Quantum executions sharing a provenance record can produce different output distributions. We ask which execution details to retain, and at what precision, to guarantee agreement within a declared Hellinger tolerance. Under a specified channel model, we combine independent stochastic mechanisms and coherent rotation differences to certify recording precision without output simulation or explicit dependence on register dimension. For four-qubit quantum approximate optimization at depth one, a 13-bit comparison record suffices at tolerance 0.05 over a declared continuous noise range, uniformly in programmed angles for each fixed task. Pairwise separation constraints characterize sufficient field sets on finite context classes, exposing omissions that pass individually but fail jointly. Separately, six-qubit hardware measurements over six tested contexts certify minimum label counts of two for raw outputs and one after a fixed decoder, at tolerance 0.15 with 97\% simultaneous confidence under stationary independent shots. Execution records thereby carry checkable agreement guarantees.
Comments
v3: Revised abstract and introduction; clarified relation to prior work; added references. Scientific results unchanged
Quantum executions sharing a provenance record can produce different output distributions. We ask which execution details to retain, and at what precision, to guarantee agreement within a declared Hellinger tolerance. Under a specified channel model, we combine independent stochastic mechanisms and coherent rotation differences to certify recording precision without output simulation or explicit dependence on register dimension. For four-qubit quantum approximate optimization at depth one, a 13-bit comparison record suffices at tolerance 0.05 over a declared continuous noise range, uniformly in programmed angles for each fixed task. Pairwise separation constraints characterize sufficient field sets on finite context classes, exposing omissions that pass individually but fail jointly. Separately, six-qubit hardware measurements over six tested contexts certify minimum label counts of two for raw outputs and one after a fixed decoder, at tolerance 0.15 with 97% simultaneous confidence under stationary independent shots. Execution records thereby carry checkable agreement guarantees.
How much internal quantum memory is needed to reproduce a stationary stochastic process, and how does reducing that memory change the environmental resources required when output records are retained? This paper establishes exact quantum-source constructions and resource bounds under explicitly specified readout and retained-output conditions. A general completion theorem identifies the minimum Hilbert-space dimension for stationary triple data in the orthogonal-current-symbol source class: it equals the sum of the complex positive-semidefinite ranks of the middle-symbol slices. An attaining construction reproduces the entire canonical stationary order-two process. For the delayed trine process, the minimum classical hidden Markov model requires nine states at every non-independent parameter value. Quantum instruments reproduce the same complete output law using four-dimensional memory throughout the parameter range, with four dimensions proved necessary in a certified low-noise regime and three dimensions both sufficient and necessary on an explicit high-noise interval. The paper then separates reusable memory dimension from environmental dilation cost. On the certified high-noise interval, the exact one-step Pareto frontier consists of a three-dimensional source with recorded-channel Choi rank nine and a four-dimensional source with rank three. Here, Choi rank measures the minimum initially pure environmental remainder dimension for the channel retaining both the emitted classical symbol and final quantum memory. A separate mixed-support qutrit construction at noise parameter 0.99 improves complete-history costs and gives upper bounds close to universal lower bounds. These results show why optimizing a single step need not optimize a retained multi-step history. The analysis also distinguishes complete channels, fixed-input states and archive-only outputs, and demonstrates how access to a retained reference can distinguish generators with identical passive output statistics. Full-word proofs, algebraic complete-positivity certificates, a technical companion and reproducibility materials accompany the results. The exact intermediate memory boundary and complete-history optimum remain open. All results use standard quantum probabilities and apply to the stated operational resource models; they do not establish total hardware, energy or thermodynamic minima.
