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QILP-0: Constructing Observational Declarative Twins of Quantum Circuits

Marina de la Cruz Echeandía, César Luis Alonso, Tony Ribeiro, Alfonso Ortega de la Puente

arXiv:2609.01049v1cs.AIquant-ph

TL;DR

Quantum circuit representations remain difficult to explain when only restricted measurements are exposed. QXymb and its QILP-0 specialization construct a target-independent, provenance-preserving declarative twin within a declared observational scope, and every reported validation relation is reconstructed completely and without conflict at strict accuracy one.

  • Problem

    Restricted quantum measurements can leave the observable relation produced by a circuit opaque despite familiar surrounding learning algorithms.

  • Method

    QILP-0 incrementally analyzes declared observables with fixed-reference geometry and maps retained structure back to original columns before target-audited symbolic induction.

  • Results

    Every reported Bars & Stripes and Low-Depth MNIST relation achieved complete, conflict-free reconstruction with strict reconstruction accuracy equal to one.

  • Takeaways & Limitations

    QILP-0 provides a finite declarative representation whose equivalence, provenance, and uncertainty can be inspected within an explicitly delimited observational interface.

  • Takeaways & Limitations

    Claims remain relative to the declared observational scope and do not establish complete quantum behaviour beyond the observed relation.

Abstract

from arXiv · show

This paper introduces QXymb, a general framework for constructing observational declarative twins of quantum circuits, and develops QILP-0, its first complete order-0 specialization. QILP-0 constructs a finite multi-valued propositional logic program from observed circuit behaviour within a declared observational scope. The pipeline traverses a declared family of quantum observables incrementally according to a reproducible structural grading and a declared observational reference horizon. Progress is quantified through reference-relative coverage against a fixed target-independent reference. Observable responses are organized through target-independent geometry, while retained latent structure is mapped deterministically back to original observable columns before symbolic processing, preserving observational semantics and provenance. Selected observable profiles are converted into a finite relation through admissible target-independent discretization. The target is used only afterwards to audit twin-admissibility and induce the declarative theory. A theory is certified as an exact observational declarative twin when it completely and correctly reconstructs the resulting finite task-conditioned discrete relation. Logical exactness is therefore separated from numerical, backend, provider, and discretization uncertainty, which is retained as audit metadata. Validation uses two complementary QML settings. Exhaustive Bars & Stripes experiments compare product and grid-CZ embeddings from 16 to 100 qubits and exercise the native-discrete branch. Low-Depth MNIST analyses all 14,708 digit-0/1 instances before and after a trained variational quantum transformation and exercises continuous discretization. In every reported relation, the induced QILP-0 theory achieves complete, conflict-free reconstruction with strict accuracy equal to one.

1. Introduction

QILP-0 addresses opacity in quantum representations by constructing a scope-qualified declarative twin from observed circuit behaviour without using targets to shape the observational vocabulary. Its pipeline preserves provenance, separates logical exactness from upstream uncertainty, and achieves exact reconstruction in two QML validations.

  • Motivation: QML representations can remain opaque because quantum embeddings, parametrized circuits, and restricted measurements expose only selected aspects of processed states.This creates explanatory questions beyond interpreting the surrounding classical learning component.
  • Observational declarative twin: QILP-0 constructs a finite multi-valued propositional program for the relation observed within a declared dataset, observable family, structural grading, and reference horizon.It does not claim complete circuit behaviour across all inputs, states, measurements, or execution conditions.
  • Certification: Logical exactness concerns the finite discrete relation produced by the pipeline, while sampling, numerical, backend, and provider uncertainty remains separate audit metadata.An exact twin must cover every analysed row without conflicting predictions.
  • Pipeline: Target-independent observable generation, geometry, coverage, original-column selection, and discretization precede target-based twin-admissibility auditing and declarative induction.This ordering prevents the explanation vocabulary from being retrospectively engineered around later classes.
  • Validation: Every reported relation achieved complete, conflict-free reconstruction with strict reconstruction accuracy equal to one across Bars & Stripes and Low-Depth MNIST.Bars & Stripes covered product and grid-CZ embeddings from 16 to 100 qubits; MNIST used all 14,708 digit-0/1 instances before and after a trained variational transformation.

2. Background and Related Work

QILP-0 occupies a complementary explainability level: it reconstructs scoped observed quantum relations as provenance-preserving declarative theories rather than producing feature attributions, gate scores, visualizations, or physically meaningful latent factors.

