The Sample Complexity of Quantum Entanglement Allocation

• A three-qubit example shows that entangling the first pair lowers optimal prediction error from 25% to 12.5% while preserving the stored bit.For queries ZZI, XXX, and IZZ with probabilities 1/4, 1/2, and 1/4, request frequencies determine which pair is preferable.Source: §1 · p. 1, Figure 1 · p. 2
• For independent commuting X- and Z-type Pauli queries, Theorem 1 characterizes attainable prediction contrasts and reduces encoder learning to choosing supported query sets.This covers arbitrary allowed partition mixtures, including mixed states.Source: §1 · p. 1
• Theorem 2 shows that paths add allocation choices as they grow, whereas boundedly connected biclique regions can grow without increasing sample requirements.The contrast depends on the choices created by query geometry.Source: §1 · p. 1
• Imperfect preparation creates a separate calibration budget, and extra detector calls can substitute for entangled preparation; the learning law also transfers to transaction co-location.A native 15-qubit commissioning favors the full-chain allocation over fixed even/odd product allocations.Source: §1 · p. 2, Figure 1 · p. 2

2 THE MEMORY AND ITS DETECTOR

• The encoder stores G = C ⊕ Z in a d-qubit state, and the decoder predicts C using only the request and detector report.The detector has no additional copy, correlated input, or later unrestricted measurement.Source: §2 · p. 2
• Allowing partitions with blocks of size at most k constrains the stored states, while the same collective detector remains available to every encoder.Codewords may use different partitions and mixtures within the convex hull of allowed product states.Source: §2 · p. 2
• Orthogonal codeword supports recover G globally with error h and information 1 − H2(h) bits about C, and an optimum can preserve the bit.The optimization also permits information-losing codes, but optimality need not sacrifice bit preservation.Source: §2 · p. 2

2.1 WHICH REPORTS PRESERVE A PARITY SECTOR?

• A detector preserving every parity sector must implement the Lüders channel, which retains all coherence within each sector.Refining individual outcomes can preserve the parity eigenvalue while violating this channel condition.Source: §2.1 · p. 2
• Every compatible report is only a stochastic postprocessing of the parity, regardless of alphabet size, because the shared detector must preserve unknown coherent inputs.The all-input requirement applies even when a particular client stores a classical bit; approximate preservation is treated separately by continuity bounds and a diamond-distance threshold.Source: §2.1 · pp. 2–3

3 WHICH QUERIES CAN AN ENCODING SUPPORT?

• For independent commuting CSS generators, the attainable contrast region is exactly characterized, and every nonempty independent-set vertex has an explicit encoding with orthogonal supports.Binary contrasts reduce to single-state absolute correlations; mixtures fill the region, although mixing preserving pairs need not preserve the bit.Source: §3 · p. 3
• The optimal risk is 1/2 −(D/2) maxΠ∈F αa(HΠ), so a best independent-set vertex is optimal and its encoding preserves the bit.Mixtures can enlarge feasibility without improving this linear workload optimum; the characterization relies on CSS structure and generator independence.Source: §3 · p. 3

3.1 WHICH QUBITS SHOULD SHARE ENTANGLEMENT?

• Graph-state allocation reduces to deleting a minimum-weight set of query vertices so every surviving connected component has at most k vertices, then entangling within those components.Hadamards convert one bipartite color to CSS form, and the deletion rule follows from weighted bipartite vertex cover.Source: §3.1 · pp. 3–4
• Exact sharp-report recovery requires every positive-weight query component to fit the budget, while constant-size groups yield fixed excess error independent of d.Exact recovery of G through every path query requires depth d; each query touches at most three qubits, limiting independent local-error visibility factors to three.Source: §3.1 · p. 4

• The learner observes only m past query identities, chooses an encoding before future requests, and minimizes worst-case excess risk relative to the best depth-k allocation.No prediction labels are provided, and the supremum ranges over all query distributions.Source: §4 · p. 4
• Empirical maximization over inclusion-maximal feasible query sets gives a finite-class excess bound O((q + log(1/δ))/m) with probability 1 − δ.The bound assumes exact empirical optimization, which may be combinatorial; counting undominated allocation choices sharpens the analysis.Source: §4 · p. 4
• On a path with d vertices and depth k, minimax excess error after m requests is proportional to k^-1 min{1, sqrt{d log(k+1)/m}}.The constants are uniform in d, k, and m, for 2 ≤ k < d; if d ≤ k, allocation excess is zero.Source: §4 · p. 4
• For r connected biclique regions with at most p connection vertices per region, the sample law is uniform in r, k, and region sizes.A single unconnected region recovers the corresponding complete-bipartite rate, so larger regions need not increase sample demand when structural bounds remain fixed.Source: §4 · p. 4
• The lower bounds arise from independent choices among separated path segments or regional capacity constraints, while information-preserving codes attain the rates.Paths use random periodic allocation at small budgets and empirical maximization at larger budgets; regional empirical maximization attains the regional rate.Source: §4 · pp. 4–5
• The laws are worst-case: workloads with a clear allocation gap can be learned faster, while regional savings trade off against larger physical biclique queries.Paths and chains of regions permit efficient exact empirical optimization, whereas the general graph theorem assumes an optimization oracle.Source: §4 · p. 5

