We study how hardware noise changes the p=1 Quantum Approximate Optimization Algorithm (QAOA) landscape for cardinality-constrained portfolio problems on ibm_fez (IBM Heron r2): seven instances from n=6 to 16 qubits, each on a 7x7 parameter grid at 4,096 and 8,192 shots per point. The first version of this paper proposed Landscape Span Compression (LSC), the fractional loss of energy range across the grid, as a device benchmark, based on stable values of 0.24-0.30 at n<=8. The larger study shows why that does not hold. Hardware LSC rises to 0.72-0.81 at n=16, growing with circuit size (R^2=0.96 and 0.88 for an exponential decay in CZ count). Because the landscape energy averages over all shots, it is nearly linear in the feasible fraction, and a statistic built only from that fraction reproduces LSC (Pearson r=0.954). For these penalty-encoded problems, landscape compression is a circuit-fidelity effect expressed as loss of feasible samples, as noise-flattening models of QAOA predict. The same mechanism biases the usual test of parameter transfer: grid verdicts on whether the optimum moves agree across shot counts for only 2 of 7 instances, and hardware optima move toward higher feasibility. Even in noiseless simulation, p=1 QAOA samples the feasible set of these instances almost uniformly, so best-feasible approximation ratios (>=0.9993) are uninformative and the only effect noise can remove is a raised feasible fraction. We withdraw the benchmark and noise-model-gap claims of the first version, correct its solution-quality figures, and describe a protocol that separates feasibility loss from solution quality.
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7 pages, 4 figures, 7 tables. v2: substantially revised after a scaling study to n=16; withdraws the device-benchmark claim of v1 (changes listed in Sec. V). Data: doi:10.5281/zenodo.23040298. Code: https://github.com/dmeliksetian/qaoa-noise-landscape
We study how hardware noise changes the p=1 Quantum Approximate Optimization Algorithm (QAOA) landscape for cardinality-constrained portfolio problems on ibm_fez (IBM Heron r2): seven instances from n=6 to 16 qubits, each on a 7x7 parameter grid at 4,096 and 8,192 shots per point. The first version of this paper proposed Landscape Span Compression (LSC), the fractional loss of energy range across the grid, as a device benchmark, based on stable values of 0.24-0.30 at n<=8. The larger study shows why that does not hold. Hardware LSC rises to 0.72-0.81 at n=16, growing with circuit size (R^2=0.96 and 0.88 for an exponential decay in CZ count). Because the landscape energy averages over all shots, it is nearly linear in the feasible fraction, and a statistic built only from that fraction reproduces LSC (Pearson r=0.954). For these penalty-encoded problems, landscape compression is a circuit-fidelity effect expressed as loss of feasible samples, as noise-flattening models of QAOA predict. The same mechanism biases the usual test of parameter transfer: grid verdicts on whether the optimum moves agree across shot counts for only 2 of 7 instances, and hardware optima move toward higher feasibility. Even in noiseless simulation, p=1 QAOA samples the feasible set of these instances almost uniformly, so best-feasible approximation ratios (>=0.9993) are uninformative and the only effect noise can remove is a raised feasible fraction. We withdraw the benchmark and noise-model-gap claims of the first version, correct its solution-quality figures, and describe a protocol that separates feasibility loss from solution quality.
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.
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.
Increasing renewable-energy penetration heightens power-system variability and complicates disturbance containment. Controlled islanding mitigates cascading failures by partitioning a stressed network to limit disrupted power transfer while preserving each island's operational integrity, but this constrained partitioning problem is NP-hard. Although QAOA offers a complementary search strategy, limited near-term qubit capacity restricts conventional formulations. This paper presents a qubit-efficient hybrid quantum framework combining a physics-informed compact encoding with Lagrangian constraint handling and classical feasibility refinement. The encoding exploits grid structure while formally preserving the original feasible solution space and objective. For a fixed island count on sparse working graphs, the formulation reduces phase-separator and per-layer gate complexity from quadratic to linear scaling with system size. Tests on eight IEEE systems ranging from 9 to 89 buses and multiple quantum-provider backends produce feasible, high-quality islanding solutions under practical circuit and sampling budgets. Factorial ablation attributes resource and runtime improvements to the complementary effects of compact encoding and qubit-efficient constraint handling. Noise analysis shows stable solution quality under tested device noise, while landscape diagnostics reveal smoother, more consistently scaled QAOA cost surfaces and improved parameter-optimization behavior. These results offer a transferable approach for scaling constrained quantum optimization toward larger real-world applications on near-term hardware.