  • Existing explainability families: Quantum explainability spans feature attribution, circuit-component relevance, intrinsically interpretable models, and visual analytics across hybrid quantum–classical pipelines.These families respectively explain inputs, gates, concepts or analytical structure, and coordinated circuit, parameter, and measurement views.
  • Adjacent approaches: Classical surrogates and shadow models primarily produce predictive models, whereas QILP-0 induces a logical theory whose literals remain tied to original observables.This distinction concerns the explanatory object rather than a preference for one learning algorithm.
  • Representation learning: Representation-learning approaches such as QDisc seek compact, physically meaningful latent factors and analytical descriptors for discovered quantum-data structure.QILP-0 instead reconstructs an explicitly scoped observed relation while retaining observable-vocabulary traceability.
  • QILP-0's position: QILP-0 is a post-hoc, dataset-level, representation-oriented pipeline that traverses observables by structural support, measures fixed-reference coverage, and maps retained geometry back to original columns before induction.Its records connect observational geometry, observables and qubit supports, and symbolic descriptions of the observed relation.

3. Continuous observational layer: methodological conditions for target-independent construction

The continuous observational layer represents finite circuit behaviour numerically, explores semantically traceable observables under reproducible structural conditions, and uses target-independent geometry while preserving original observable semantics.

  • Scope conditions: The construction remains relative to the declared scope, including analysed inputs, provider, structural horizon, backend or estimator, numerical tolerances, and discretization policy.Another provider is admissible only if it supplies traceable observables, stable identifiers, reproducible finite grading, and an explicit horizon.
  • Conditions 1–2: QILP-0 begins with a real-valued observational dataset for finitely many analysed input situations and a declared family of semantically identifiable observables.Rows represent input-specific observable profiles, while columns record each observable across the dataset.
  • Target independence: The target is excluded from Conditions 1–8 and first enters during the twin-admissibility audit of Condition 9.Thus the observational matrix is constructed independently of supervised labels.
  • Observable family: The Pauli family supplies the current observable language because its tensor-product operators form a complete n-qubit operator basis in the non-truncated case.Complete non-identity Pauli expectation values together with trace one determine the density operator.
  • Target-independent geometry: SVD organizes redundancy and numerically independent directions, while reference-relative coverage measures represented normalized geometry within the declared observational horizon.These quantities operationalize observational structural richness rather than physical energy, Shannon information, computational complexity, or target-dependent relevance.
  • Original-vocabulary preservation: Retained latent geometry is deterministically mapped to original observable columns, so latent coordinates never become symbolic variables.Geometric reduction selects which original observables to preserve while maintaining semantic traceability.

4. Declarative reasoning layer: from continuous observations to a finite relation

QILP-0 converts target-independent continuous observations into a finite symbolic relation, then uses the target only to audit determinism and induce a declarative theory. Its discretization preserves observable semantics while respecting empirical and provider-specific resolution.

  • Declarative reasoning layer: QILP-0 separates target-independent representation construction from post-discretization target-consistency auditing and rule induction.The observable vocabulary, geometry, selection, and discretization are fixed before the target is used.
  • Resolution control: Continuous profiles receive finite symbolic domains whose granularity is limited by empirical distinguishability and available uncertainty or effective-resolution information.QILP-0 avoids recovering presumed true states and avoids distinctions finer than the reported observational resolution.
  • Discretization policy: Discretization is global, marginal, static by column, and unsupervised, with cuts fitted independently from each complete analysed profile.The supervised target does not participate in determining symbolic states.
  • Column treatment: Each selected observable is classified as constant, native discrete, or continuous before discretization.Native discrete values are encoded bijectively, while only genuinely continuous columns enter bin construction.
  • Finite relation: The resulting task-conditioned observational relation couples discretized observable states with targets under the declared finite scope.It is not a reconstruction of complete circuit semantics across arbitrary inputs, states, measurements, or execution conditions.

5. Constructive realization and correctness of QILP-0

The constructive algorithms implement QILP-0’s contracts and return a finite logic program when finite-scope evaluation, admissible discretization, and target determinism hold. Correctness is defined for the reported discrete relation, while backend uncertainty remains provenance metadata.