Quantum records may be well encoded upstream yet fail to support downstream inference when routing, decoder resources, memory, or comparison channels are restricted. We analyze finite routed quantum-record networks using task-specific Bayes risk. Three exact results provide the constructive core. First, a single Bell-coded pair supplied as the same complete input to GLOBAL and finite-round LOCC-late decoders yields the risk gap qμ/2, where q is pair-delivery probability and μ retained coherence; memory policy determines whether this access is preserved. Second, fixing local binary experiments with error rates 0.20 and 0.30, we determine the entire compatible joint-risk interval [0.15, 0.20] and an attaining decoder with minimax risk 0.20 that does not know the coupling. Third, a binary sufficient-statistic delivery law is attained by a three-record majority compression with nonzero raw risk, yielding a tight physical reconstruction defect. A generated-outcome model gives an exact receiver-availability boundary, while four reduced-description witnesses and continuity bounds delineate when source, state/resource, local-coupling, or clock/memory information is insufficient. The results distinguish record formation, receiver use, memory retention, and task-adequate comparison within standard finite quantum processes. Author manuscript, preprint version 1.0. This manuscript is not a peer-reviewed journal publication. A separate technical Supplement provides complete model definitions, proofs, channel specifications and numerical details. Reproducibility materials are available separately as Software release 1.3 on OSF: https://doi.org/10.17605/OSF.IO/5TDRB. The preprint and reproducibility record are separate, linked scholarly outputs.
Calibrated support observations can constrain a complete quantum instrument without identifying its action by unrestricted tomography. We study what those observations guarantee when the noise law is not supplied. On a fixed four-dimensional incidence-record experiment, we define the set of all complete two-outcome completely positive instruments compatible with the exact raw rejection and outcome probabilities, retaining the classical flags. Exact ideal data identify the target, but an explicit family of complete instruments reproduces the nonzero-data category laws of a low-error contamination device while admitting coherent error of square-root order. A target-ray mass certificate and a common complete-dilation comparison give a universal upper, while an exactly feasible rational instrument supplies a stronger lower witness. At the diagnostic parameter r = 1/300000000, the unhalved target-relative worst-case radius lies between 0.000258065115532250 and 0.000951566242566. The result separates avoidable inequality conservatism from information absent from the measured data. Fixed-functional and fixed-map optimizations have separately certified scopes; they do not solve the global radius. The conclusions are conditional on the supplied carrier, calibration and common-device model. They do not establish an apparatus, device-independent certification, a finite-sample power guarantee or complete formal verification. Selected supporting declarations have one recorded successful Lean 4.19.0 macOS execution and declaration-specific source, type and axiom review; this is not an end-to-end formalization.
This paper develops task-relative reader certification within Shadow Theory’s source–readout programme. It asks how many source copies are needed to validate a fixed quantum reader, and when access to its retained measurement records reduces that cost compared with observing only its final correct-or-incorrect decisions. The reader is fixed before validation and must return a sign after each complete experiment. A trusted evaluator knows the preparation label for scoring, while the reader does not. The analysis distinguishes terminal-error observations, records with trusted ancillary information, and readable measurement trajectories governed by a shared physical response model. For terminal-error validation, the paper establishes exact finite-budget tests and sequential likelihood tests, together with nearly matching information-theoretic lower bounds on expected source consumption. An unrestricted-record result shows why additional archived data alone need not improve certification: an advantage requires specified information or response constraints. A separate trusted-seed construction gives an exact information-projection test. The central quantitative comparison concerns a fixed fourteen-copy reader in a trusted shared-visibility family. To certify a maximum label error of one percent, a trajectory-based validator achieves a false-certificate probability of at most one in 1,048,576 and a nominal acceptance probability of at least 95 percent, using no more than 720,160 source copies. Under the same preparation and physical-response promises, every validator restricted to terminal errors and meeting the same acceptance requirement needs more than 1,992,371 source copies in expectation. At the exact nominal device, an uncapped trajectory-based test accepts almost surely with expected consumption below 254,746 copies. A separate robustness analysis extends likelihood-test validity to a quantified class of conditional response deviations. Finite-cap guarantees and expected stopping costs remain explicitly distinguished. The results demonstrate a certified resource advantage from access to retained classical measurement trajectories, rather than merely an improvement over one chosen terminal-error test. Supporting materials provide the fixed reader, exact certificates and reproducibility procedures. All guarantees depend on the stated preparation, response, information-access and deployment-stability assumptions. Certification concerns the nominated reader’s decision error; it does not establish complete quantum-instrument equivalence or identify an underlying source. The analysis uses standard quantum probabilities within Shadow Theory’s research programme.
