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arXiv:2502.04021

Variational Quantum Optimization with Continuous Bandits

Marc Wanner, Johan Jonasson, Carlsson, Emil, Devdatt Dubhashi

Published Feb 6, 2025• 1 citation
arXivHTMLPDF

Abstract

We introduce a novel approach to variational Quantum algorithms (VQA) via continuous bandits. VQA are a class of hybrid Quantum-classical algorithms where the parameters of Quantum circuits are optimized by classical algorithms. Previous work has used zero and first order gradient based methods, however such algorithms suffer from the barren plateau (BP) problem where gradients and loss differences are exponentially small. We introduce an approach using bandits methods which combine global exploration with local exploitation. We show how VQA can be formulated as a best arm identification problem in a continuous space of arms with Lipschitz smoothness. While regret minimization has been addressed in this setting, existing methods for pure exploration only cover discrete spaces. We give the first results for pure exploration in a continuous setting and derive a fixed-confidence, information-theoretic, instance specific lower bound. Under certain assumptions on the expected payoff, we derive a simple algorithm, which is near-optimal with respect to our lower bound. Finally, we apply our continuous bandit algorithm to two VQA schemes: a PQC and a QAOA quantum circuit, showing that we significantly outperform the previously known state of the art methods (which used gradient based methods).

DOI
10.48550/arxiv.2502.04021

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We introduce a novel approach to variational Quantum algorithms (VQA) via continuous bandits. VQA are a class of hybrid Quantum-classical algorithms where the parameters of Quantum circuits are optimized by classical algorithms. Previous work has used zero and first order gradient based methods, however such algorithms suffer from the barren plateau (BP) problem where gradients and loss differences are exponentially small. We introduce an approach using bandits methods which combine global exploration with local exploitation. We show how VQA can be formulated as a best arm identification problem in a continuous space of arms with Lipschitz smoothness. While regret minimization has been addressed in this setting, existing methods for pure exploration only cover discrete spaces. We give the first results for pure exploration in a continuous setting and derive a fixed-confidence, information-theoretic, instance specific lower bound. Under certain assumptions on the expected payoff, we derive a simple algorithm, which is near-optimal with respect to our lower bound. Finally, we apply our continuous bandit algorithm to two VQA schemes: a PQC and a QAOA quantum circuit, showing that we significantly outperform the previously known state of the art methods (which used gradient based methods).
Variational quantum algorithms (VQAs) promise efficient use of near-term quantum computers. However, training VQAs often requires an extensive amount of time and suffers from the barren plateau problem where the magnitude of the gradients vanishes with increasing number of qubits. Here, we show how to optimally train VQAs for learning quantum states. Parameterized quantum circuits can form Gaussian kernels, which we use to derive adaptive learning rates for gradient ascent. We introduce the generalized quantum natural gradient that features stability and optimized movement in parameter space. Both methods together outperform other optimization routines in training VQAs. Our methods also excel at numerically optimizing driving protocols for quantum control problems. The gradients of the VQA do not vanish when the fidelity between the initial state and the state to be learned is bounded from below. We identify a VQA for quantum simulation with such a constraint that thus can be trained free of barren plateaus. Finally, we propose the application of Gaussian kernels for quantum machine learning.
Variational quantum algorithms (VQAs) are promising methods that leverage noisy quantum computers and classical computing techniques for practical applications. In VQAs, the classical optimizers such as gradient-based optimizers are utilized to adjust the parameters of the quantum circuit so that the objective function is minimized. However, they often suffer from the so-called vanishing gradient or barren plateau issue. On the other hand, the normalized gradient descent (NGD) method, which employs the normalized gradient vector to update the parameters, has been successfully utilized in several optimization problems. Here, we study the performance of the NGD methods in the optimization of VQAs for the first time. Our goal is two-fold. The first is to examine the effectiveness of NGD and its variants for overcoming the vanishing gradient problems. The second is to propose a new NGD that can attain the faster convergence than the ordinary NGD. We performed numerical simulations of these gradient-based optimizers in the context of quantum chemistry where VQAs are used to find the ground state of a given Hamiltonian. The results show the effective convergence property of the NGD methods in VQAs, compared to the relevant optimizers without normalization. Moreover, we make use of some normalized gradient vectors at the past iteration steps to propose the novel historical NGD that has a theoretical guarantee to accelerate the convergence speed, which is observed in the numerical experiments as well.
