Feedback-based methods have gained significant attention as an alternative training paradigm for the Quantum Approximate Optimization Algorithm (QAOA) in solving combinatorial optimization problems such as MAX-CUT. In particular, Quantum Lyapunov Control (QLC) employs feedback-driven control laws that guarantee monotonic non-decreasing objective values, can substantially reduce the training overhead of QAOA, and mitigate barren plateaus. However, these methods might require long control sequences, leading to sub-optimal convergence rates. In this work, we propose a hybrid method that incorporates per-layer gradient estimation to accelerate the convergence of QLC while preserving its low training overhead and stability guarantees. By leveraging layer-wise gradient information, the proposed approach selects near-optimal control parameters, resulting in significantly faster convergence and improved robustness. We validate the effectiveness of the method through extensive numerical experiments across a range of problem instances and optimization settings.
Feedback-based methods have gained significant attention as an alternative training paradigm for the Quantum Approximate Optimization Algorithm (QAOA) in solving combinatorial optimization problems such as MAX-CUT. In particular, Quantum Lyapunov Control (QLC) employs feedback-driven control laws that guarantee monotonic non-decreasing objective values, can substantially reduce the training overhead of QAOA, and mitigate barren plateaus. However, these methods might require long control sequences, leading to sub-optimal convergence rates. In this work, we propose a hybrid method that incorporates per-layer gradient estimation to accelerate the convergence of QLC while preserving its low training overhead and stability guarantees. By leveraging layer-wise gradient information, the proposed approach selects near-optimal control parameters, resulting in significantly faster convergence and improved robustness. We validate the effectiveness of the method through extensive numerical experiments across a range of problem instances and optimization settings.
Abstract The paper introduces a novel hybrid Feedback-Based Quantum Optimization (FBQO) framework that integrates reinforcement learning (RL) for adaptive parameter control and Kalman filters for noise mitigation to enhance quantum optimization on noisy intermediate-scale quantum (NISQ) devices. Unlike traditional quantum Approximate optimization algorithm (QAOA), the proposed method dynamically tunes parameters through learned policies and statistically filtered feedback, enabling faster convergence and greater noise resilience. Key contributions include the formulation of quantum optimization as a Markov Decision Process, integration of deep quantum networks (DQNs) for adaptive control, and the use of Kalman filtering for robust state estimation. Experimental results on the Max-Cut problem demonstrate superior performance in convergence rate, stability, and optimization accuracy over existing techniques. The main finding is that RL-FBQO achieves convergence in 10–20 iterations, compared to 20–30 for FBQO and 40–50 for QAOA, with superior stability.
Optimisation of training still faces challenges due to an ill-conditioned loss surface and noisy gradient signals; at the same time, limited computational resources also restrict training options. Quantum Gradient Descent as a Hybrid Optimisation Strategy for Neural Network Training. The proposed noise-resilient quantum gradient descent framework maps selected gradient components to parameterised quantum circuits, estimates directional derivatives via parameter-shift measurements, and returns stable updates to the classical optimizer. It is not directly substituted for backpropagation; instead, it is a type of controllable gradient module that can be added to sensitive layers or low-dimensional projection heads. Add adaptive shot scheduling, damped momentum and layer-wise trust constraints to reduce estimator variance and prevent unstable quantum updates. Representative classification tasks are selected as the subjects of experiments for convergence speed, validation accuracy, gradient fidelity, measurement cost and noise robustness. Based on the above analysis, the new optimizer outperforms standard SGD in terms of final accuracy by 1.6% - 3.2% when using the same number of epochs; at the same time, it reduces the median gradient variance by 28.4% and maintains stable convergence even with an increase in simulated depolarizing noise from 0.001 to 0.015. Based on the above, it can be seen that Quantum Gradient Descent is not an all-purpose optimizer but rather a target-specific training accelerator for gradient-sensitive modules that can trade off sampling cost for more informative descent directions in measurement-aware update design.
The prospect of using quantum computers to solve combinatorial optimization problems via the quantum approximate optimization algorithm (QAOA) has attracted considerable interest in recent years. However, a key limitation associated with QAOA is the need to classically optimize over a set of quantum circuit parameters. This classical optimization can have significant associated costs and challenges. Here we provide an expanded description of Lyapunov control-inspired strategies for quantum optimization, as presented in [Magann et al., Phys. Rev. Lett. 129, 250502 (2022)], that do not require any classical optimization effort. Instead, these strategies utilize feedback from qubit measurements to assign values to the quantum circuit parameters in a deterministic manner, such that the combinatorial optimization problem solution improves monotonically with the quantum circuit depth. Numerical analyses are presented that investigate the utility of these strategies towards MaxCut on weighted and unweighted 3-regular graphs, both in ideal implementations and in the presence of measurement noise. We also discuss how how these strategies compare with QAOA and how they may be used to seed QAOA optimizations in order to improve performance for near-term applications, and we explore connections to quantum annealing.
Quantum optimization, for both classical and quantum functions, is one of the most well-studied applications of quantum computing, but recent trends have relied on hybrid methods that push much of the fine-tuning off onto costly classical algorithms. Feedback-based quantum algorithms, such as FALQON, avoid these fine-tuning problems but at the cost of additional circuit depth and a lack of convergence guarantees. In this work we take the local greedy information collected by Lyapunov feedback control and develop an analytic framework to use it to perturbatively update previous control layers, similar to the global optimal control achievable using Pontryagin optimal control. This perturbative methodology, which we call feedback-based optimally controlled quantum states (FOCQS), can be used to improve the results of feedback-based algorithms, like FALQON, providing better convergence but still no formal guarantees of convergence. Furthermore, this perturbative method can be used to push smooth annealing-like control protocol closer to the control optimum, even providing an iterative approach, albeit with diminishing returns. In numerical testing, we show improvements in convergence and required depth due to these methods over existing quantum feedback control methods.
Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of s t o c h a s t i c gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with k measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of k . In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values -- originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances -- and we show that in these cases k -shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain ``doubly stochastic'' gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.
Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of stochastic gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with k measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of k. In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values - originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances - and we show that in these cases k-shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain "doubly stochastic" gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.