Variational quantum algorithms (VQAs) are a promising approach for leveraging near-term quantum computers, but trainability and quantum-resource costs remain major challenges. Here we address these challenges with the \textbf{H}ardware \textbf{E}fficient and dynamical \textbf{LI}e algebra supported \textbf{A}nsatz (HELIA) and two training methods combining the classical-enhanced \g method with the quantum-based Parameter-Shift Rule (PSR). This distributes gradient estimation and training across classical and quantum hardware. We evaluate ground-state estimation of 6- to 18-qubit Hamiltonians using the Variational Quantum Eigensolver (VQE) and quantum phase classification for Hamiltonians up to 12 qubits. For VQE, our methods improve accuracy and success rates, with a mean reduction in estimated logical quantum evaluations of up to $60\%$ relative to Full-PSR training of the same ansatz. For classification, the largest improvement in mean peak test accuracy relative to Full-PSR is $2.8$ percentage points. We also use these methods as a warm-start pretraining strategy for Hamiltonians with exponential-dimensional dynamical Lie algebras. Gradient diagnostics for the studied Hamiltonian configurations show slower gradient-variance decay than the selected deep hardware-efficient reference over the simulated system sizes, potentially paving the way for training large-qubit quantum models.
Variational quantum algorithms (VQAs) are a promising approach for leveraging near-term quantum computers, but trainability and quantum-resource costs remain major challenges. Here we address these challenges with the \textbf{H}ardware \textbf{E}fficient and dynamical \textbf{LI}e algebra supported \textbf{A}nsatz (HELIA) and two training methods combining the classical-enhanced \g method with the quantum-based Parameter-Shift Rule (PSR). This distributes gradient estimation and training across classical and quantum hardware. We evaluate ground-state estimation of 6- to 18-qubit Hamiltonians using the Variational Quantum Eigensolver (VQE) and quantum phase classification for Hamiltonians up to 12 qubits. For VQE, our methods improve accuracy and success rates, with a mean reduction in estimated logical quantum evaluations of up to 60% relative to Full-PSR training of the same ansatz. For classification, the largest improvement in mean peak test accuracy relative to Full-PSR is 2.8 percentage points. We also use these methods as a warm-start pretraining strategy for Hamiltonians with exponential-dimensional dynamical Lie algebras. Gradient diagnostics for the studied Hamiltonian configurations show slower gradient-variance decay than the selected deep hardware-efficient reference over the simulated system sizes, potentially paving the way for training large-qubit quantum models.
We introduce a general framework called neural-network- (NN) encoded variational quantum algorithms (VQAs), or NNVQA for short, to address the challenges of implementing VQAs on noisy intermediate-scale quantum (NISQ) computers. Specifically, NNVQA feeds input (such as parameters of a Hamiltonian) from a given problem to a neural network and uses its outputs to parameterize an ansatz circuit for the standard VQA. Combining the strengths of NN and parameterized quantum circuits, NNVQA can accelerate the training process of VQAs and handle a broad family of related problems with varying input parameters with the pretrained NN. To concretely illustrate the merits of NNVQA, we present results on a NN variational quantum eigensolver (VQE) for solving the ground state of parameterized XXZ spin models in one and two dimensions. Our results demonstrate that NNVQE is able to estimate the ground-state energies of parameterized Hamiltonians with high precision without fine tuning, and significantly reduce the overall training cost to estimate ground-state properties across the phases of the given Hamiltonian. We also employ an active learning strategy to further increase the training efficiency while maintaining prediction accuracy. These encouraging results demonstrate that NNVQAs offer an alternative hybrid quantum-classical paradigm to utilize NISQ resources for solving more realistic and challenging computational problems.
Variational quantum algorithms, such as the variational quantum eigensolver (VQE), have a very high cost due to the large number of measurement outcomes that must be acquired from the quantum device during training. This is due in large part to the use of the parameter shift rule for estimating the gradient at each step, which requires two circuit evaluations per parameter. It was recently shown that if all the gate generators in the ansatz commute with each other, then the complete gradient vector can be found from a single evaluation of an augmented circuit (Bowles et al., 2023), and this has the potential to significantly reduce the quantum cost. Unfortunately, this is quite a severe restriction, as it prevents the ansatz from having a nontrivial dynamical Lie algebra (DLA). In order to obtain a more expressive ansatz than permitted by the commutation requirement, we construct it iteratively and train it in a layerwise fashion using the parallel gradient method. For VQE training of an 8-qubit transverse field Ising model (TFIM) on an ideal simulator, we show that this approach is able to reach a lower energy using fewer circuit evaluations than the parameter shift rule with a more conventional ansatz.