Error-aware allocation

• Preparation errors alter neighboring-query reliability, so retaining an edge changes workload risk rather than contributing only a fixed cost.The risk depends on reporting visibility and edge-dependent error factors, with neighboring-query effects induced by retained edges.Source: §5 · p. 5
• An odd-sized retained component preserves the encoded bit despite retained-edge ZZ errors, whereas an isolated two-site run fails.The stabilizer product has eigenvalue (−1)^G, commutes with retained-edge ZZ errors, and keeps noisy codewords in orthogonal sectors.Source: §5 · p. 5

Optimization

• A dynamic program optimizes the noisy lossless circuit family using O(dk) transitions, with an independent O(dk^2) interval implementation for verification.The noisy optimization is restricted to the circuit family, unlike the ideal theorem’s optimization over the full physical class.Source: §5 · p. 5

Statistical requirements

• Calibration supplies separate estimates of edge reliabilities and native query visibilities at a cost of B(2d−1) additional observations.The comparator is the best allocation within the same noisy circuit family; direct binary-response calibration and channel-mismatch bounds are treated separately.Source: §5 · p. 5
• The sharp two-resource minimax excess rate on depth-three paths depends jointly on m requests and B calibration trials per parameter.The request term is forced by four-site allocation choices, while the calibration term is forced by nearly indistinguishable three-site channels with different optimal allocations.Source: §5 · p. 5

6.1 WHAT CHANGES WHEN THE MEMORY GROWS?

• Scaling excess by k and requests by d log(k + 1) produces the predicted square-root slope, while empirical learning improves on fixed workloads and periodic allocation remains flat.Connected-region curves remain similar when region size changes at fixed m/(rk), and selective repair plus local improvement is within 0.145 error points of exact optimization.Source: §6.1 · p. 6, Figure 2 · p. 6

6.2 REQUESTS AND CALIBRATION LIMIT DIFFERENT DECISIONS

• 1.21 points separate Joint from Task only at 1,008 requests and 256 calibration trials, reducing error from 9.79% to 8.58%.At B = 4, increasing requests from 16 to 4,032 leaves a 1.30-point calibration floor, whereas at m = 16, even 1,024 calibration trials leave 5.14 points.Source: §6.2 · p. 7

6.3 TESTING THE ENCODINGS ON A QUANTUM PROCESSOR

• 15 graph queries were implemented on a 15-data-qubit ibm_marrakesh path using separate preparation and extraction compilation with two parallel CZ layers.Seven query centers had degree two, and parity-extraction circuits borrowed and ideally restored data qubits, preserving the local conflicts defining the learning problem.Source: §6.3 · p. 7
• 5.57% error makes Full chain the best fixed encoding, versus 14.27% for depth-three and 25.05% or 30.42% for product colors.This comparison used 7,680 commissioning shots; learning and calibration on the processor were not measured because preliminary jobs timed out, and a coherence check found an incorrect witness sign.Source: §6.3 · p. 7

Changing the detector interface

• At one call, diagnosis power regret equals 2(e1 −e0)/D times prediction regret, so detector diagnosis inherits the same request-learning laws.The optimal N-call experiment is E_p*^⊗N, where p* is the maximum supported mass.Source: §7 · p. 8
• At equal operating performance, entanglement reduces lifetime cost only when its extra startup cost is smaller than accumulated preparation and detection savings.Prior IBM diagnosis runs reported GHZ power gains of 9.11 and 16.34 points, but differing calibrations prevent a matched-cost hardware comparison with the adaptive policy.Source: §7 · p. 8

Classical transaction co-location

• Θ(min{1, rk/m}) minimax locality regret holds for row/column transactions on r disjoint s × s tables under the stated capacity and size conditions.The transfer uses capacity C = (s −1)k + 1, s ≥ max{2, k, ⌊(k −1)2/4⌋+ 2}, and allows rs shards.Source: §7 · p. 9
• All 15,360 checks of actual record placements reproduce the predicted masks, showing that larger tables need not require more requests despite growing transaction width and storage requirements.Figure 5 reports exact empirical co-location on structured tables with requests scaled by rk.Source: §7 · p. 9, Figure 5 · p. 9
• 10.90% locality for basket-based search, 11.37% for same-budget frequency grouping, and 10.35% for fixed key-range placement at capacity 512 and 512 training baskets; the simpler frequency control wins.The full study evaluates all 3,992 held-out invoices without projection, and the comparison appears in Figure 5 alongside structured-table results.Source: §7 · p. 9, Figure 5 · p. 9

