Achieving both high precision and large dynamic range remains a central challenge in quantum metrology, as improving local sensitivity typically reduces the unambiguous estimation range. Variational quantum interferometers enhance precision but are generally limited to narrow operating regimes. Here we introduce a hybrid variational quantum--classical neural network interferometer (VQ-CNNI), where a shallow quantum circuit encodes phase-dependent measurement statistics and a neural network performs nonlinear phase reconstruction. Joint optimization enables accurate and unambiguous phase estimation over $[-π,π)$ without loss of precision. We show that this performance requires co-optimization of quantum encoding and classical decoding. Visualization of the learned representation geometry links global estimation to well-conditioned measurement statistics across the full phase range, enabling stable inversion. Odd-symmetric activations further improve robustness by promoting global consistency. These results suggest that global quantum metrology can be understood through the learnability of the quantum--classical representation, providing a practical route to programmable interferometers with both high precision and large dynamic range.
Achieving both high precision and large dynamic range remains a central challenge in quantum metrology, as improving local sensitivity typically reduces the unambiguous estimation range. Variational quantum interferometers enhance precision but are generally limited to narrow operating regimes. Here we introduce a hybrid variational quantum--classical neural network interferometer (VQ-CNNI), where a shallow quantum circuit encodes phase-dependent measurement statistics and a neural network performs nonlinear phase reconstruction. Joint optimization enables accurate and unambiguous phase estimation over [−π,π) without loss of precision. We show that this performance requires co-optimization of quantum encoding and classical decoding. Visualization of the learned representation geometry links global estimation to well-conditioned measurement statistics across the full phase range, enabling stable inversion. Odd-symmetric activations further improve robustness by promoting global consistency. These results suggest that global quantum metrology can be understood through the learnability of the quantum--classical representation, providing a practical route to programmable interferometers with both high precision and large dynamic range.
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 physical stability through three requirements: self-contained training, polynomial resource scaling, and variational consistency. Its constructive realization, termed 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 polynomial-range reweighting exactly matches only an exponentially small fraction of Haar-random target distributions from a fixed ansatz, and exact matching for structured ansatz--target pairs can require exponentially large dynamic range 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.
With increasing maturity of quantum computers, standard methods for characterizing global properties of their output quantum states via direct measurements and classical post-processing are becoming increasingly impractical due to large measurement costs. Although quantum neural networks could directly process quantum states to identify underlying characteristics with reduced measurement efforts, they often require deep quantum circuits that cannot be implemented on existing devices. To overcome these challenges, we introduce a hybrid quantum-classical neural network that consists of a shallow parameterized quantum circuit, measurements, and a classical neural network. The parameterized quantum circuit performs a nonlocal transformation of the measurement basis, which is jointly trained with the classical neural network to maximize the statistical distance between data obtained by measuring different quantum states. Using supervised learning, we demonstrate that the hybrid neural network distinguishes the topological phase of the surface code from a symmetry-enriched topological phase and random product states. Moreover, this hybrid neural network reduces both inference and training sample complexities of recognizing the topological phase by approximately one order of magnitude compared to a classical neural network trained on randomized Pauli measurements. As this hybrid neural network features a shallow quantum circuit that can be readily implemented on existing quantum computers, it enables the efficient characterization of complex quantum states.
