The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for near-term quantum advantage in combinatorial optimization. A central design parameter is circuit depth p, with theoretical analyses suggesting higher p monotonically improves solution quality. This paper presents empirical evidence that this theoretical relationship does not hold on current noisy intermediate-scale quantum (NISQ) hardware for fault tree analysis applications. Across 120 QAOA executions on 60 QUBO-encoded fault tree instances (N ∈ {12, 16, 20, 24, 28, 32}) on IBM quantum hardware, circuit depths p = 1 and p = 2 were directly compared. The mean change in optimal state sampling probability from doubling circuit depth was Δmax_prob = 1.22 × 10⁻⁵ with a median Δ of zero, making the difference statistically indistinguishable from noise across all evaluated metrics. Output distribution entropy approaches the theoretical maximum (mean = 12.74 bits), indicating hardware noise dominates algorithmic signal. These findings establish that for fault tree minimal cut set identification on current NISQ devices, computational resources are more effectively allocated to increased shot counts than to circuit depth. Complete experimental artifacts with SHA-256 verification are provided, enabling full reproducibility.
The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for near-term quantum advantage in combinatorial optimization. A central design parameter is circuit depth p, with theoretical analyses suggesting higher p monotonically improves solution quality. This paper presents empirical evidence that this theoretical relationship does not hold on current noisy intermediate-scale quantum (NISQ) hardware for fault tree analysis applications. Across 120 QAOA executions on 60 QUBO-encoded fault tree instances (N ∈ {12, 16, 20, 24, 28, 32}) on IBM quantum hardware, circuit depths p = 1 and p = 2 were directly compared. The mean change in optimal state sampling probability from doubling circuit depth was Δmax_prob = 1.22 × 10⁻⁵ with a median Δ of zero, making the difference statistically indistinguishable from noise across all evaluated metrics. Output distribution entropy approaches the theoretical maximum (mean = 12.74 bits), indicating hardware noise dominates algorithmic signal. These findings establish that for fault tree minimal cut set identification on current NISQ devices, computational resources are more effectively allocated to increased shot counts than to circuit depth. Complete experimental artifacts with SHA-256 verification are provided, enabling full reproducibility.
The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for near-term quantum advantage in combinatorial optimization. A central design parameter is circuit depth p, with theoretical analyses establishing asymptotic convergence under idealized conditions. This paper presents a limits study demonstrating that this theoretical relationship does not translate to measurable improvement on current noisy intermediate-scale quantum (NISQ) hardware for fault tree analysis applications. Across 120 QAOA executions on 60 QUBO-encoded fault tree instances (N ∈ {12, 16, 20, 24, 28, 32}) on IBM Quantum hardware (ibm_torino, 133-qubit Heron processor), circuit depths p = 1 and p = 2 were directly compared. The mean change in optimal state sampling probability from doubling circuit depth was Δmax_prob = 1.22 × 10⁻⁵ with a median Δ of zero. For 8,192 shots, the binomial sampling uncertainty is O(10⁻²); the observed mean delta is therefore approximately three orders of magnitude below the shot-noise scale, making the difference statistically indistinguishable from zero. Output distribution entropy approaches the theoretical maximum of log₂(8192) ≈ 13 bits, indicating hardware noise dominates algorithmic signal. These findings establish that for fault tree minimal cut set identification on current NISQ devices, computational resources are more effectively allocated to increased shot counts than to circuit depth. Complete experimental artifacts with SHA-256 verification are provided, enabling full reproducibility.
This paper analyzes why increased QAOA circuit depth does not improve sampling outcomes on contemporary superconducting hardware. Using a fixed experimental corpus of 60 fault tree QUBO instances spanning problem sizes N = 12, 16, 20, 24, 28, and 32, we evaluate ibm_torino executions at depths p = 1 and p = 2 with 8,192 shots per circuit, for 120 circuit publications total. We connect canonical distribution metrics from the companion limits study—including maximum probability, Shannon entropy, and top-10 mass—to circuit-level indicators derived from archived circuit renderings. Across all problem sizes, increasing depth from p = 1 to p = 2 does not yield consistent concentration gains. The median improvement is exactly zero for all three metrics; mean changes are three orders of magnitude below the shot-noise scale. Entropy remains close to the theoretical maximum of log₂(8192) ≈ 13 bits, indicating noise-dominated operation. Circuit-level analysis reveals that all 10 circuits within each job share identical structure, with no depth-dependent structural diversification observed. These results support a mechanism-level interpretation: additional depth increases circuit exposure to hardware noise without producing usable structure in the sampled distribution under the studied compilation and device conditions. This paper provides audit-ready tables and figures linking depth study outcomes to circuit-level characteristics while keeping claims strictly within empirical behavior for this backend, circuit family, and execution configuration.
