewords and constructs an optimal code preserving the classical bit.
  • Logarithmic loss: Contrast does not determine every code’s log loss: two codes with binary prediction error 1/4 have log losses 0.811278 and 0.688722 bits.Output bias matters for codewise information, even when both encodings preserve the original bit.
  • Learning guarantees: The learning laws extend to log-loss excess with J0 replacing D, while uniform constants and zero allocation excess hold when k ≥d.The upper bound uses information-preserving codes; the lower bound permits arbitrary mixed physical encodings.

K.3 RETAINED INFORMATION AND CORRELATED ERRORS

Correlated retained-edge errors preserve a shared information witness, whereas independent single-qubit phase errors can destroy losslessness and alter access comparisons. The section also documents interface constraints, calibration bounds, and controls for interpreting device results.

  • Correlated retained-edge errors: Arbitrary joint distributions of retained-edge ZZ errors preserve the odd-component witness, although independence is needed for the factorized prediction formula.A common Bernoulli shock provides an explicit correlated-error control while global decoding remains exact.
  • Independent single-qubit errors: At phase-error probability 0.03, global errors were 0.874% for Joint, 0.026% for Product, and 0.295% for Full chain, while restricted errors were 15.14%, 19.52%, and 13.32%.None of these representations remained lossless at this noise level, so their access comparison was no longer information matched.
  • Scope of physical evidence: Device observations compare fixed implementations in a different measurement family and do not establish learned allocation, an all-input detector guarantee, or exact information preservation in noisy states.The physical experiment tests response prediction rather than validating the four-qubit service’s coherence-preserving instrument.
  • Interface controls: The benchmark charges one detector call per probe because request-aware preparation and a second nondestructive service call are separate controls.A request-aware product preparation and a product probe followed by a second ZZI call can both recover the bit in the ideal commuting case.
  • Calibration: Direct response calibration estimates signed report contrasts that include preparation, gate, assignment, and report errors, while balanced-label risk cancels output bias.The finite-catalog guarantee uses independent calibration shots for each encoder, request, and label, with total cost Q = 2AqB.
  • Calibration: The finite-catalog calibration bound is conservative: A = 4, q = 3, m = 128, and B = 2048 yield an excess-risk bound of 0.208 from 49,152 healthy calibration shots.The bound guarantees consistency but does not certify the much smaller empirical improvement.

M.3 CIRCUIT EXPERIMENTS AND ESCAPE CONTROLS

Circuit experiments test allocation-based diagnosis and response calibration under declared noise models, while controls distinguish empirical prediction gains from stronger coherence-preservation claims. The results support calibration and an exact diagnostic transfer law but retain important scope boundaries.

  • Fault diagnosis: At an empirical 5% false-alarm rate, the left pair reached 70.70% power versus 54.30% for the X-product, a 16.4-point paired-bootstrap gain.The descriptive ROC threshold differs from the independently calibrated operating threshold; all episodes were offline replays.
  • Fault diagnosis: The weaker fault separated probes, whereas the extreme fault was nearly saturated: both pair probes detected all 256 faults and the X-product detected 255.The saturated condition supplied little evidence of a large advantage.
  • Response model: The direct-minus-factorized mean log score was −1.2 × 10−5 nats per shot, with conditional interval [−4.4, 2.0] × 10−5.This experiment did not resolve a predictive difference between the models.
  • Physical calibration: Mean absolute probability error fell from 0.00942 for ideal circuits to 0.00365 for measured responses, while held-out Brier score improved by 0.000114.All 20 overlapping calibration/test splits showed positive Brier gains, supporting response calibration without identifying isolated noise parameters.
  • Scope of evidence: The physical experiment tests response prediction, not the four-qubit service’s coherence-preserving instrument, and archived split blocks are not demonstrated later-day tests.The archived IBM calibration split was selected after original aggregate results were known, with unavailable per-circuit timestamps.
  • Diagnostic transfer: The optimal N-call diagnostic experiment is Blackwell-equivalent to repeated calls of the single-call experiment E⊗N.A fixed signed-projector probe supported on a maximizing compatible query set attains the relevant bound.
  • Diagnostic transfer: After m training requests, one-call diagnostic-power loss equals D RF(m), transferring the sharp-report prediction regret exactly when the detector hypotheses are known.Estimating a healthy baseline or operating threshold requires separate calibration data.

N.2 THE COMPLETE SEPARABLE BENCHMARK AND ITS RESOURCE COST

The section establishes exact separable benchmarks for fixed-horizon tests, then compares entangled and product probes under ideal and perturbed conditions. It shows that calibration, preparation noise, and query access materially affect both statistical performance and resource cost.

