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.
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.
The variational quantum eigensolver (VQE) is one of the most representative quantum algorithms in the noisy intermediate-scale quantum (NISQ) era, and is generally speculated to deliver one of the first quantum advantages for the ground-state simulations of some nontrivial Hamiltonians. However, short quantum coherence time and limited availability of quantum hardware resources in the NISQ hardware strongly restrain the capacity and expressiveness of VQEs. In this Letter, we introduce the variational quantum-neural hybrid eigensolver (VQNHE) in which the shallow-circuit quantum Ansatz can be further enhanced by classical post-processing with neural networks. We show that the VQNHE consistently and significantly outperforms the VQE in simulating ground-state energies of quantum spins and molecules given the same amount of quantum resources. More importantly, we demonstrate that, for arbitrary postprocessing neural functions, the VQNHE only incurs a polynomial overhead of processing time and represents the first scalable method to exponentially accelerate the VQE with nonunitary postprocessing that can be efficiently implemented in the NISQ era.
In the emergent realm of quantum computing, the Variational Quantum Eigensolver (VQE) stands out as a promising algorithm for solving complex quantum problems, especially in the noisy intermediate-scale quantum (NISQ) era. However, the ubiquitous presence of noise in quantum devices often limits the accuracy and reliability of VQE outcomes. This research introduces a novel approach to ameliorate this challenge by utilizing neural networks for zero noise extrapolation (ZNE) in VQE computations. By employing the Qiskit framework, we crafted parameterized quantum circuits using the RY-RZ ansatz and examined their behavior under varying levels of depolarizing noise. Our investigations spanned from determining the expectation values of a Hamiltonian, defined as a tensor product of Z operators, under different noise intensities to extracting the ground state energy. To bridge the observed outcomes under noise with the ideal noise-free scenario, we trained a Feed Forward Neural Network on the error probabilities and their associated expectation values. Remarkably, our model proficiently predicted the VQE outcome under hypothetical noise-free conditions. By juxtaposing the simulation results with real quantum device executions, we unveiled the discrepancies induced by noise and showcased the efficacy of our neural network-based ZNE technique in rectifying them. This integrative approach not only paves the way for enhanced accuracy in VQE computations on NISQ devices but also underlines the immense potential of hybrid quantum-classical paradigms in circumventing the challenges posed by quantum noise. Through this research, we envision a future where quantum algorithms can be reliably executed on noisy devices, bringing us one step closer to realizing the full potential of quantum computing.
The variational quantum eigensolver (VQE) is a hybrid algorithm that has the potential to provide a quantum advantage in practical chemistry problems that are currently intractable on classical computers. VQE trains parameterized quantum circuits using a classical optimizer to approximate the eigenvalues and eigenstates of a given Hamiltonian. However, VQE faces challenges in task-specific design and machine-specific architecture, particularly when running on noisy quantum devices. This can have a negative impact on its trainability, accuracy, and efficiency, resulting in noisy quantum data. We propose variational denoising, an unsupervised learning method that employs a parameterized quantum neural network to improve the solution of VQE by learning from noisy VQE outputs. Our approach can significantly decrease energy estimation errors and increase fidelities with ground states compared to noisy input data for the H2 , LiH, and BeH2 molecular Hamiltonians and the transverse field Ising model. Surprisingly, it only requires noisy data for training. Variational denoising can be integrated into quantum hardware, increasing its versatility as an end-to-end quantum processing for quantum data. Published by the American Physical Society 2024
In the noisy intermediate-scale quantum (NISQ) era, the Variational Quantum Eigensolver (VQE) comes out as an effective algorithm for addressing complex quantum challenges. However, the presence of noise in quantum devices frequently reduces the accuracy and reliability of VQE outcomes. This paper presents an innovative method to address this issue by using a neural network-based extrapolation technique in VQE calculation. By utilizing the Qiskit framework, we designed parameterized quantum circuits using the RY-RZ ansatz, and their performance was analyzed under different levels of depolarizing noise with bit-flip errors, phase-flip errors, and amplitude damping errors. Our inquiry involved analyzing the expected outcomes of a Hamiltonian across various levels of noise intensity with the objective of deducing the ground state energy (GSE). To connect the observed noisy outcomes with the ideal noise-free condition, a Feedforward Neural Network (FFNN) was trained using the error probabilities and their corresponding expectation values. This model accurately predicted the VQE results in an ideal noise-free scenario. Comparison of the result of simulation and real quantum hardware executions revealed noise-induced inconsistencies, highlighting the effectiveness of this neural network-based extrapolation approach in correcting them. This comprehensive method improves the accuracy of VQE calculation on NISQ devices and highlights the significant potential of blending quantum and classical methods to address the threats imposed by quantum noise. The comparison of results between FFNN, convolutional neural network (CNN), and long short-term memory (LSTM) network reveals that FFNN predicts outcomes with more accuracy but in less time.
In the Noisy Intermediate-Scale Quantum (NISQ) era, using variational quantum algorithms (VQAs) to solve optimization problems has become a key application. However, these algorithms face significant challenges, such as choosing an effective initial set of parameters and the limited quantum processing time that restricts the number of optimization iterations. In this study, we introduce a new framework for optimizing parameterized quantum circuits (PQCs) that employs a classical optimizer, inspired by Model-Agnostic Meta-Learning (MAML) technique. This approach aim to achieve better parameter initialization that ensures fast convergence. Our framework features a classical neural network, called Learner, which interacts with a PQC using the output of Learner as an initial parameter. During the pre-training phase, Learner is trained with the meta objective function. In the adaptation phase, the framework requires only a few PQC updates to converge to a more accurate value, while the learner remains unchanged. This method is highly adaptable and is effectively extended to various Hamiltonian optimization problems. We validate our approach through experiments, including distribution function mapping and optimization of the Heisenberg XYZ Hamiltonian. The result implies that the Learner successfully estimates initial parameters that generalize across the problem space, enabling fast adaptation.
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.