Related work

• The constructive region and sharp learning laws extend prior stabilizer-polytope, graph-generator, and connectivity results for selected entangled-state families (Jafarizadeh et al., 2008; Zander & Becker, 2025; Knips et al., 2016; Li et al., 2026).The analysis also uses stable-set integrality and component-order connectivity (Drange et al., 2016).Source: §8 · p. 9
• The work extends predictor-dependent information to physically restricted observations with unrestricted classical decoders and identifies the optimal experiment and allocation-learning cost for a fixed query menu (Xu et al., 2020; Dubois et al., 2020).It connects this perspective to data hiding and locally indistinguishable product ensembles and to entanglement-assisted channel discrimination (Piani & Watrous, 2009).Source: §8 · p. 9

Implications and limitations

• Memory size alone does not determine allocation-learning cost: paths create more choices as they grow, whereas biclique regions need not, with the same distinction appearing in classical co-location.Practical gains also depend on preparation costs and workload, which may favor simpler choices.Source: §8 · p. 9
• General measurements remain uncharacterized, and the learning laws have not yet been tested across independent hardware jobs.These limitations define open directions for extending the framework and validating its statistical predictions.Source: §8 · p. 9

REPRODUCIBILITY STATEMENT

• Ancillary materials include raw observations, seeds, encoders, resource counts, replay code, full proofs, and an installable package for the path optimizer and online sampler.The README identifies entry points for hardware, classical, and interface evidence while distinguishing recorded observations from unacquired experiments.Source: REPRODUCIBILITY STATEMENT · p. 10

• Generative AI assisted with theoretical models, proofs, writing, hypothesis refinement, experiments, implementation, synthetic data, processing, interpretation, literature review, and figures; the author remains responsible for the final manuscript and artifacts.Source: AI USE STATEMENT · p. 10

A.1 EXACT COMPATIBILITY

• Exact detector compatibility requires reports to be classical postprocessings of sharp parity while preserving coherence within parity sectors; approximate channel calibration alone does not ensure informative prediction.The one-outcome instrument has δ = 0 but risk 1/2, whereas δ < 1/64 suffices for the three-vertex gadget comparison independently of gadget count.Source: §A.1 · p. 14, §A.2 · p. 15

A.4 OPERATIONAL RESOURCE ACCOUNT

• The model is explicitly an access-service contract: preparation may select encodings from past requests, but prediction sees only the query and fixed detector report, so the results do not establish total-cost quantum advantage.Retained classical copies, correlated inputs, or different detectors can change the optimum, and the hardware interpretation remains limited to stated measurement and control assumptions.Source: ETHICS STATEMENT · p. 10, §A.4 · p. 15

B PROOF OF THE EXACT CONTRAST REGION

• The exact contrast region is a stable-set polytope, and constructive encodings attain every feasible contrast while preserving the encoded bit at an optimal vertex.For allowed partition mixtures, the six-qubit example yields conv(CA ∪ CB) = {c ∈ [0, 1]3 : c1 + c2 + c3 ≤ 2}, including a lossless mixed point with errors (1/4, 1/4, 0).Source: §B.1 · p. 16, §B.2 · p. 17

B.3 RELATION TO SINGLE-STATE WITNESSES

• For independent commuting Pauli queries, the binary-contrast region equals the single-state absolute-correlation region, rather than requiring a separate two-codeword construction.A Pauli conjugation maps any allowed state to a pair of codewords with the same absolute correlations.Source: §B.3 · p. 18

B.4 A SUFFICIENT CONDITION BEYOND CSS

• A sufficient extension beyond CSS holds when each conflict graph is perfect and every nontrivial maximal clique has two blocks supporting pairwise anticommuting restrictions.The condition is sufficient rather than a characterization; the independent family {XXI, YYZ, ZZI} satisfies it, while dependent or unsupported-clique examples show failures.Source: §B.4 · pp. 18–19

B.5 GRAPH COMPONENTS AND DEPTH

• For graph-state generators, depth-k allocation reduces to deleting vertices so remaining components fit within blocks; on a path, uniform-weight optimum deletes ⌊d/(k + 1)⌋ vertices.Deleting every (k + 1)th vertex attains the bound, and sharp zero-excess risk requires every active component to fit within a block.Source: §B.5 · p. 19

C.1 FINITE-CLASS UPPER BOUND

• The finite-class upper bound applies to all allowed mixed physical encoders, with expected excess risk scaling as D√(log(2N)/(2m)) for N feasible indicators.The learner uses only stipulated query samples and known interface parameters; exact risk completeness supplies the physical comparator.Source: §C.1 · p. 19

C.2 DEPTH-TWO LOWER BOUND

• Depth-two gadgets give separable risk h + D/4 versus oracle risk h + D(1 − θ)/8, while learning the endpoint choices incurs an Assouad lower bound.Each gadget encodes an unknown endpoint choice; a wrong choice costs excess Dθ/(4r), reducing the problem to r binary decisions.Source: §C.2 · p. 20
• The testing bound permits correlated estimator choices and arbitrary randomized physical learners, and revealing the common θ does not invalidate it.Source: §C.2 · p. 20
• At most 3r full allocations with Eq. (10) prove the local-gadget rate, while no observations yield a separate constant binary-testing lower bound.Source: §C.2 · p. 20