  • Certification: An exact certificate requires target consistency, complete row coverage, no conflicting predictions, and strict reconstruction accuracy equal to one.The certificate also records provenance, numerical tolerances, backend information, and diagnostic warnings.
  • Constructive correctness: QILP-0 terminates and returns a finite multi-valued logic program when Conditions 1–7, admissible discretization, and twin-admissibility hold.The returned program satisfies the observational declarative twin definition within the declared scope.
  • Finite construction: Finite datasets and finite observable horizons yield finite selected vocabularies and finite discretized state spaces.Deterministic, target-independent column mappings ensure each observed response maps to a state in a finite domain.
  • Logic induction: PRIDE produces a finite program that is complete and correct for the observed discrete relation while preserving literal-level provenance.Provenance traces each rule literal to its state, original observable, structural grade, qubit support, and evaluation metadata.
  • Computational scope: For fixed K, the Pauli observable count grows as O(n^K), whereas unrestricted traversal through K = n contains 4^n − 1 non-identity words.The declared support-bounded reference horizon is therefore computationally distinct from the full Pauli family.
  • Computational scope: The 100-qubit Bars & Stripes executions demonstrate complete support-≤2 reference construction, not affordable evaluation of the unrestricted K_phys = 100 horizon.A cross-provider performance study remains separate future work.

6. Experimental Validation

Validation exercises QILP-0 in complementary native-discrete and continuous QML settings rather than benchmarking predictive performance. Bars & Stripes varies embedding and system size, while Low-Depth MNIST compares representations before and after variational transformation.

  • Bars & Stripes: Bars & Stripes provides an exhaustive native-discrete setting comparing fixed product and grid-CZ embeddings across increasing system sizes.The experiments span 16 to 100 qubits.

6.1. Validation objectives and certificate criteria

Validation reports continuous-stage coverage and geometry alongside discretization, relation consistency, and row-level declarative reconstruction. All quantities and exactness claims are restricted to the declared observational scope and qualified by uncertainty metadata.

  • Validation criteria: Validation reports the executed structural horizon, geometric coverage, retained dimension, discretization audit, relation consistency, and row-level reconstruction.These measures separate continuous observational processing from declarative reconstruction.
  • Scope: All reported quantities are interpreted only within the declared observational scope.The scope includes the configured dataset, observables, horizon, provider, and discretization choices.
  • Certificate criteria: An exact certificate requires every analysed row to be covered, no conflicting prediction, and strict reconstruction accuracy equal to one.Symbolic literals must remain traceable to their discrete states and original observables.
  • Uncertainty: Backend or provider uncertainty qualifies correspondence to ideal observable values but does not make reconstruction of the reported discrete relation logically approximate.Uncertainty and numerical metadata are retained for domain-expert audit.

6.2. Native-discrete validation: Bars & Stripes

Native-discrete Bars & Stripes validation compares product and grid-CZ embeddings across 16–100 qubits using exact observable evaluation and reference-relative coverage. QILP-0 reconstructs every reported finite relation completely and without conflicts, while geometric and declarative refinement can diverge.

  • Experimental setting: The exhaustive binary benchmark evaluates Bars & Stripes patterns with product and grid-CZ embeddings across 16–100 qubits.Both maps share local angle encoding; grid-CZ additionally applies nearest-neighbour CZ gates.
  • Incremental coverage: The executed reference horizons for L = 4, … , 10 are 6, 4, 3, 3, 3, 2, 2, with resource criteria becoming active before the support-8 inspection cap.These horizons were identical for both Bars & Stripes routes.
  • Incremental coverage: At 100 qubits, reference-relative coverage is 1.000000 for product and 0.999874 for grid-CZ against the declared support-≤2 reference.These values do not quantify observable geometry beyond the declared reference horizon.
  • Incremental coverage: For product embeddings, the 0.99 criterion is reached at support 3 for L = 4, 5, 6 and support 2 for L = 9, 10; L = 7, 8 stop at resource-bounded endpoints.For L = 7 and 8, C_ref(2) equals 0.978420 and 0.977275, respectively, and must not be read as convergence.
  • Incremental coverage: Grid-CZ reaches 0.99 reference coverage at support 2 for every evaluated size except L = 8, where C_ref(2) = 0.987322 before resource rejection.The next complete downstream grade at L = 8 was rejected by the configured resource policy.
  • Methodological interpretation: Resource endpoints reflect downstream cumulative feature-budget rejection after geometric analysis, not failed observable evaluation or exhausted RAM.The next exact-support block was analyzed geometrically, but its selected original observables were not admitted.
  • Declarative reconstruction: The product theory is stable at support 1, whereas grid-CZ changes between supports 1 and 2 for L = 4, 5, 7, 8 but not for L = 6, 9, 10.Thus, additional geometric directions need not change the declarative theory, while entanglement can make reconstruction support-dependent for some sizes.
  • Declarative reconstruction: Every reported relation is target-consistent, fully covered, conflict-free, and reconstructed with strict accuracy 1.0.At L = 5, all 60 exhaustive patterns are covered, and the additional support-2 structure does not alter the rule theory.