The exponential runtime of quantum simulators on classical machines, along with long queue times and high costs associated with real quantum devices, presents significant challenges for the efficient optimization of Variational Quantum Algorithms (VQAs) such as the Variational Quantum Eigensolver (VQE), Quantum Approximate Optimization Algorithm (QAOA), and Quantum Neural Networks (QNNs). To address these limitations, we propose a novel approach, DyPP (Dynamic Parameter Prediction), which accelerates the convergence of VQAs by leveraging regular trends in the parameter weights to update parameters efficiently. We introduce two techniques for optimal prediction performance: Naive Prediction (NaP) and Adaptive Prediction (AdaP). Through extensive experimentation and training of multiple QNN models on various datasets, we demonstrate that DyPP achieves a speedup of approximately 2.25×compared to standard training methods, while also providing improved accuracy (up to 2.3% higher) and lower loss (up to 6.1% reduction) with minimal storage and computational overhead. We further evaluate DyPP’s effectiveness in VQE for molecular ground-state energy estimation and in QAOA for graph MaxCut. Our results show that, on average, DyPP leads to a speedup of up to 3.1× for VQE and 2.91× for QAOA compared to traditional optimization, while requiring up to 3.3× fewer shots (i.e., repeated circuit executions). Even in the presence of hardware noise, DyPP outperforms existing optimization techniques, delivering up to 3.33× speedup and requiring 2.5× fewer shots, thereby enhancing the efficiency of VQAs.
Abstract Many near-term quantum computing algorithms are conceived as variational quantum algorithms, in which parameterized quantum circuits are optimized in a hybrid quantum–classical setup. Examples are variational quantum eigensolvers, quantum approximate optimization algorithms as well as various algorithms in the context of quantum-assisted machine learning. A common bottleneck of any such algorithm is constituted by the optimization of the variational parameters. A popular set of optimization methods work on the estimate of the gradient, obtained by means of circuit evaluations. We will refer to the way in which one can combine these circuit evaluations as gradient rules. This work provides a comprehensive picture of the family of gradient rules that vary parameters of quantum gates individually. The most prominent known members of this family are the parameter shift rule (PSR) and the finite differences method. To unite this family, we propose a generalized PSR that expresses all members of the aforementioned family as special cases, and discuss how all of these can be seen as providing access to a linear combination of exact first- and second-order derivatives. We further prove that a PSR with one non-shifted evaluation and only one shifted circuit evaluation does not exist, and introduce a novel perspective for approaching new gradient rules.
In this study, we investigated the Variational Quantum Eigensolver (VQE) application for the Ising model as a testbed model, in which we thoroughly delved into several optimizers, both classical and quantum, and analyzed the extent to which each of these methods would offer a benefit. We then investigated a new combinatorial optimization scheme, termed QN-SPSA+PSR, in which the Fubini-Study metric is approximated within the Quantum Natural Gradient (QN) framework, with its inner gradient estimated by the Simultaneous Perturbation Stochastic Approximation (SPSA), while the outer gradient of the cost function is evaluated exactly by the Parameter-Shift Rule (PSR). The QN-SPSA+PSR method integrates the QN-SPSA computational efficiency with the precise gradient computation of the PSR, improving the stability of QN-SPSA-based and convergence speed per parameter update while maintaining low computational consumption. Our results provide a potential performance improvement in the VQAs' optimization subroutine, even in Quantum Machine Learning's optimization section, and enhance viable paths toward efficient quantum simulations on Noisy Intermediate-Scale Quantum Computing (NISQ) devices. Additionally, we also conducted a detailed study of quantum circuit ansatz structures in order to find the one that would work best with the Ising model and NISQ, in which we utilized the properties of the investigated model.
Determining the ground state of a many-body Hamiltonian is a central problem across physics, chemistry, and combinatorial optimization, yet it is often classically intractable due to the exponential growth of Hilbert space with system size. Even on fault-tolerant quantum computers, quantum algorithms with convergence guarantees -- such as quantum phase estimation and quantum subspace methods -- require an initial state with sufficiently large overlap with the true ground state to be effective. Variational quantum eigensolvers (VQEs) are natural candidates for preparing such states; however, standard VQEs typically exhibit poor generalization, requiring retraining for each Hamiltonian instance, and often suffer from barren plateaus, where gradients can vanish exponentially with circuit depth and system size. To address these limitations, we propose an end-to-end representation learning framework that combines a graph autoencoder with a classical neural network to generate VQE parameters that generalize across Hamiltonian instances. By encoding interaction topology and coupling structure, the proposed model produces high-overlap initial states without instance-specific optimization. Through extensive numerical experiments on families of one- and two-local Hamiltonians, we demonstrate improved generalization and trainability, manifested as reduced test error and a significantly milder decay of gradient variance. We further show that our method substantially accelerates convergence in quantum subspace-based eigensolvers, highlighting its practical impact for downstream quantum algorithms.