C.3 ALLOCATION GEOMETRY CHANGES THE LEARNING RATE

• The maximal feasible allocation class has VC dimension exactly k, whereas the full downward-closed class has dimension max(a, b), so counting dominated allocations can overstate statistical difficulty arbitrarily.For complete bipartite measurement graphs, only the frontier {A, B} plus mixed k-sets is statistically relevant; maximal sets suffice under nonnegative workloads.Source: §C.3 · pp. 20–21

C.4 BOUNDED QUERY LOCALITY FORCES DIMENSIONAL LEARNING COST

• At depth two, every connected bipartite family with fixed maximum degree ∆ has rate Θ∆(D min{1, d/m}), and bounded query support yields the same dimensional learning cost.A dimension-free rate along a growing connected family requires leaving the bounded-degree regime; growing degree alone is not sufficient.Source: §C.4 · p. 22

C.5 THE LOCALITY LAW AT EVERY FIXED DEPTH

• At every fixed depth k, connected bipartite graphs obey matching upper and lower rates in d and m, with lower coefficients uniform for fixed k and ∆.The result uses connected k + 1-vertex gadgets and remains valid for strictly positive workloads without a common probability floor; the simultaneous dependence on growing k or ∆ is not sharp.Source: §C.5 · p. 24

C.6 CONNECTED REGIONS AND THE EFFECTIVE ALLOCATION DIMENSION

• Connected complete-bipartite regions have rate Θ(D min{1, sqrt(rk/m)}) with constants independent of region sizes, so increasing region size need not increase sample demand while increasing region count does.The bound depends on at most p connection vertices per region; if connection count grows, fixed r and k no longer guarantee size-independent rates.Source: §C.6 · pp. 24–26
• A two-region positive-weight example achieves optimal errors 3/38 at depth two and 3/19 at depth one, while preserving the encoded bit with projector codes.The depth-two optimum retains weight 32 of 38, and exhaustive enumeration of all 212 query subsets reproduces both error values.Source: §C.6 · p. 25
• Connected-region optimization is exact with O(rk^2) scalar work after O(d log d) sorting, using boundary states that track the selected component size across regions.Prefix sums reduce mixed-color choices to O(k) resulting options, and induction establishes the recurrence; floating-point comparisons do not certify nearly equal objective orderings.Source: §C.6 · p. 26

D.1 EXACT CIRCUIT-FAMILY OBJECTIVE

• The circuit-family objective models graph-state contrasts under independent retained-edge ZZ errors and classical reporting noise, with expectation factors constrained by 0 ≤ γ_e, v_i ≤ 1.Reporting noise is postprocessing of parity reports rather than destructive local measurement; for path queries, each v_i has at most three site-rate factors.Source: §D.1 · p. 26

D.2 DYNAMIC PROGRAMMING

• Dynamic programming retains the best partial score for each current run length, rejects lengths above k, and solves the allocation in O(dk) transitions with backpointers.The primary family adds one Boolean flag to require at least one positive odd run, doubling the state count while preserving the O(dk) bound; integer masks support deterministic ties at additional bit-operation cost.Source: §D.2 · pp. 26–27
• An interval-scheduling verifier independently reproduces the dynamic-programming objective by summing weighted noisy contrasts over allowed runs, separating optimization checks from direct Born-rule evaluation.The verifier either skips a site or retains a run of allowed length before continuing after its separator.Source: §D.2 · p. 27

D.3 INFORMATION EQUALITY AFTER PREPARATION ERRORS

• Odd preparation-error components preserve the exact classical experiment because retained-edge ZZ errors commute with the global stabilizer, whereas even components can mix codewords and destroy information.A fixed architecture must be selected before future trials; at p = .2, the even-component counterexample has trace overlap .32.Source: §D.3 · p. 27

D.5 A HOMOGENEOUS PREPARATION-ERROR THRESHOLD

• Under homogeneous noise, for γ ≤ 2/3 every longer run is inferior to singleton density 1/2, whereas for γ > 2/3 density increases with length; without a depth cap, intermediate groups improve neither product nor full-chain alternatives.Source: §D.5 · p. 27

E PREDICTION UNDER LOGARITHMIC LOSS

• For logarithmic loss, the optimal prediction value is conditional mutual information I(C; Y | T), and a projector code attains the optimal bound while preserving the classical bit.With sharp reports, the path learning laws transfer after replacing contrast D by J0 = 1 − H2(h), although near h = 1/2 the scales are not interchangeable.Source: E PREDICTION UNDER LOGARITHMIC LOSS · p. 28, §E.2 · pp. 28–29

E.1 WHAT CONTRAST DOES AND DOES NOT DETERMINE

• Contrast alone does not determine logarithmic loss: accuracy selects ZZ, while log loss selects XX in a two-qubit product-code example, even though both encodings preserve the bit perfectly.The two choices have weighted contrasts 0.35 versus 0.30, but information values 0.30 bits versus 0.132105 bits.Source: §E.1 · p. 28