  • Complete separable benchmark: At (0.2, 0.6, 0.2), depth two raises the optimum from p1 = 0.6 to p2 = 0.8, whereas uniform workloads give p1 = p2.The comparison uses full-class optima, including arbitrary mixed states and adaptation.
  • Complete separable benchmark: At 32 calls, optimal power is 82.39% for separable policies and 88.45% at depth two; achieving 80% power requires 30 versus 22 calls.These results use e0 = 0.02, e1 = 0.164, and α = 0.05.
  • Resource cost and preparation noise: A 1% Bell-probe parity flip reduces 32-call power from 88.45% to 84.84%, while 1.6% lowers it to 82.00%, below the ideal separable value of 82.39%.Preparation error can erase the detector-call advantage before calibration costs are amortized.
  • Resource cost and preparation noise: Early query access removes the preparation problem, while the factory contract forbids rebuilding a probe after the current request is selected.The fixed-horizon cost model excludes calibration and dynamic-control latency.
  • Optimized probes under errors: At s = 0.08, the Bell probe exceeds the full separable upper bound by more than 2.19 percentage points; at s = 0.20, an optimized product outperforms both fixed Bell probes.The separable comparison is supplied by an upper bound, not optimizer convergence.
  • Physical acquisition: Physical validation found 51.74% power for GHZ versus 43.23% for Z product, but the selected probes did not demonstrate a learning gain over fixed GHZ.Calibration estimates selected the probes for held-out comparison, and the witnesses test the stated input family rather than a complete instrument.

O.5 MEASURED OUTCOMES AND RESOURCE ACCOUNTING

The measurements compare entanglement, adaptive querying, coherence, and classical allocation costs under specified operating assumptions. Results show strong device-level GHZ gains, measurable coherence, exact fresh-call tradeoffs, and a retail advantage for frequency grouping, while calibration and job variation constrain interpretation.

  • Hardware detection comparison: 16.34 percentage points higher detection power for GHZ than Selected product in cycle 2, with conservative 95% interval [10.77,21.83] and one-sided Fisher p = 5.79 × 10−21.Cycle 1 also shows a 9.11-point gain, with both cycles passing the prespecified superiority rule.
  • Coherence through the dynamic service: 56 of 56 target-request coherence cells have positive simultaneous lower bounds, with the smallest equal to 0.302.Expected-parity agreement ranges from 78.80% to 95.63% across the cells.
  • Fresh-request conversion: 30 fresh product calls versus 22 Bell calls achieve 80% power at 5% false alarms for (e0, e1) = (.02, .164), giving a finite-horizon ratio of 30/22.Equal expected informative counts alone do not imply equivalent fixed-horizon performance.
  • Retained-probe policy: 14 probes provide 80.2647% power at 5% false alarms, using 19.6 calls on average and at most 28, but exceed 22 calls with probability 5.8319%.Thus the policy is not a hard-budget counterexample to a 22-call comparison.
  • Resource accounting: GHZ is cheaper than the repeated-query policy exactly when per-call overhead x exceeds 22.5, before extra startup cost.The comparison concerns two attaining policies rather than an optimized adaptive frontier.

Q.1 STRUCTURED EXPERIMENTS AND A PUBLIC NEGATIVE CONTROL

The experiments evaluate learned allocation and partitioning methods on structured synthetic workloads, a quantum processor, and public purchase baskets. Results favor request-informed methods in some settings, but expose capacity-dependent reversals, calibration limits, and hardware-validation failures.

  • Q.2 OBSERVED PURCHASE BASKETS: At 512 requests, learned grouping reaches 67.25% versus 63.57% for frequency grouping in the 12-item schema, and 52.21% versus 48.19% in the 24-item schema.Fixed key-range placement reaches 62.08% and 47.70%, respectively; the learned fits use exact MILP optimization on the reduced schemas.
  • Q.3 COMPLETE BASKETS ON THE FULL CATALOGUE: The full-catalogue study uses a heuristic candidate search rather than exact empirical maximization, and the data are observed purchases rather than a captured SQL trace.The heuristic evaluates five training-only candidates and scores complete baskets under shard-capacity constraints.
  • Q.3 COMPLETE BASKETS ON THE FULL CATALOGUE: At capacity 512, frequency grouping beats basket search by 0.470 points at 512 requests and 0.528 points at 2,048 requests, while basket search wins at capacity 128.The comparison reverses with capacity, and selecting by training locality does not always select the best future partition.
  • R.1 NATIVE-PATH DEVICE TEST: In 7,680 commissioning shots, Full chain has 5.57% uniform-request error versus 14.27% for fixed depth-three and 25.05% or 30.42% for product encodings.The fixed-encoding comparison favors the full chain, but exploratory learning jobs timed out without usable observations.
  • R.2 A LIMITED COHERENCE CHECK EXPOSES A SIGN REVERSAL: Query seven reverses the expected coherence sign, query six is inconclusive, and repeated extraction intervals implicate the added extraction or measurement interval rather than preparation alone.The reversal prevents claiming that the coherence check validates preservation across the whole path and does not identify a specific faulty gate.
Loading 2609.10141v1…