Combining variational quantum process with classical neural learning offers a flexible route to improve ground-state estimation. We establish an integrated framework that connects neural transformations, measurement statistics, and guarantees physical stability through three requirements: self-contained training, polynomial resource scaling, and variational consistency. Its constructive realization, termed \emph{unitary variational quantum-neural hybrid eigensolver}~(U-VQNHE), couples a variational quantum circuit to a neural phase function through norm-preserving post-processing. The learned transformation is evaluated from measurement records, preserves the exact variational bound, and admits range-independent concentration guarantees for independent finite-shot evaluations. Complementing this construction, we characterize the statistical and representational constraints of amplitude reweighting: sampled-support mismatch can destabilize empirical normalization, while exact distribution matching can require exponentially large dynamic range for Haar-random targets and structured ansatz--target pairs under specified near-tensorizability and mismatch conditions. Finite-shot simulations on Ising and disordered XYZ spin models demonstrate improved energy accuracy over the underlying variational quantum eigensolver and greater robustness than the amplitude-reweighting baselines. Together, these results provide a principled foundation for quantum--neural eigensolvers in which physical consistency, expressive capacity, and measurement cost are treated as a single design problem.
Combining variational quantum process with classical neural learning offers a flexible route to improve ground-state estimation. We establish an integrated framework that connects neural transformations, measurement statistics, and guarantees physical stability through three requirements: self-contained training, polynomial resource scaling, and variational consistency. Its constructive realization, termed \emph{unitary variational quantum-neural hybrid eigensolver}~(U-VQNHE), couples a variational quantum circuit to a neural phase function through norm-preserving post-processing. The learned transformation is evaluated from measurement records, preserves the exact variational bound, and admits range-independent concentration guarantees for independent finite-shot evaluations. Complementing this construction, we characterize the statistical and representational constraints of amplitude reweighting: sampled-support mismatch can destabilize empirical normalization, while exact distribution matching can require exponentially large dynamic range for Haar-random targets and structured ansatz--target pairs under specified near-tensorizability and mismatch conditions. Finite-shot simulations on Ising and disordered XYZ spin models demonstrate improved energy accuracy over the underlying variational quantum eigensolver and greater robustness than the amplitude-reweighting baselines. Together, these results provide a principled foundation for quantum--neural eigensolvers in which physical consistency, expressive capacity, and measurement cost are treated as a single design problem.
We propose the unitary variational quantum-neural hybrid eigensolver (U-VQNHE), which improves upon the original VQNHE by enforcing unitary neural transformations. The non-unitary nature of VQNHE causes normalization issues and divergence of the loss function during training, leading to exponential scaling of measurement overhead with qubit number. U-VQNHE resolves these issues, significantly reduces required measurements, and retains improved accuracy and stability over standard variational quantum eigensolvers.
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.
Abstract Variational quantum eigensolvers (VQEs) are garnering significant attention for their ability to estimate the ground-state energy of target Hamiltonians. In particular, digital-analog quantum computation (DAQC) stands out as a crucial platform for VQEs in the noisy intermediate-scale quantum era, leveraging single-qubit operations and natural time evolution governed by parameterized Hamiltonians. On the other hand, the variational quantum-neural hybrid eigensolver is also a vital technique that enhances the accuracy of estimating expectation values, utilizing parameterized quantum circuits combined with neural networks. This hybrid approach typically requires the implementation of several controlled gates before measurements and post-processing. However, implementing controlled gates within the DAQC framework poses challenges. To address this, we propose a hybrid architecture that integrates DAQC with neural networks, substituting controlled gates with the classical shadow method. Our approach maintains a polynomial measurement cost as long as the number of qubits with Pauli X or Y operators in each term of the target Hamiltonian is constant or O ( log N q ) , where N q denotes the total number of qubits. We demonstrate the efficacy of our method through numerical simulations, applying it to estimate the ground-state energy of the H2 molecule.
Abstract Quantum error mitigation (QEM) is crucial for obtaining reliable results on quantum computers by suppressing quantum noise with moderate resources. It is a key factor for successful and practical quantum algorithm implementations in the noisy intermediate scale quantum (NISQ) era. Since quantum‐classical hybrid algorithms can be executed with moderate and noisy quantum resources, combining QEM with quantum‐classical hybrid schemes is one of the most promising directions toward practical quantum advantages. This work shows how the variational quantum‐neural hybrid eigensolver (VQNHE) algorithm, which seamlessly combines the expressive power of a parameterized quantum circuit with a neural network, is inherently noise resilient with a unique QEM capacity, which is absent in vanilla variational quantum eigensolvers (VQE). The study carefully analyzes and elucidates the asymptotic scaling of this unique QEM capacity in VQNHE from both theoretical and experimental perspectives. Finally, a variational basis transformation is proposed for the Hamiltonian to be measured under the VQNHE framework, yielding a powerful tri‐optimization setup, dubbed as VQNHE++. VQNHE++ can further enhance the quantum‐neural hybrid expressive power and error mitigation capacity.
Numerous methodologies have been proposed to implement imaginary-time evolution (ITE) on quantum computers. Among these, variational ITE (VITE) methods for noisy intermediate-scale quantum (NISQ) computers have attracted much attention, as they use parameterized quantum circuits to mimic non-unitary dynamics. Although widely studied, conventional variational quantum algorithms face challenges in achieving high accuracy due to their strong dependence on the choice of ansatz quantum circuits. Recently, the variational quantum-neural hybrid eigensolver (VQNHE), which combines neural networks (NN) with a variational quantum eigensolver, has been proposed. This approach enhances the expressive power of variational states and improves the estimation of expectation values. Motivated by this idea, we explore the hybridization of VITE with an NN-based non-unitary operator. In this study, we propose a method named variational quantum-neural hybrid ITE (VQNHITE). By combining the NN with a parameterized quantum circuit, our proposal enhances the expressive power compared to conventional approaches, enabling more accurate tracking of imaginary-time dynamics. In addition, to mitigate the instability arising from randomly initialized NN parameters, we introduce an initial-parameter optimization procedure at a small imaginary-time step, which stabilizes the subsequent variational evolution. We tested our approach with numerical simulations on Heisenberg spin chains under both nearest-neighbor and all-to-all circuit connectivities. The results demonstrate that VQNHITE consistently achieves higher fidelity with the exact ITE state compared to VITE.
While quantum computers are a very promising tool for the far future, in their current state of the art they remain limited both in size and quality. This has given rise to hybrid quantum-classical algorithms, where the quantum device performs only a small but vital part of the overall computation. Among these, variational quantum algorithms (VQAs), which combine a classical optimization procedure with quantum evaluation of a cost function, have emerged as particularly promising. However, barren plateaus and ill-conditioned optimization landscapes remain among the primary obstacles faced by VQAs, often leading to unstable convergence and high sensitivity to initialization. Motivated by this challenge, we investigate whether a purely classical remedy, standard L2 squared-norm regularization, can systematically stabilize hybrid quantum-classical optimization. Specifically, we augment the Variational Quantum Eigensolver (VQE) objective with a quadratic penalty proportional to the squared norm of the parameters, without modifying the quantum circuit or measurement process. Across all tested Hamiltonians, H2, LiH, and the Random Field Ising Model (RFIM), we observe improved performance over a broad window of the regularization strength. Our large-scale numerical results demonstrate that classical regularization provides a robust, system-independent mechanism for mitigating VQE instability, enhancing the reliability and reproducibility of variational quantum optimization without altering the underlying quantum circuit.