F A SHARP PATH LAW WHEN ENTANGLEMENT DEPTH GROWS

• For sharp reports on a d-qubit path, depth k reduces worst-case excess allocation error to order 1/k even without requests; constants are uniform in d, k, and m, and excess is zero for k ≥ d.Learning a sufficiently small fixed fraction of that scale requires order d log(k + 1) requests. The upper bound is attained by information-preserving codes, while the lower bound permits arbitrary mixed physical encodings.Source: F A SHARP PATH LAW WHEN ENTANGLEMENT… · p. 29, §F.2 · p. 30

G COMPLETING AN EMPIRICAL ALLOCATION

• Completing an empirical allocation to a maximal feasible set is statistically equivalent to full-feasible ERM under nonnegative weights, but poor tie rules can cause arbitrarily large sample penalties.In the two-color example, completion has excess at most (D/4)e^-m/8 independent of a, whereas the sparse rule’s excess is at least (3D/8)(1 − m/a).Source: G COMPLETING AN EMPIRICAL ALLOCATION · p. 31

H TWO STATISTICAL RESOURCES UNDER PREPARATION NOISE

• Calibration uncertainty adds a model-error term to request-learning excess, requiring B(2d − 1) scalar calibration observations in addition to m requests.The guarantee concerns prediction contrasts within the same circuit family and does not certify information preservation under an unspecified true channel.Source: H TWO STATISTICAL RESOURCES UNDER… · p. 31

H.1 A SHARP JOINT LAW ON DEPTH-THREE PATHS

• With depth-three one-odd circuits, request and calibration resources contribute separate terms, and Theorem 4 gives a sharp joint minimax law over workloads and preparation factors.The request lower bound remains nonzero when calibration is uninformative, while a separate construction makes the preferred allocation depend on unknown calibration parameters.Source: §H.1 · p. 32
• The sharp depth-three path law is rate-sharp for the path family, while its calibration term becomes (2d −1)/Q and may improve under adaptive calibration or additional shared parameter structure.The stated sharpness does not claim the displayed constants are optimal.Source: §H.1 · p. 33

I ONLINE ALLOCATION ON A PATH

• The online learner uses exponential weighting with full-information feedback, comparing against the best fixed allocation and computing each round in O(dk) scalar operations.Fresh records are encoded each round, query observations reveal every candidate’s reward under the known model, and the method does not claim dynamic regret.Source: I ONLINE ALLOCATION ON A PATH · p. 33

J.1 GROWING-DEPTH SCALING DIAGNOSTIC

• Normalized empirical excess ranged from 0.0656 to 0.1578 on hard-family cells and from 0.0028 to 0.0071 at the largest budget on fixed Dirichlet workloads.The fixed-workload controls are not minimax-hard populations, and the diagnostics test a proved rate rather than a universal learning-curve exponent.Source: §J.1 · p. 34

J.3 TWO-RESOURCE FACTORIAL DESIGN

• The factorial study crossed request budgets with calibration budgets, using Q = B(2d −1) total calibration observations separately from m requests across 76,800 fitted rows per method.Controlled synthetic noise included low, heterogeneous, and stress regimes; no physical exchange rate among resources was supplied.Source: §J.3 · p. 34

J.5 UNCERTAINTY, ADVERSE CELLS, AND DEPTH SENSITIVITY

• The primary matched comparison found a 1.214-point gain for Joint smoothing, with Joint error 8.576% versus 8.320% for the same-family oracle.In the raw-count comparison, Joint error was 8.618% versus 9.616% for Task only; smoothing helped Joint but increased Task only’s error.Source: §J.5 · p. 36, Table 2 · p. 36

J.6 COMPLETION AND A CALIBRATION-HARD DIAGNOSTIC

• Calibration-hard three-site experiments produced B-scaled excess from 0.0387 to 0.0411, measuring finite-sample switching behavior rather than the lower-bound constant.The construction changes channel separation with B, so these are not fixed-channel learning curves.Source: §J.6 · p. 36, Figure 7 · p. 37

K.1 SELECTIVE REPAIR OF CONNECTED ALLOCATIONS

• 14.998% exact-allocation error rose only to 15.170% with repair and 15.143% with local improvement, versus 5.242 points for deleting every connection query.At the primary connected-region condition, repair differences were 0.172 points ([0.093, 0.252]) and 0.145 points ([0.075, 0.220]).Source: §K.1 · p. 37

K.2 ONLINE PROTOCOL AND RESULTS

• Smoothed follow-the-leader had lower mean cumulative loss than Gibbs in all six online conditions, although every sequence satisfied the Gibbs bound.The comparison used expected additional mistakes against the best single allocation for the realized sequence; it does not establish dynamic regret or Gibbs superiority.Source: §K.2 · pp. 37–38

K.3 RETAINED INFORMATION AND CORRELATED ERRORS

• At ρ = 0.1, correlated retained-edge shocks yield 9.47% error for the chosen allocation versus 8.23% for the correlated-model oracle, while global decoding remains exact.The odd-component invariant holds for arbitrary joint retained-edge ZZ errors; independence is needed only for the factorized prediction formula.Source: §K.3 · p. 38
• At phase-error probability 0.03, global errors are 0.874% for Joint, 0.026% for Product, and 0.295% for Full chain, while restricted errors are 15.14%, 19.52%, and 13.32%.Independent single-qubit Z errors need not commute with the witness, so none of these representations remains lossless and their access comparison is not information matched.Source: §K.3 · p. 38