Zero-noise extrapolation (ZNE) is usually validated by the accuracy of mitigated observables, yet variational quantum optimization uses those observables to select parameters. We ask whether, and by how much, an accuracy gain from ZNE coincides with a worse decision. In depth-one simulations of the quantum approximate optimization algorithm (QAOA) for MaxCut on four to six qubits, each sampled 120×120 landscape yields both its mean-squared error (MSE) and the ideal regret of its selected grid minimum, comparing Richardson orders one to three with unmitigated estimation at the same total shot budget per grid point. In an exploratory screen of 675 conditions, whose per-point random seeds collide, a familywise (Holm) analysis retains only four of 540 replication conditions as showing both directional MSE improvement and regret increase. A frozen confirmation with 100 fresh paired repetitions per condition reproduces all four: median MSE improvements of 7.65-73.09% accompany median normalized regret increases of only 0.0031-0.0073. We distinguish a bias route, in which residual bias reorders the mean landscape, from a variance route, in which Richardson weights amplify finite-shot noise. Exact first and second moments on a bounded control grid find no bias-route reversal: mitigated mean landscapes preserve the ideal minima, and ZNE can more than halve expected MSE, yet finite-shot selections incur higher mean regret. Decision losses that survive strict testing are few and small in this benchmark, but reproducible and not revealed by aggregate MSE. We present a paired benchmark and recommend reporting decision regret alongside estimation error when mitigation is assessed for optimization. Results concern simulated exhaustive grids, not hardware or optimizer loops.
On a cloud-accessed quantum processor the classical optimizer of a variational algorithm communicates with the device through submitted jobs, and each job carries a queueing and turnaround overhead that does not depend on how many circuits it contains. We study a simple consequence of this for the quantum approximate optimization algorithm (QAOA). An optimizer whose next set of trial parameters is known before any result returns can evaluate the whole set in one job, whereas a sequential optimizer such as COBYLA spends one job per function evaluation. We compare a batched pattern search with COBYLA on IBM's \ibmfez\ processor for cardinality-constrained portfolio selection with six to twelve assets, giving both optimizers the same number of jobs and the same number of shots. At every size the batched search reaches, after its first job, a parameter quality that the serial optimizer takes several jobs to match, and the two methods converge to comparable final values; repeating the eight-asset comparison from four starting points gives the same ordering each time. The instances are small enough to be solved exactly, which lets us check that the device's output distribution is correlated with the true ranking of portfolios and is clearly separated from a control circuit of the same depth with the cost layer removed. A short scan over circuit depth at classically optimal parameters shows that on this device the measured solution quality peaks at two or three QAOA layers. The protocol is described in enough detail to be reproduced, and the notebooks and raw counts are released.
Despite rapid progress in quantum optimization, broad real-hardware benchmarks comparing multiple algorithmic families across diverse combinatorial problems under a common protocol remain limited. We benchmark gate-based quantum optimization on four NP-hard problems: multi-dimensional knapsack (MDKP), maximum independent set (MIS), quadratic assignment (QAP), and market-share (MSP). We study VQE, CVaR-VQE, standard, multi-angle, and warm-start QAOA, together with qubit-efficient PCE and QRAO, on IBM Heron r1/r2 processors using resilience-level-2 mitigation. To our knowledge, this includes the first real-hardware QRAO results and the first multi-problem PCE hardware benchmark.
Across 247 method-instance combinations, we report transpiled circuit size, hardware outcomes, and an independent-error gate-count fidelity proxy, Fest. For MDKP and MIS, an empirical operating point near Fest≈0.1, corresponding to about 770 two-qubit gates at the median Heron-r2 CZ error rate, marks the onset of noise-dominated execution. QAP exposes a separate bottleneck: dense one-hot encodings and an exponentially sparse feasible manifold, with feasible fraction 10!/2100 at n=10; no tested hardware method produces a feasible assignment. Compiled QAOA-family circuits are generally noise dominated, and a matched uniform-random control shows that most feasible low-fidelity outcomes fall within the random range, apart from one finite-sample MIS warm-start exception. A SWAP-aware, fractional-gate, Nighthawk-topology compilation counterfactual reduces two-qubit counts but leaves all circuits below Fest=10−3. These conclusions apply to the tested implementations rather than QAOA in general. Qubit-efficient methods extend runnable instance sizes, but only within the empirical fidelity budget.