A small quantum arrangement — a reader qubit Q₁ coupled to a remainder qubit Q₂ through one fixed gate, with declared operations on Q₁ only — has access to some distinctions of the remainder and not to others. This manuscript follows that arrangement through fourteen research units — thirteen with a model, exact proofs and an executable checker (with certified interval enclosures wherever a propagation is read), and one, the maintenance criterion, a definition with checked decision logic — and organizes what they establish around four distinctions. First, protecting an access restriction is different from preserving the distinctions a reader can use: a universal leakage bound and an exact one-gate value are proved; a reset-based maintenance pulse prolongs limited leakage at the price of previously accessible information, while a reversible pulse prolongs it and preserves the full declared input class. Second, meeting protection criteria does not identify a mechanism: a prepared finite clock-controller meets four declared criteria at four sizes, and so does a static field with the clock decoupled, and so does the clock's free mean field at the largest size. Third, restoring a supporting condition does not necessarily recover the capability: coupling alignment is shown to be a supporting condition of the capability, and a restoring process that improves alignment in six matched realizations meets the full restoration standard in one, misses it narrowly in one and fails in four, while useful access is preserved in none. Fourth, retaining a relation, delivering it to a reader and using it internally are different achievements, and a supplied design's advantage can reverse when its context changes without the comparison identifying the mechanism: a sector record restores access at three of four endpoints under an added permission; an internal coupling improves the primary endpoint without reaching the reference; a two-by-two of axis term and gating cell reverses both contrasts. Each result is stated with what it does not establish. No conclusion about self-maintenance, ownership or experience is drawn. The deposit is a technical manuscript, not peer-reviewed, that consolidates fourteen closed research notes and adds no theorem; the supporting proofs are in the notes. The package contains the manuscript (PDF and Markdown), the fourteen notes with two errata, every checker and its run record, the failure-path harness with its recorded outputs and a release adapter, the two earlier baseline notes, a claim-to-proof map, a correction ledger, a source lock and a manifest with the SHA-256 of every file. Every number printed in the manuscript was read by script from a packaged run record. Running checks/run_all.sh reproduces every checker's recorded summary line with the declared dependencies only (Python 3.11 or later; sympy 1.14, mpmath 1.3, numpy); the harness coverage, including the checkers it does not cover, is stated in §10 and Appendix A. Drafting and computation were AI-assisted under the author's direction. Archive SHA-256: 119d835b2d2e19463f3eb6478ea34ddbdf4deea220a5d6d6fe528dcf7061bedb.
How much quantum storage must a hidden implementation retain when experimental observations leave its structure undetermined? This paper establishes exact minimum quantum-storage requirements across all implementations compatible with specified channel observations. The resource is the worst-input expected logarithm of retained quantum dimension across one controlled temporal cut, with unlimited classical records available at no cost. For the channel dimension measure introduced by Cope and Uola, the paper proves their suggested minimax equality relating minimum storage to output Schmidt measure and extends it to every nonempty compact convex set of compatible channels. One common input enforces the resource minimum across all compatible implementations. A finite optimal implementation exists, although an optimal entangled probe need not. For qubit channels, four measured coefficients determine an exact, attained minimum without assuming channel covariance or unitality. An explicit five-Kraus construction attains 99/169, while a separate analytical certificate proves that every compatible implementation must use nontrivial quantum storage with at least that probability on the same specified input. A second exact formula uses only two Pauli-transfer coefficients and yields a minimum storage requirement of 0.58 when both coefficients are jointly constrained to the interval [0.79, 0.81]. The paper also presents semidefinite optimization over compatible experimental uncertainty regions. The results require calibrated quantum inputs and outputs, concealed preparation labels, and accounting for every quantum carrier across the storage cut. They constrain storage resources without reconstructing the hidden implementation or certifying environmental non-Markovianity. Motivated by Shadow Theory’s source/readout distinction, the proofs use standard quantum mechanics and quantum-information mathematics. Version 1.1 includes complete proofs, explicit attaining constructions, worked examples, and discussion of experimental assumptions and limitations. No experimental data are reported.