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arXiv:2610.36597

Effective Noise Thresholds for Reliable Quantum Approximation Optimisation Algorithm under Noisy Environment

Yozef Tjandra, Hendrik Santoso Sugiarto, Celine Vania Setiadi, Joanna Gracia Tan, Andriyan Bayu Suksmono

quant-phphysics.comp-phSubmitted Oct 27, 2026• Announced Oct 28, 2026• 0 citations
arXivHTMLPDF

Abstract

Variational quantum algorithms are leading candidates for practical utilisation on noisy intermediate-scale quantum (NISQ) processors, but accumulating noise bounds the usefulness of circuit depths. For the quantum approximate optimisation algorithm (QAOA), this trade-off has mostly been addressed by searching for the best depth for a given device. In this work, we consider the inverse question: how much noise each depth can tolerate. Using GPU-accelerated noisy QAOA simulations for MaxCut on three graph families with 6 to 18 vertices, depths up to 10 and three noise channels, we fit the approximation ratios with a generalised logistic surface. We propose the effective noise threshold as the inflection boundary of the logistic surfaces, a noise boundary separating reliable and unreliable QAOA approximation regimes. In general, the threshold decreases as depth rises, indicating that deeper circuits demand for stricter noise requirement for reliable approximation. Its best and worst values over depth divide the noise axis into three regions: every depth remains effective, depth must be chosen with care, or no depth induces effective optimisation. Assessing on the ibm_marrakesh processor, although modelled gate errors understate factual noise, the threshold modeling remains beneficial in informing qualitative depth-dependence structure and identifying dominating noise channel in noisy QAOA.

AI Summary

The paper asks how much noise a fixed QAOA depth can tolerate, rather than which depth is optimal at a given noise rate. It defines an effective threshold $r^\star(p)$ through a generalized logistic fit to performance versus depth and $-\log_{10}r$, using the fitted midpoint boundary to separate higher-performance and noise-dominated regimes. GPU-accelerated simulations study MaxCut on Erdős–Rényi, path, and complete balanced bipartite graphs with 6, 9, 12, and 18 vertices, depths $p=1,\ldots,10$, and eight noise rates from $10^{-4}$ to $0.5$. Depolarizing, bit-flip, or amplitude-damping noise is applied independently after every gate. Each configuration uses ten runs. The reported approximation ratio is the cut value of the most frequently sampled output divided by the exact optimum, with ties favoring larger cuts—not the expected cut value. Fits have mean absolute error below 0.1 in most configurations. Thresholds at $p=10$ are below those at $p=1$ in 35 of 36 configurations, with median ratio 0.33, although intermediate-depth behavior need not be monotonic. The best-to-worst threshold ratio across depths has median 3.2 and range 1.08–8.5; graph connectivity and noise channel affect these values. An 18-qubit IBM hardware assessment yields approximation ratios 0.67–0.85, below predictions based on median calibrated gate error. Stated limitations include one hardware instance, one execution per depth, a single calibration snapshot, different optimization budgets, and omitted routing, readout, and composite-noise effects. The hardware study illustrates threshold use rather than validating its quantitative predictions.

Some signs of AI-generated textIncrementalOverclaimsStrong assumptionsPossible error

Central claim may not hold: The proposed threshold condition sets a quadratic logistic predictor to zero, yielding a midpoint contour that is not generally an inflection or maximum-noise-sensitivity contour. The channel-dependent fitted midpoints define model-relative performance regimes, but do not establish a common reliability threshold or justify declaring optimisation ineffective above them. The single-instance hardware comparison cannot identify a dominating noise channel because falling below a simulated mean-minus-standard-deviation band does not rule out unital noise.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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15 pages, 8 figures

