Robust Algorithm Optimizes Combinatorial Problems Despite Control Input Errors

Quantum optimisation seeks efficient solutions to complex problems that are currently intractable for classical computers, with potential applications spanning logistics, materials science, and financial modelling. Mirko Legnini and Julian Berberich, alongside colleagues, investigate the resilience of a specific quantum algorithm, the Feedback-based Algorithm for Optimisation (FALQON), to imperfections in the control signals used to manipulate quantum bits, or qubits. Their analysis, detailed in the article “Robust feedback-based quantum optimisation: analysis of coherent control errors”, demonstrates asymptotic robustness against systematic errors and establishes bounds for independent errors, ultimately proposing a modified version of FALQON designed to mitigate these control inaccuracies. The Lyapunov function, central to their approach, represents a mathematical tool used to assess the stability of a dynamic system, in this case, the optimisation process.

Quantum computation is progressing rapidly, necessitating algorithms that are tolerant to inherent hardware imperfections. Researchers are currently investigating the Feedback-based Algorithm for Optimisation (FALQON) to assess its performance under realistic noise conditions, with a specific focus on coherent control errors that impact qubit manipulation. This study rigorously examines FALQON’s stability and efficiency when subjected to both systematic, predictable errors and independent, random disturbances, providing crucial insights for practical implementation on noisy intermediate-scale quantum (NISQ) devices. The findings demonstrate FALQON’s inherent robustness against predictable errors and highlight a modified version, Robust FALQON, which significantly improves performance in the presence of random noise.

The exploration begins by establishing FALQON’s foundational stability, confirming its ability to converge to a solution even when consistent errors distort control inputs. This inherent stability stems from the algorithm’s design, rooted in Lyapunov principles, which guarantees convergence despite perturbations. Researchers meticulously analyse FALQON’s behaviour under systematic errors, demonstrating its asymptotic stability, meaning it approaches a stable solution over time, even with persistent disturbances.

Further investigation delves into the more challenging scenario of independent coherent control errors, where each qubit experiences uncorrelated, random disturbances. Experiments utilising a maximum circuit depth of 200 layers reveal that a standard implementation of FALQON exhibits sensitivity to these random errors, leading to performance degradation. Researchers then introduce Robust FALQON, a modified version incorporating a penalty term within the algorithm’s objective function, designed to counteract the effects of random errors.

Robust FALQON achieves improved performance by discouraging solutions sensitive to noise, effectively enhancing its ability to find optimal or near-optimal solutions despite the presence of errors. The introduction of this penalty term functions by regularizing the learned parameters, preventing overfitting to noisy data and promoting more stable and generalizable solutions. Quantitative analysis, gathered from 50 repetitions with varying noise samples, confirms that Robust FALQON not only maintains accuracy but also potentially accelerates the convergence rate compared to the standard implementation.

The design of Robust FALQON draws inspiration from classical optimisation techniques employing regularization to enhance stability and generalisation. By controlling the strength of this penalty via the parameter λ, the algorithm achieves a balance between minimising the original cost function and maintaining robustness against noise. A value of λ = 1.0 consistently outperforms the standard FALQON (λ = 0.5) in the presence of independent coherent control errors, demonstrating the effectiveness of the parameter tuning.

These findings hold significant implications for the practical implementation of FALQON on NISQ devices, offering a pathway towards more reliable quantum algorithms for tackling complex combinatorial optimisation problems. By demonstrating resilience to coherent control errors and providing a method for further enhancement through parameter tuning, this research contributes to the development of robust quantum solutions.

The research demonstrates the asymptotic stability of FALQON when subjected to systematic coherent control errors. The algorithm maintains convergence even with consistent errors affecting the control input, although the rate of convergence may vary, highlighting the importance of error characterisation and mitigation. Investigations into independent coherent control errors reveal that standard FALQON exhibits sensitivity to random, layer-specific errors, necessitating the development of noise-resilient strategies. Robust FALQON demonstrably improves noise rejection and accelerates convergence, stemming from the incorporation of a penalty term into the algorithm’s cost function, effectively regularising the learned parameters and mitigating the impact of stochastic errors.

Quantitative analysis, supported by computational experiments, confirms the improved performance of Robust FALQON, exhibiting reduced variability in its final solution and a faster reduction in the cost function over successive layers, particularly when subjected to increasing levels of noise.

Future research will focus on extending these findings to more complex noise models and exploring the potential for combining Robust FALQON with other error mitigation techniques. Researchers plan to investigate the impact of different penalty functions and optimisation algorithms on the algorithm’s performance and robustness. Furthermore, they aim to apply Robust FALQON to a wider range of combinatorial optimisation problems, demonstrating its versatility and practical applicability. This ongoing research promises to further enhance the reliability and efficiency of quantum optimisation algorithms, paving the way for breakthroughs in various fields, including materials science, drug discovery, and financial modelling.