L REPRODUCTION AND HISTORICAL EVIDENCE

• The installable allocation package contains the path optimizer, Gibbs distribution and sampler, input validation, and runnable example; its reproduction entry point verifies frozen acquisition hashes, regenerates request and calibration prefixes, checks population risks, and compares the main optimizer against an independent interval implementation.Source: L REPRODUCTION AND HISTORICAL EVIDENCE · p. 38

M A PARITY-SERVICE HEALTH TEST AND PHYSICAL RESPONSE

• The parity-service experiment tests response prediction and archived IBM calibration, but does not validate the four-qubit service’s coherence-preserving instrument.The archived 215,040-shot device observations concern fixed implementations in a different measurement family and do not establish learned-allocation or noisy-state information preservation.Source: L REPRODUCTION AND HISTORICAL EVIDENCE · p. 38, M A PARITY-SERVICE HEALTH TEST AND… · p. 39

M.1 THE SERVICE AND ITS HEALTH TEST

• The service accepts three-qubit probes for ZZI, XXX, or IZZ; the Z queries are repetition-code stabilizers, whereas XXX is a logical-state diagnostic.Parity repeatability alone is insufficient because local outcomes can be measured and their parity reported perfectly without preserving coherence within parity sectors.Source: §M.1 · p. 39
• Depth-two candidates include product, left-pair, right-pair, and X-product codes with ideal contrast vectors (1, 0, 1), (1, 1, 0), (0, 1, 1), and (0, 1, 0).Compute–copy–uncompute uses three CNOTs for Z checks and five for X checks; all-input Kraus checks recover (I ± P)/2 with residual at most 4.5 × 10^-16.Source: §M.1 · p. 39

M.2 CALIBRATION FROM DIRECTLY OBSERVED RESPONSES

• Direct response calibration incorporates preparation, gate, assignment, and report errors, selecting among A encoders and q requests from independent calibration shots without factorization assumptions.For A = 4, q = 3, m = 128, B = 2048, D = 1, and δ = 0.05, the bound is 0.208 using 49,152 healthy calibration shots.Source: §M.2 · p. 40
• The calibration guarantee adds Dκ when test and calibration contrasts differ by at most κ, but observed blocks cannot certify that condition for future device operation.The separate likelihood alarm uses healthy and faulty response fits plus a healthy threshold block, guaranteeing marginal false-alarm probability at most 0.05 for the specified injected fault.Source: §M.2 · p. 40

M.3 CIRCUIT EXPERIMENTS AND ESCAPE CONTROLS

• The circuit simulations use declared hypothetical noise channels, and the back-action results do not constitute a physical instrument certificate because preparation and analysis errors also require characterization.The asymmetric channel combines gate depolarization, amplitude damping, asymmetric assignment errors, and phase damping while omitting spectator measurement crosstalk.Source: §M.3 · pp. 41–42
• Calibration lowers mean held-out recovery error from 16.88% to 16.28% in the asymmetric model, while the symmetric control changes by +0.035 points with no resolved benefit.The asymmetric gain interval is [0.323, 0.928] percentage points; the alarm comparison with one fixed history tests probe choice rather than an additional calibration gain.Source: §M.3 · p. 41
• 16.4 points is the weaker-fault ROC-power gain for the left-pair probe over the X-product at an empirical 5% false-alarm rate, with interval [3.9, 28.9].Across request histories, calibrated, request-only, and learned-product selectors achieve 69.14%, 64.36%, and 54.23% descriptive ROC power, respectively.Source: §M.3 · p. 41, Figure 8 · p. 42

M.4 PHYSICAL CALIBRATION ON ARCHIVED IBM OBSERVATIONS

• 0.00942 to 0.00365 is the decrease in mean absolute probability error from ideal circuits to measured responses on archived IBM observations.The held-out Brier score also decreases from 0.070311 to 0.070197, a paired gain of 0.000114 with interval [0.000064, 0.000164].Source: Figure 8 · p. 42, §M.4 · p. 43

N.1 AN EXACT DIAGNOSTIC EXPERIMENT FOR EVERY CSS ALLOCATION CLASS

• Theorem 5 establishes that every optimal N-call diagnostic experiment for a CSS allocation class is Blackwell-equivalent to N independent erasure-and-flip experiments generated by a maximizing compatible set.A fixed signed-projector probe attains the experiment, and the resulting alarm does not require knowledge of the request distribution π.Source: §N.1 · pp. 43–44
• The diagnostic-power regret equals the prediction-regret loss scaled by e1 −e0, while prediction regret is scaled by D/2 over the same weighted-coverage loss.This identity follows because the feasible mean absolute-correlation vectors and their tagged signed-projector realizations are shared by both tasks.Source: §N.1 · p. 44
• The transferred sharp-report learning law remains proportional to k^-1 min{1, sqrt(d log(k + 1)/m)} uniformly for 2 ≤ k < d, with the same connected-region conditions.This concerns probe-allocation learning with known e0 and e1; estimating a healthy baseline or operating threshold requires separate calibration data.Source: §N.1 · p. 44
• For arbitrary N, a learned compatible set of mass bp obeys the same-level transfer bound by coupling it to the optimal mass p∗, with one-use total-variation error (e1 −e0)(p∗−bp).The bound compares optimal population tests rather than estimated thresholds.Source: §N.1 · p. 44

