This paper evaluates whether feasible-subspace-preserving variational circuit families can compete with the frozen QAOA+ baseline from Peters (2026) for Constraint-Level QUBO (CL-QUBO) fault tree minimal cut set identification. Six alternative constructions—an independently parameterized XX+YY generator family, a cost-phase-augmented variant, a transpiled-matched variant, a structure-ordered variant, an n = 6 restricted screen, and a warm-start variant—were tested against the QAOA+ comparator on a frozen pilot cohort of 14 reactor-scale fault tree subtrees (10 n = 6, 4 n = 8a instances) under noiseless statevector simulation with deterministic multi-restart optimization. The directly materialized family constructions passed feasible-subspace closure audits, and the screen-only variants inherited closure-qualified source families. These audits confirmed that the excitation-number-conserving XX+YY construction remained inside the intended fixed-Hamming-weight sector for the tested CL-QUBO instances. The failure mechanism is identified through per-instance analysis: the alternative families concentrate essentially all statevector probability (≈99.99%) on a single computational basis state that does not correspond to a minimal cut set, while QAOA+ distributes probability across the feasible subspace with sufficient structure to place MCS among the highest-ranked outcomes. The independently parameterized generator family lacks the problem-specific cost–mixer alternation that gives QAOA+ its concentration advantage, and adding a single cost phase layer does not rescue this deficit. The result is a disciplined negative finding: within the tested family definitions and frozen screening protocol, no approved alternative produced evidence strong enough to justify hardware execution. The contribution is a careful narrowing of the viable variational design space for CL-QUBO fault tree optimization, establishing that feasible-subspace preservation alone is insufficient for competitive MCS enrichment. Complete experimental artifacts with SHA-256 verification are provided.
This paper evaluates whether feasible-subspace-preserving variational circuit families can compete with the frozen QAOA+ baseline from Peters (2026) for Constraint-Level QUBO (CL-QUBO) fault tree minimal cut set identification. Six alternative constructions—an independently parameterized XX+YY generator family, a cost-phase-augmented variant, a transpiled-matched variant, a structure-ordered variant, an n = 6 restricted screen, and a warm-start variant—were tested against the QAOA+ comparator on a frozen pilot cohort of 14 reactor-scale fault tree subtrees (10 n = 6, 4 n = 8a instances) under noiseless statevector simulation with deterministic multi-restart optimization. The directly materialized family constructions passed feasible-subspace closure audits, and the screen-only variants inherited closure-qualified source families. These audits confirmed that the excitation-number-conserving XX+YY construction remained inside the intended fixed-Hamming-weight sector for the tested CL-QUBO instances. The failure mechanism is identified through per-instance analysis: the alternative families concentrate essentially all statevector probability (≈99.99%) on a single computational basis state that does not correspond to a minimal cut set, while QAOA+ distributes probability across the feasible subspace with sufficient structure to place MCS among the highest-ranked outcomes. The independently parameterized generator family lacks the problem-specific cost–mixer alternation that gives QAOA+ its concentration advantage, and adding a single cost phase layer does not rescue this deficit. The result is a disciplined negative finding: within the tested family definitions and frozen screening protocol, no approved alternative produced evidence strong enough to justify hardware execution. The contribution is a careful narrowing of the viable variational design space for CL-QUBO fault tree optimization, establishing that feasible-subspace preservation alone is insufficient for competitive MCS enrichment. Complete experimental artifacts with SHA-256 verification are provided.
This paper presents an audit-ready benchmark, comparison, and freeze decision for alternative CL-QUBO builder and variational family paths applied to fault tree minimal cut set (MCS) identification. The study evaluates whether cost-phase-augmented variational families from prior work can challenge the incumbent CL-QUBO builder path (v5) on a widened 9-case tiny validation corpus covering OR, AND, NOT, and nested gate structures. A disciplined promotion pipeline, comprising role mapping, contract extraction, execution coverage adjudication, semantic grid screening across 1,305 parameter configurations, and formal synthesis, was exercised to ensure that challengers were genuinely advanced rather than dismissed prematurely. Two evidence tiers structure the comparison: the incumbent v5 was evaluated through exact combinatorial audit (exhaustive projected-MCS-family recovery), while challengers were evaluated through best-of-grid statevector screening (top-1 MCS family membership and total MCS mass). The incumbent v5 achieved 9 of 9 primary target hits (minimum-cardinality MCS under the minimize-basic-events objective) and 8 of 9 exact all-MCS family recovery hits, with the single miss occurring on a case where the minimum-cardinality MCS is a strict subset of the full MCS family. The independent cost-phase challenger achieved 3 of 9 primary hits and 3 of 9 all-MCS top-1 hits at best-of-grid. The block-shared cost-phase challenger achieved 3 of 8 primary and 3 of 8 all-MCS top-1 hits on applicable cases, with one case (TVC_007_NOT_A) structurally not applicable because the block-shared generator schedule requires a retained prefix structure that the one-qubit case does not provide. The completion gate passed all eight checks, and v5 was frozen as the final promoted path within the widened tiny benchmark scope studied. This paper does not claim quantum advantage.