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Variational quantum algorithms are leading candidates for practical utilisation on noisy intermediate-scale quantum (NISQ) processors, but accumulating noise bounds the usefulness of circuit depths. For the quantum approximate optimisation algorithm (QAOA), this trade-off has mostly been addressed by searching for the best depth for a given device. In this work, we consider the inverse question: how much noise each depth can tolerate. Using GPU-accelerated noisy QAOA simulations for MaxCut on three graph families with 6 to 18 vertices, depths up to 10 and three noise channels, we fit the approximation ratios with a generalised logistic surface. We propose the effective noise threshold as the inflection boundary of the logistic surfaces, a noise boundary separating reliable and unreliable QAOA approximation regimes. In general, the threshold decreases as depth rises, indicating that deeper circuits demand for stricter noise requirement for reliable approximation. Its best and worst values over depth divide the noise axis into three regions: every depth remains effective, depth must be chosen with care, or no depth induces effective optimisation. Assessing on the ibm_marrakesh processor, although modelled gate errors understate factual noise, the threshold modeling remains beneficial in informing qualitative depth-dependence structure and identifying dominating noise channel in noisy QAOA.
The paper asks how much noise a fixed QAOA depth can tolerate, rather than which depth is optimal at a given noise rate. It defines an effective threshold r⋆(p)r^\star(p)r⋆(p) through a generalized logistic fit to performance versus depth and −log⁡10r-\log_{10}r−log10​r, using the fitted midpoint boundary to separate higher-performance and noise-dominated regimes.
GPU-accelerated simulations study MaxCut on Erdős–Rényi, path, and complete balanced bipartite graphs with 6, 9, 12, and 18 vertices, depths p=1,…,10p=1,\ldots,10p=1,…,10, and eight noise rates from 10−410^{-4}10−4 to 0.50.50.5. Depolarizing, bit-flip, or amplitude-damping noise is applied independently after every gate. Each configuration uses ten runs. The reported approximation ratio is the cut value of the most frequently sampled output divided by the exact optimum, with ties favoring larger cuts—not the expected cut value.
Fits have mean absolute error below 0.1 in most configurations. Thresholds at p=10p=10p=10 are below those at p=1p=1p=1 in 35 of 36 configurations, with median ratio 0.33, although intermediate-depth behavior need not be monotonic. The best-to-worst threshold ratio across depths has median 3.2 and range 1.08–8.5; graph connectivity and noise channel affect these values.
An 18-qubit IBM hardware assessment yields approximation ratios 0.67–0.85, below predictions based on median calibrated gate error. Stated limitations include one hardware instance, one execution per depth, a single calibration snapshot, different optimization budgets, and omitted routing, readout, and composite-noise effects. The hardware study illustrates threshold use rather than validating its quantitative predictions.
Abstract In the noisy intermediate-scale quantum (NISQ) era, quantum approximate optimisation algorithms are important tools for solving NP-hard problems such as combinatorial optimisation. However, most QAOA implementations ignore the inherent noise in quantum circuit environments. This study proposes a noisy quantum approximate optimisation algorithm (NQAOA) and analyses its performance under noisy conditions through numerical simulations. We compare the approximate ratios (AR) of various graphs under different optimisation strategies and noise conditions, and assess the impact of circuit depth, node count, edge count, and noise coefficient on the maximum cut (MaxCut). Experimental results show that noise leads to overfitting, and the approximate ratio ( AR ) exhibits fluctuating changes as the edge count e increases. The AR values under ideal conditions and the expected results are both superior to those under noisy conditions. This study is the first to systematically integrate multiple types of quantum noise into the QAOA framework, quantitatively analysing the synergistic effects of noise type, optimiser performance, and graph structure. Through theoretical derivations and numerical simulations, we established linear/exponential theoretical boundaries for AR decay related to noise coefficients, providing a critical reference for designing noise-resistant quantum optimisation algorithms on NISQ devices.
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 solving combinatorial optimization problems on near-term quantum hardware. However, its practical deployment remains severely limited by hardware-induced noise and decoherence. In the absence of full fault tolerance, Quantum Error Mitigation (QEM) techniques offer a viable approach to enhance QAOA performance on noisy intermediate-scale quantum (NISQ) devices. In this work, we present a unified classification of QEM strategies specifically tailored for QAOA, systematically categorized into preprocessing, during execution, and post-processing phases. We further classify QAOA deployment settings—such as hardware-constrained devices, stochastic noise-dominated environments, and dense graphs problem—and recommend appropriate QEM techniques for each. A comparative analysis highlights the trade-offs among mitigation accuracy, resource overhead, and robustness to different noise types. By synthesizing recent advances and identifying open challenges, our paper provides both a comprehensive survey and a practical framework for designing robust, noise-resilient QAOA pipelines on near-term quantum platforms.
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 variational quantum algorithm for Near-term Intermediate-Scale Quantum computers (NISQ) providing approximate solutions for combinatorial optimization problems. The QAOA utilizes a quantum-classical loop, consisting of a quantum ansatz and a classical optimizer, to minimize some cost function, computed on the quantum device. This paper presents an investigation into the impact of realistic noise on the classical optimizer and the determination of optimal circuit depth for the Quantum Approximate Optimization Algorithm (QAOA) in the presence of noise. We find that, while there is no significant difference in the performance of classical optimizers in a state vector simulation, the Adam and AMSGrad optimizers perform best in the presence of shot noise. Under the conditions of real noise, the SPSA optimizer, along with ADAM and AMSGrad, emerge as the top performers. The study also reveals that the quality of solutions to some 5 qubit minimum vertex cover problems increases for up to around six layers in the QAOA circuit, after which it begins to decline. This analysis shows that increasing the number of layers in the QAOA in an attempt to increase accuracy may not work well in a noisy device.
Structured variational quantum algorithms such as the Quantum Approximate Optimisation Algorithm (QAOA) have emerged as leading candidates for exploiting advantages of near-term quantum hardware. They interlace classical computation, in particular optimisation of variational parameters, with quantum-specific routines, and combine problem-specific advantages -- sometimes even provable -- with adaptability to the constraints of noisy, intermediate-scale quantum (NISQ) devices. While circuit depth can be parametrically increased and is known to improve performance in an ideal (noiseless) setting, on realistic hardware greater depth exacerbates noise: The overall quality of results depends critically on both, variational parameters and circuit depth. Although identifying optimal parameters is NP-hard, prior work has suggested that they may exhibit regular, predictable patterns for increasingly deep circuits and depending on the studied class of problems. In this work, we systematically investigate the role of classical parameters in QAOA performance through extensive numerical simulations and suggest a simple, yet effective heuristic scheme to find good parameters for low-depth circuits. Our results demonstrate that: (i) high-quality parameters often deviate substantially from expected patterns; (ii) QAOA performance becomes progressively less sensitive to specific parameter choices as depth increases; and (iii) iterative component-wise fixing performs on par with, and at shallow depth may even outperform, several established parameter-selection strategies. We identify conditions under which structured parameter patterns emerge, and when deviations from the patterns warrant further consideration. These insights for low-depth circuits may inform more robust pathways to harnessing QAOA in realistic quantum computing scenarios.