N.2 THE COMPLETE SEPARABLE BENCHMARK AND ITS RESOURCE COST

• At 32 calls, depth-two policies reach 88.45% power versus 82.39% for all separable policies, while an independent 1% Bell-probe flip reduces power to 84.84%.The fixed-horizon counts for 80% power are 22 calls at depth two and 30 for separable policies; at 1.6% flip, Bell power falls to 82.00%, below the separable value.Source: §N.2 · p. 45

N.3 A CERTIFICATE FOR ARBITRARY BINARY DETECTOR EFFECTS

• The arbitrary-effect certificate upper-bounds every separable alarm by λα + U(λ) and yields a dominating binary experiment whose product tensorizes across classically adaptive calls.The construction applies to arbitrary healthy and faulty binary effects, including revealed classical mixtures, but scales exponentially in outcome/query decisions and Hilbert-space dimension.Source: §N.3 · p. 46

N.4 OPTIMIZED PROBES UNDER CORRELATED AND NON-PAULI ERRORS

• At noise strength s = 0.08, the Bell probe exceeds the full separable upper bound by more than 2.19 percentage points, whereas at s = 0.20, an optimized product beats both fixed Bell probes.The comparison uses declared correlated coherent errors and biased responses; the upper bound may be loose, and its excess over the exact separable optimum at s = 0 is 0.023 points.Source: §N.4 · p. 46, Table 7 · p. 47, Figure 9 · p. 47

N.5 FROM IDEAL PLANNING TO PHYSICAL ACQUISITION

• Ideal planning was fragile: the Bell-versus-product rule passed in 88.69% of model cycles, but Bell preparation flips reduced this to 40.79% at 0.5% and 8.53% at 1%.Physical commissioning therefore superseded the ideal acquisition plan; limited calibration cannot provide a full separable device certificate without tomography and preparation-error confidence regions.Source: §N.5 · p. 47

N.6 RELATION TO OTHER LEARNING PROBLEMS

• The contribution differs from request-distribution access codes and preparation games by learning one allocation from classical request identities under fixed-detector and depth constraints.Its common dominating experiment permits tensorizing a one-use bound, while the action class follows from physical feasibility rather than general constrained-preparation test optimization.Source: §N.6 · pp. 47–48

O PROSPECTIVE EVALUATION ON A QUANTUM PROCESSOR

• On ibm_marrakesh, the prospective evaluation tests calibration transfer and one-call probe comparison under the nominal (0.2, 0.6, 0.2) workload, not the theoretically established learning rate.The request arrives after probe preparation, and the detector service uses data qubits (98, 111, 110) with ancilla 112.Source: O PROSPECTIVE EVALUATION ON A QUANTUM… · p. 48, §O.1 · p. 48

O.2 CALIBRATION SELECTS A PROBE; A SECOND JOB CHECKS IT

• Direct validation selected Z product and GHZ: in the first cycle they achieved 43.23% and 51.74% estimated power, while the unconstrained product candidate reached only 15.01%.The second cycle again selected Z product and GHZ, with estimates of 40.19% and 58.54%; probe choice and scores were frozen before held-out testing.Source: §O.2 · pp. 48–49

O.3 FROZEN ALARMS AND INDEPENDENT TEST JOBS / O.4 COHERENCE WITHIN BOTH PARITY SECTORS

• The physical tests used frozen, separately calibrated cycles, but shared backend, layout, implementation, and acquisition day, so they establish observation separation rather than independent physical noise.Coherence measurements covered both parity sectors and phases using absolute residual witnesses with Bonferroni-corrected Clopper–Pearson intervals; they test the stated input family, not a complete instrument.Source: §O.3 · p. 49, §O.4 · pp. 49–50

O.5 MEASURED OUTCOMES AND RESOURCE ACCOUNTING

• 16.34 percentage points was the cycle-2 power difference, versus 9.11 points in cycle 1, while healthy upper bounds remained near 2–3%.Cycle 2 had one-sided Fisher p = 5.79 × 10−21 and cycle 1 p = 7.20 × 10−8; the acquisition used 1,482,496 shots and 522 charged QPU seconds across 40 jobs.Source: §O.5 · p. 50

O.6 COHERENCE THROUGH THE DYNAMIC SERVICE

• 56 of 56 target-request coherence cells had positive simultaneous lower bounds, with the smallest bound 0.302 and expected-parity agreement ranging from 78.80% to 95.63%.The follow-up analyzed complete request-conditioned extraction branches without selecting shots by detector agreement or witness value.Source: §O.6 · p. 52