This work evaluates Constraint-Level QUBO (CL-QUBO) circuit execution on IBM Quantum hardware for 34 reactor-scale fault tree subtrees spanning 5–8 basic events, selected from the 776-member quantum-feasible candidate pool established in Paper 9 (Peters, 2026). Circuits were constructed using the CL-QUBO encoding and QAOA Alternating Operator Ansatz at depth p = 1, transpiled to both ibm_torino (133 qubits) and ibm_marrakesh (156 qubits), and executed under three IBM Runtime Sampler configurations (RL0, RL1, RL2) with identical bound parameters (β = 0.2, γ = 0.2) across all resilience levels. A total of six hardware jobs consumed 476 seconds (7.93 minutes) of quantum runtime across 8,192 shots per circuit. The results demonstrate that at this circuit scale, 10–16 logical qubits transpiling to 78–150 two-qubit gates at depths of 165–338, hardware output distributions are overwhelmingly noise-dominated by standard distributional metrics. Neither RL1 nor RL2 produces improvement; RL1 actively degrades output quality relative to RL0 on both backends. However, direct comparison of hardware top-probability outcomes against SCRAM-validated prime implicants from Paper 9 reveals substantial above-random minimal cut set (MCS) enrichment relative to the uniform-random baseline in the n = 8 groups (6.2× and 3.8× expected random rate in the top-200 outcomes), demonstrating that CL-QUBO circuits produce detectable MCS-correlated structure even in distributions that appear noise-dominated by aggregate metrics. These findings characterize the current observed performance ceiling for this CL-QUBO configuration on the tested IBM backends and provide both distributional and MCS-correctness baselines against which future hardware improvements can be measured.
This paper presents a reproducible, audit-ready quantum hardware experiment for minimal cut set (MCS) inference on reactor fault tree subtrees using IBM Quantum hardware. Eight two-qubit subtrees were selected from a Phase 2B reactor-scale fault tree corpus spanning four nuclear hazard families (EQK, LOCA, TRANS, SGTR). Classical baselines were computed with SCRAM 0.16.2, yielding two order-1 MCS per instance. Shallow p = 1 Quantum Alternating Operator Ansatz (QAOA+) circuits were constructed using the Constraint-Level QUBO (CL-QUBO) encoding from Peters (2026b), transpiled to ibm_torino, and executed as batch jobs totaling 57,344 shots across four experiments. Raw hardware results showed apparent infeasible leakage of 13.2% on the hw = 0 state. A dedicated four-circuit readout calibration protocol revealed that this leakage was almost entirely attributable to readout error: after confusion-matrix correction (κ = 1.56), the baseline experiment concentrates 98.3% of probability mass on the MCS manifold (hw = 1), with residual infeasible mass of 0.5%. Extension experiments substantially broaden the empirical foundation. Cross-pair validation on three additional qubit pairs (ibm_torino pairs 61–62 and 36–48; ibm_marrakesh pair 6–7) confirms that corrected MCS fidelity exceeds 95% across all tested physical qubit assignments (range 95.2%–98.3%), with the best pairs achieving 98.2% (ibm_marrakesh 6–7) and 97.9% (ibm_torino 61–62). A QAOA depth sweep (p ∈ {1, 2, 3}) on ibm_torino pair 0–1 quantifies noise–depth tradeoffs: because the mixer is inactive (β = 0), the observed redistribution from hw = 2 toward hw = 1 (7.8% at p = 1, 25.6% at p = 2, 40.9% at p = 3) is attributable to gate noise rather than productive mixing, with infeasible hw = 0 mass rising concurrently from 0.2% to 7.2%. A three-qubit exactly-one constraint experiment on ibm_torino triple 0–1–2 achieves 92.2% corrected hw = 1 mass from a feasible initialization, demonstrating that the CL-QUBO framework extends beyond two-qubit OR gates to richer constraint types on quantum hardware. Hard-start initialization experiments confirm that p = 1 circuits do not induce sufficient mixing to recover MCS states from non-minimal initializations, with this conclusion stable across independent replication and all tested qubit pairs. Complete experimental artifacts with SHA-256 verification are provided under the DOI above.