6.3. Continuous validation: Low-Depth MNIST

Low-Depth MNIST evaluates QILP-0 on all 14,708 digit-0/1 instances before and after a trained variational transformation. The comparison separates predictive performance, observable geometry, and exact reconstruction of the resulting discrete relations.

  • Dataset and representations: 14,708 digit-0/1 instances are evaluated on 11 qubits using the depth-4 preparation and its trained-VQC transformation.The dataset contains 6,912 zeros and 7,796 ones; the preparation uses 171 input-dependent rotations and 251 fixed operations.
  • Training and geometry: 0.465330 to 0.971108: validation accuracy increases after supervised variational training on the fixed 80/20 split.This predictive metric is distinct from QILP-0’s later strict reconstruction accuracy.
  • Training and geometry: 0.355399 to 0.667402: support 1 coverage increases, while accumulated support 2 coverage rises from 0.809468 to 0.901905 after training.Coverage is interpreted relative to the declared observational reference and reflects the evaluated normalized observable-response geometry.
  • Training and geometry: 1,819 to 2,092: accumulated retained geometric dimension increases, while selected original observables rise from 2,168 to 2,498.The final coverages are almost identical over the complete evaluated support-3 horizon; this does not measure circuit complexity.
  • Declarative reconstruction: 1.0 strict reconstruction accuracy is achieved in every evaluated accumulated-support layer, with complete and conflict-free theories over all 14,708 rows.The certified relation is the finite discrete observational relation, not the VQC’s held-out predictive task.
  • Declarative reconstruction: Final rule bodies use observables of q0 and q10, while intermediate-qubit observables do not occur in those bodies.This restriction concerns final symbolic rules, not the relevance of intermediate qubits to continuous observational geometry.
  • Declarative reconstruction: The post-VQC theory is more extensively instantiated, although its observable-level schemas change only from 21 to 24.Independently fitted agnostic discretization changes available states and observable-state atoms, so raw rule counts require care.
  • Discretization audit: Every selected continuous column passes admissible-discretization checks, while sample-sufficiency warnings remain diagnostic rather than certificate failures.No selected column is constant, cuts do not collapse numerically, every effective bin is occupied, and minimum occupancy is positive.

6.4. Cross-experiment certification

The cross-experiment certificate applies one auditable reconstruction contract to native-discrete Bars & Stripes relations and continuously discretized Low-Depth MNIST relations. Within each declared observational scope, the reported relations support exact observational declarative twins.

  • Cross-experiment scope: Bars & Stripes exercises preservation of native discrete states, whereas Low-Depth MNIST exercises the continuous admissible-discretization contract.
  • Common certificate: All four experimental routes use target-independent symbolic-vocabulary construction, complete row-level audits, and exact PRIDE reconstruction.The certificate fields are shared across the reported routes.
  • Certification outcome: Within each declared finite observational scope, the results support exact observational declarative twins for the corresponding reported discrete relations.Coverage metadata remain separate and relative to each run’s declared observational reference horizon.

6.5. Interpreting certificates beyond the exact reported cases

Beyond the exact reported cases, QILP-0 distinguishes logical reconstruction from uncertainty in observable values and from failures of target-consistency. Approximate twins would require a separate explicit approximation contract.

  • Estimated observations: Estimated observable values can still support exact reconstruction of a target-consistent reported discretized relation when uncertainty and resolution metadata are retained.
  • Twin-admissibility boundary: If identical discrete observational states have incompatible targets, QILP-0 preserves the relation but makes no exact observational-twin claim.The consistency failure prevents deterministic target reconstruction for that configuration.
  • Scope boundary: A resource-limited traversal changes the declared observational scope without making the logical reconstruction approximate by itself.
  • Approximation boundary: A certified approximate observational twin would require an explicit contract specifying admitted discrepancies and their bounds; the reported exact experiments make no such claim.

6.6. Additional implementation validation

Additional checks exercised QILP-0 across heterogeneous circuit families, while the reported figures document the two main embedding and before/after validation settings.