O.7 ACQUISITION PROVENANCE AND TIMING AUDIT

• The provenance audit retained physical records and found swapped timing payload identities in one job, but independent probe signatures supported the intended assignments without changing counts or frozen alarm rules.Unverified per-record timing cannot establish duration matching.Source: §O.7 · pp. 52–53

P THE COST OF CHANGING THE DETECTOR INTERFACE

• The detector-interface comparison treats Blackwell equivalence as the criterion, requiring equivalent experiments under both detector-health hypotheses rather than matching power at one operating point.The analysis separates fresh calls, chosen extra queries, retained probes, preparation, detector calls, calibration, and operating costs.Source: P THE COST OF CHANGING THE DETECTOR… · p. 54

P.1 FRESH REQUESTS AND OPTIMAL EXPECTED CALL CONVERSION

• A target experiment requiring q informative probability from probes with probability p requires at least Nq/p fresh calls in expectation, and a stopped policy attains equality.The result assumes fresh probes before independent requests, no retained quantum system, and conditional one-call domination by the baseline experiment.Source: §P.1 · p. 54
• At (e0, e1) = (.02, .164), 80% power at 5% false alarms requires 30 fresh product calls or 22 Bell calls, while the Bell/product Chernoff-exponent ratio is 1.33860.For the (ZZI, XXX, IZZ) menu, coverage increases from .6 to .8 with expected cost 4N/3 and maximum 2N.Source: §P.1 · p. 54

P.2 CHOSEN EXTRA QUERIES AND RETAINED PROBES

• A retained probe can diagnose detector noise through repeated unsupported queries, but at the same 5% false-alarm level 14 probes reach 80.2647% power using 19.6 average calls and at most 28.The policy exceeds 22 calls with probability 5.8319%, so it is not a hard-budget counterexample; chosen commuting repair queries attain coverage q at cost 1 + q −p.Source: §P.2 · p. 55

P.3 PREPARATION, CALLS, AND CALIBRATION

• GHZ is cheaper than repeated queries exactly when call overhead x exceeds 22.5, comparing 19.6x + 75.6 with 18x + 111.6 before startup costs.This is an attaining-policy comparison, not an optimized adaptive frontier, and observed physical powers from differently calibrated studies cannot be substituted into it.Source: §P.3 · pp. 55–56

Q.1 STRUCTURED EXPERIMENTS AND A PUBLIC NEGATIVE CONTROL

• The public negative control has locality .672727 for every one of 15,400 balanced packings, so learning cannot improve it.The replay uses published transaction footprints and does not execute BenchBase or measure throughput.Source: §Q.1 · p. 57

Q.2 OBSERVED PURCHASE BASKETS

• The basket study uses an exact set-partition MILP over three-record shards, with 128 zero-gap fits and nested request samples of m ∈ {8, 32, 128, 512}.The objective includes singleton baskets and unavoidable failures larger than three items.Source: §Q.2 · p. 57
• At 512 requests, mean test locality is 67.25% and 52.21% for 12- and 24-item schemas, versus 63.57% and 48.19% for frequency grouping.Fixed hash gives 62.08% and 47.70%; validation-trained comparisons give 68.02% and 53.02% after additional observed baskets.Source: §Q.2 · p. 57

Q.3 COMPLETE BASKETS ON THE FULL CATALOGUE

• At capacity 512, frequency grouping beats basket search in 14/16 streams at m = 512 and 15/16 at m = 2048, by 0.470 and 0.528 points on average.At capacity 128, basket search has higher mean locality at every budget, so the better method depends on capacity.Source: §Q.3 · p. 58, Figure 13 · p. 58

R NATIVE 15-QUBIT PATH: EXECUTION AND PENDING LEARNING STUDY

• The full-chain encoding records 107 errors in 1,920 trials, versus 274 for fixed depth three, 481 for product even, and 584 for product odd.Independent-shot Hoeffding bounds give depth-three minus full error 8.70 points, even minus full 19.48, and odd minus full 24.84.Source: R NATIVE 15-QUBIT PATH: EXECUTION AND… · p. 59

R.1 FROZEN LEARNING DESIGN AND EXECUTION STATUS

• Confirmatory request-learning and calibration curves were not completed because two exploratory shared-response jobs timed out and returned no usable observations.The frozen protocol proposed testing a positive paired advantage over both extremes on every workload at m ≥128.Source: §R.1 · p. 59

R.2 A LIMITED COHERENCE CHECK EXPOSES A SIGN REVERSAL

• The limited coherence check finds mean signed coherence 0.602 for extraction and 0.029 for refinement, but query seven reverses the expected sign at −0.656.This diagnostic covers neither both parity sectors nor the all-input detector contract, so it cannot validate preservation across the whole path.Source: §R.2 · p. 60, Figure 15 · p. 60

R.3 THE REVERSAL REPEATS ON BOTH BITS AND WITNESS PHASES

• In the focused replication, all four extraction intervals are negative while all preparation-only intervals are positive and all refinement intervals contain zero.The result implicates the added extraction/measurement interval, without identifying a faulty gate or distinguishing measurement backaction from coherent evolution.Source: §R.3 · pp. 60–61, Table 12 · p. 60