Abstract Characterizing entanglement in high-dimensional continuous-variable (CV) systems remains a formidable challenge, particularly when full quantum state tomography is prohibitively resource-intensive. In this work, a hybrid quantum–classical framework is introduced that learns nonlinear entanglement witnesses directly from sparse measurement data via CV quantum neural networks (CV-QNNs). The architecture combines variational interferometers, squeezing operations, and non-Gaussian Kerr nonlinearities with a classical neural head to output a scalar witness value. To rigorously isolate the information-theoretic sample-complexity advantage, a Coherence Trap benchmark is devised in which genuinely entangled and separable states are engineered to exhibit identical photon-number statistics, thereby rendering them indistinguishable to classical classifiers operating on intensity histograms. Numerical simulations show that while classical models fail to classify these states without full tomography, the CV-QNN achieves 100 % accuracy by extracting phase-dependent quantum correlations prior to measurement. To address scalability and optimization under realistic finite-shot measurement noise, a variance-reduced simultaneous perturbation stochastic approximation algorithm is employed, demonstrating sustained convergence and consistent performance across the primary tested cutoffs up to a Fock space cutoff of d = 5 for the 3 -mode CV system, alongside a proof-of-concept indication of algorithmic trainability at d = 7 . Moreover, the generalization of this framework to the detection of bound entangled states with zero negativity is demonstrated, establishing a fundamental advantage in sample complexity over standard linear witnesses. Robustness to photon loss and statistical noise is further quantified under realistic finite-shot measurement conditions, confirming the model’s stability in noisy environments. Collectively, these findings provide a rigorous proof-of-principle for the necessity of non-Gaussian operations and demonstrate a clear information-theoretic advantage in data-driven quantum state characterization.
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.
Variational quantum algorithms hold great promise for unlocking the power of near-term quantum processors, yet high measurement costs, barren plateaus, and challenging optimization landscapes frequently hinder them. Here, we introduce sVQNHE, a neural-guided variational quantum algorithm that decouples amplitude and sign learning across classical and quantum modules, respectively. Our approach employs shallow quantum circuits composed of commuting diagonal gates to efficiently model quantum phase information, while a classical neural network learns the amplitude distribution and guides circuit optimization in a bidirectional feedback loop. This hybrid quantum-classical synergy not only reduces measurement costs but also achieves high expressivity with limited quantum resources and improves the convergence rate of the variational optimization. We demonstrate the advancements brought by sVQNHE through extensive numerical experiments. For the 6-qubit J1-J2 model, a prototypical system with a severe sign problem for Monte Carlo-based methods, it reduces the mean absolute error by 98.9% and suppresses variance by 99.6% relative to a baseline neural network, while requiring nearly 19x fewer optimization steps than a standard hardware-efficient VQE. Furthermore, for MaxCut problems on 45-vertex Erdos-Renyi graphs, sVQNHE improves solution quality by 19% and quantum resource efficiency by 85%. Importantly, this framework is designed to be scalable and robust against hardware noise and finite-sampling uncertainty, making it well-suited for both current NISQ processors and future high-quality quantum computers. Our results highlight a promising path forward for efficiently tackling complex many-body and combinatorial optimization problems by fully exploiting the synergy between classical and quantum resources in the NISQ era and beyond.
We propose a high-precision phase estimation scheme in a hybrid interferometer by synergistically combining a Kerr nonlinear phase shifter and multi-photon subtraction operations. Using a coherent state and a vacuum state as input resources, we systematically evaluate the phase sensitivity via homodyne detection and analyze the quantum Fisher information as well as the quantum Cramér-Rao bound under both ideal and lossy conditions. Our results show that the joint integration of Kerr nonlinearity and multi-photon subtraction yields remarkable advantages over either technique used alone. The proposed scheme enables the phase sensitivity to surpass the standard quantum limit, exceed the conventional Heisenberg scaling (1/N), and approach the super-Heisenberg scaling (1/N2)-a direct consequence of Kerr nonlinearity. More precisely, the super-Heisenberg scaling $\propto $ 1/N2 is the ultimate precision limit permitted by the k=2 Kerr nonlinearity and does not violate the fundamental Heisenberg limit for linear phase accumulation. Even under moderate internal photon loss, the system maintains high precision and exhibits enhanced robustness to decoherence. The Kerr nonlinearity introduces an intensity-dependent phase shift proportional to the squared photon number, while multi-photon subtraction tailors non-Gaussian states to strengthen phase information extraction. Compared with existing schemes based on hybrid interferometers or SU(1,1) interferometers, our architecture achieves superior precision and stronger loss resilience. All components are experimentally accessible with current quantum optical technologies. This work provides a promising route for practical high-precision quantum metrology and quantum sensing.