This paper presents a noise-scaled simulation study to characterize whether increased QAOA circuit depth provides measurable improvement for fault tree QUBO instances under varying two-qubit gate fidelity conditions. Using 60 fault tree instances with circuits loaded via QPY serialization, we executed 960 simulations spanning depths p = 1 and p = 2 across a fidelity sweep from F₂ = 0.900 to F₂ = 0.992, with fixed single-qubit fidelity F₁ = 0.9995 and readout fidelity Fro = 0.97. The primary finding is that output distributions are noise-dominated and shot-limited at all tested fidelity levels. Maximum probability values cluster near the shot floor of 1/1024, and Shannon entropy approaches the measurement ceiling of 10 bits. The median difference between depth p = 2 and depth p = 1 is exactly zero for maximum probability at all fidelity levels. No consistent improvement in distribution concentration is observed as fidelity increases within the tested range. These results establish that no depth benefit is detectable through F₂ = 0.992 under this noise model and shot budget. Practitioners should not expect depth scaling to improve output quality until gate fidelity exceeds the upper bound of this sweep. This completes the investigation arc from prior work: depth does not help on current hardware because noise accumulation exceeds algorithmic signal, and this finding persists through the tested fidelity range.
The Quantum Approximate Optimization Algorithm (QAOA) is widely regarded as a promising near-term quantum algorithm for addressing hard combinatorial optimization problems on Noisy Intermediate-Scale Quantum (NISQ) devices; however, its real-world effectiveness is severely constrained by hardware noise, including gate infidelities, decoherence, and measurement errors. This work explores an error mitigation–driven approach to enhance the performance and robustness of QAOA without altering its algorithmic structure or requiring fault-tolerant quantum resources. Measurement error mitigation and zero-noise extrapolation are incorporated into the execution and post-processing stages of QAOA circuits and evaluated using the Minimum Vertex Cover problem as a representative benchmark. Performance is assessed through approximation ratios, solution stability, and noise sensitivity across multiple circuit depths. The results indicate that error mitigation not only improves approximation accuracy but also stabilizes output distributions and delays performance degradation at increased circuit depths. Additionally, the mitigated executions demonstrate better consistency across repeated runs, highlighting improved resilience to stochastic noise. These findings emphasize that lightweight and hardware-efficient error mitigation techniques can significantly extend the practical usability of QAOA on current NISQ hardware, providing a scalable and realistic pathway for executing quantum optimization algorithms more reliably in near-term quantum computing environments.
The quantum approximate optimization algorithm (QAOA) is a promising quantum-classical hybrid technique to solve combinatorial optimization problems in near-term gate-based noisy quantum devices. In QAOA, the objective is a function of the quantum state, which itself is a function of the gate parameters of a multi-level parameterized quantum circuit (PQC). A classical optimizer varies the continuous gate parameters to generate distributions (quantum state) with significant support to the optimal solution. Even at the lowest circuit depth, QAOA offers non-trivial provable performance guarantee which is expected to increase with the circuit depth. However, the existing analysis fails to consider non-idealities in the qubit quality i.e., short lifetime and imperfect gate operations in realistic quantum hardware. In this article, we investigate the impact of various noise sources on the performance of QAOA both in simulation and on a real quantum computer from IBM. Our analyses indicate that the optimal number of stages (p-value) for any QAOA instance is limited by the noise characteristics (gate error, coherence time, etc.) of the target hardware as opposed to the current perception that higher-depth QAOA will provide monotonically better performance for a given problem compared to the low-depth implementations.
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.