Variational quantum algorithms stand at the forefront of simulations on near-term and future fault-tolerant quantum devices. While most variational quantum algorithms involve only continuous optimization variables, the representational power of the variational ansatz can sometimes be significantly enhanced by adding certain discrete optimization variables, as is exemplified by the generalized quantum approximate optimization algorithm (QAOA). However, the hybrid discrete-continuous optimization problem in the generalized QAOA poses a challenge to the optimization. We propose a new algorithm called MCTS-QAOA, which combines a Monte Carlo tree search method with an improved natural policy gradient solver to optimize the discrete and continuous variables in the quantum circuit, respectively. We find that MCTS-QAOA has excellent noise-resilience properties and outperforms prior algorithms in challenging instances of the generalized QAOA.
Variational quantum algorithms (VQAs) have emerged as a leading paradigm in near-term quantum computing, yet their performance can be hindered by the so-called barren plateau problem, where gradients vanish exponentially with system size or circuit depth. While most existing VQA research employs simple Gaussian or zero-initialization schemes, classical deep learning has long benefited from sophisticated weight initialization strategies such as Xavier, He, and orthogonal initialization to improve gradient flow and expedite convergence. In this work, we systematically investigate whether these classical methods can mitigate barren plateaus in quantum circuits. We first review each initialization's theoretical grounding and outline how to adapt the notions from neural networks to VQAs. We then conduct extensive numerical experiments on various circuit architectures and optimization tasks. Our findings indicate that while the initial heuristics, inspired by classical initialization, yield moderate improvements in certain experiments, their overall benefits remain marginal. By outlining a preliminary exploration plan in this paper, we aim to offer the research community a broader perspective and accessible demonstrations. Furthermore, we propose future research directions that may be further refined by leveraging the insights gained from this work.
Variational quantum algorithms (VQAs) are a modern family of quantum algorithms designed to solve optimization problems using a quantum computer. Typically VQAs rely on a feedback loop between the quantum device and a classical optimization algorithm. The appeal of VQAs lies in their versatility, resistance to noise, and ability to demonstrate some results even with circuits of small depth. We primarily focus on the algorithm called variational quantum eigensolver (VQE), which takes a qubit Hamiltonian and returns its approximate ground state. We first present our numerical findings regarding VQE applied to two spin models and a variant of the Hubbard model. Next, we briefly touch the topic of quantum machine learning by developing a quantum classifier to partition quantum data. We further study the phenomenon of vanishing derivatives in VQAs, also known as barren plateaus phenomenon. We derive a new lower bound on variance of the derivatives, which depends on the causal structure of the ansatz circuit and the individual terms entering the Pauli decomposition of the problem Hamiltonian. In the final chapter of the thesis, we present our results on bounding the fidelity of experimentally prepared Clifford states using their parent Hamiltonians.
Exploring quantum applications of near-term quantum devices is a rapidly growing field of quantum information science with both theoretical and practical interests. A leading paradigm to establish such near-term quantum applications is variational quantum algorithms (VQAs). These algorithms use a classical optimizer to train a parameterized quantum circuit to accomplish certain tasks, where the circuits are usually randomly initialized. In this work, we prove that for a broad class of such random circuits, the variation range of the cost function via adjusting any local quantum gate within the circuit vanishes exponentially in the number of qubits with a high probability. This result can unify the restrictions on gradient-based and gradient-free optimizations in a natural manner and reveal extra harsh constraints on the training landscapes of VQAs. Hence a fundamental limitation on the trainability of VQAs is unraveled, indicating the essential mechanism of the optimization hardness in the Hilbert space with exponential dimension. We further showcase the validity of our results with numerical simulations of representative VQAs. We believe that these results would deepen our understanding of the scalability of VQAs and shed light on the search for near-term quantum applications with advantages.