  • Additional implementation checks: Additional implementation checks covered MQT Bench, TFIM, Heisenberg, and increasing-size Bars & Stripes circuit families.These exploratory checks exercised exact-support generation, incremental accumulation, target-independent geometry, and related prototype invariants.
  • Scope of reported evidence: Quantitative claims are restricted to the controlled Bars & Stripes and Low-Depth MNIST studies.The exploratory configurations were not all executed under a single comparative protocol.
  • Embedding configurations: Product embeddings independently encode binary pixels with RY rotations, without two-qubit interactions.The illustrative 3×3 layout differs from the reported L=4,…,10 experiments.
  • Embedding configurations: Grid-CZ embeddings add CZ gates across horizontal and vertical nearest-neighbour edges after the same local encoding.The reported experiments use corresponding L×L connectivity for L=4,…,10.
  • Observational validation: Figure 3 measures incremental reference-relative coverage, while Figure 5 compares coverage, retained geometric dimension, and selected original observables.The coverage plot marks the 0.99 criterion and identifies candidate supports rejected by the downstream feature budget.
  • Low-Depth MNIST validation: The Low-Depth MNIST architecture combines an 11-qubit depth-4 preparation with a trained variational classifier.The complete d4 layout contains 171 input-dependent RY rotations and 80 CNOT gates; the VQC block is applied over ten nearest-neighbour pairs in four sweeps.

7. Conclusions and Future Work

The paper establishes QILP-0 as an auditable, finite observational declarative twin construction, while delimiting its exactness to declared scopes and identifying scalability and execution uncertainties for future work.

  • Conclusions: QILP-0 provides a constructive affirmative answer for translating quantum-circuit behaviour into a finite declarative object within a declared observational scope.Its exactness concerns the finite task-conditioned discrete relation produced under that scope.
  • Conclusions: The continuous layer organizes target-independent observable geometry and maps retained latent structure back to semantically traceable original observables.SVD organizes the present numerical geometry without replacing the original observable vocabulary.
  • Experimental conclusions: Bars & Stripes and Low-Depth MNIST exercise complementary native-discrete and continuous branches across embedding and trained-representation settings.Bars & Stripes spans 16–100 qubits, while MNIST analyzes all 14,708 digit-0/1 instances before and after a trained VQC.
  • Conclusions: The results support separating geometric refinement from declarative refinement while retaining a logical vocabulary tied to original observables.Additional observational directions need not change the induced logical theory.
  • Limitations: QILP-0 is currently an order-0 finite multi-valued propositional construction certified only for finite datasets and declared observational reference horizons.For Pauli observables, K_ref may be smaller than the physical maximum K_phys=n, so unobserved higher-support geometry is not quantified.
  • Limitations: Full Pauli traversal has exponential growth when support is unrestricted, and the reported 100-qubit Bars & Stripes executions rely on embedding-specific algebraic structure.Those executions do not establish equivalent cost for arbitrary 100-qubit circuits.
  • Limitations: Reference horizons and downstream endpoints depend on provider, data, execution, and budget constraints.A downstream policy may still yield K_end < K_ref after the reference is fixed.
  • Future work: The two main studies use exact observable evaluation, leaving finite-shot, device-noise, hardware-error, and controlled discretization comparisons for future work.The framework records uncertainty metadata, but systematic hardware execution assessment remains outstanding.

Data and code availability

The reproducibility package provides an inspectable QILP-0 execution, with a deliberately small didactic Bars & Stripes configuration and a clean-environment reproduction record.

  • Package scope: The package exercises the complete QILP-0 path from target-independent acquisition through geometry, discretization, PRIDE induction, and certification.Outputs retain original observables and row-level observational-twin audits.
  • Didactic configuration: The executable uses an exhaustive non-uniform 3×3 Bars & Stripes instance with K_ref=2 as a didactic configuration.It demonstrates the contract after the observational reference horizon is supplied rather than reproducing the larger experiments’ resource policy.
  • Reproduction result: A clean-environment reproduction covers all 12 rows without conflicts, achieves strict reconstruction accuracy equal to one, and reproduces the reference theory hash.PRIDE is accessed through the external PyLFIT 0.5.1 package.
  • Availability: Release v1.0.0 is archived in Zenodo and the reproducibility package is available from the project repository.The passage supplies both repository and persistent-identifier locations.

CRediT authorship contribution statement

The CRediT statement assigns software, validation, writing, conceptualization, and methodology contributions across the four authors.

  • Author contributions: Marina de la Cruz Echeandía contributed software, validation, and writing.
  • Author contributions: César Luis Alonso contributed conceptualization, methodology, validation, and writing.
  • Author contributions: Tony Ribeiro contributed software, validation, and writing.
  • Author contributions: Alfonso Ortega de la Puente contributed conceptualization, methodology, software, validation, and writing.

Funding

The research was funded by the European Union, while the authors retain responsibility for the expressed views and opinions.

  • The research was funded by the European Union.
  • The authors’ views and opinions do not necessarily reflect those of the European Union or ERCEA.
  • Neither the European Union nor the granting authority can be held responsible for the stated views and opinions.
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