Quadratic Unconstrained Binary Optimization formulations for fault tree analysis encode gate logic through penalty terms, but prior work has demonstrated that these encodings fail to preserve fault tree semantics at scale: across 80 extended fault tree instances, only 7.5% of Rosenberg-based soft QUBO solutions match the constrained exact baseline, while 27.5% produce outright semantic violations. This paper develops two complementary methods to restore semantic correctness. A Hierarchical Semantic Validator and repair framework achieves 100% semantic validity across all 80 cohort instances through structure-aware bit-flip correction, improving the exact match rate from 7.5% to 28.75%. A constraint-level QUBO encoding eliminates penalty-based failures entirely by restricting optimization to the feasible subspace, reducing qubit requirements by a factor of 1.85x relative to Rosenberg encodings. Validation through Qiskit mixer circuits implementing a feasibility-preserving Quantum Alternating Operator Ansatz confirms zero probability leakage onto infeasible states for representative instances. These methods establish that semantic-preserving quantum optimization for fault tree minimal cut set identification is achievable with current algorithmic tools. Complete experimental artifacts with SHA-256 verification are provided under the DOI above.
This paper presents a formal validation framework establishing equivalence between Quadratic Unconstrained Binary Optimization (QUBO) formulations and classical minimal cut set (MCS) identification for fault tree analysis in probabilistic risk assessment. The framework develops a 21-gate hierarchical verification system that proves QUBO ground states correspond exactly to minimal cut sets computed by the SCRAM fault tree solver. The validation methodology employs exhaustive enumeration for tractable models and conservative spot checks for larger instances, achieving PASS status across all verification gates. The theoretical contribution formalizes MCS identification as a constrained Boolean satisfiability problem amenable to quantum optimization. Fault trees with OR-of-ANDs structure are encoded into QUBO form with penalty terms enforcing logic constraints. The encoding preserves semantic equivalence: valid QUBO ground states map bijectively to classical minimal cut sets. Empirical validation demonstrates the framework on a canonical test case (TOP = A ∨ B ∨ (C ∧ D)) using QAOA simulation with plateau-based convergence methodology. The quantum circuit correctly identifies valid MCS configurations with p_opt = 0.01058 (95% CI: [0.00972, 0.01151]). Hardware feasibility is demonstrated through execution on IBM's ibm_torino 133-qubit Heron processor, where all circuits passed sanity and interpretation checks despite noise distributing probability across 110-120 unique states. The framework establishes foundational infrastructure for quantum PRA research by providing: (1) formal QUBO-MCS bijection verification against ground truth, (2) a reproducible classical baseline using the SCRAM solver, (3) plateau-based convergence criteria for stochastic quantum optimization, and (4) a hierarchical verification gate system ensuring scientific defensibility. No claim of quantum advantage is made; results are evaluated solely for semantic correctness, reproducibility, and NISQ feasibility.
This paper presents a topology-aware feasibility roadmap for quantum probabilistic risk assessment (PRA) under bounded study conditions. The roadmap synthesizes three experimental phases applied to constraint-level quadratic unconstrained binary optimization (CL-QUBO) encodings of nuclear fault tree structures. Phase A establishes fidelity threshold behavior for the quantum approximate optimization algorithm with feasibility-preserving mixer (QAOA+) across four topology classes—Class A (anchored at n = 5), Class B (n = 6), Class C (n = 8, variant a), and Class D (n = 8, variant b)—derived from prior structural classification work. Phase B provides measured circuit-depth scaling evidence across target sizes from 5 to 20 basic events, including a deterministic constructor amendment that closes a previously missing Class B measured family. Phase C synthesizes measured and projected requirement rows into a 28-row requirements matrix spanning four topology classes and seven target sizes, with explicit evidence-tier marking distinguishing 19 measured rows from 9 projected rows. A public hardware capability screen maps the roadmap’s required-qubit count (equal to the number of basic events n in each encoded instance) against publicly reported qubit counts for Google Willow (105 qubits) [31], IBM Heron r2 (156 qubits) [32], IonQ Forte Enterprise (36 qubits) [33], and Quantinuum H2 (56 qubits) [34]; the required-qubit count does not exceed the published qubit counts for any mapped platform. A bounded C3 utility bridge links the roadmap outputs to local PRA importance-measure behavior across Fussell–Vesely, Risk Achievement Worth, and Birnbaum importance families, with interpretation restricted to the studied setting. The roadmap identifies topology class as the dominant differentiator of threshold and scaling behavior: Classes A and C are threshold-favorable under the tested conditions, while Classes B and D are threshold-unfavorable. Pooled treatment across topology classes would mask these real structural differences. This paper does not claim quantum advantage, does not include live IBM hardware execution, and does not make deployment readiness or commercialization forecasts.