Abstract Machine learning provides a powerful framework for predicting ground-state properties across families of quantum many-body problems, enabling amortized inference and reducing the cost of repeated simulations. Variational quantum algorithms (VQAs) are promising candidates for implementing such learnable solvers on quantum computers, yet rigorous guarantees for convergence and generalization remain scarce. Motivated by the fact that standard quantum phase estimation (QPE) provides provable performance when given a guiding state with non-trivial overlap with the ground state, we introduce a variational quantum algorithm with guiding states aiming towards predicting ground-state properties of quantum many-body systems. We then develop a proof technique—the linearization trick—that maps the training dynamics of the algorithm to those of a kernel model. This connection yields theoretical guarantees on both convergence and generalization for the VQA under the guiding state assumption. Our analysis shows that guiding states facilitate convergence, suppress finite-size error terms, and ensure stability across system dimensions. Finally, we validate our findings with numerical experiments on 2D random Heisenberg models.
Abstract Machine learning provides a powerful framework for predicting ground-state properties across families of quantum many-body problems, enabling amortized inference and reducing the cost of repeated simulations. Variational quantum algorithms (VQAs) are promising candidates for implementing such learnable solvers on quantum computers, yet rigorous guarantees for convergence and generalization remain scarce. Motivated by the fact that standard quantum phase estimation (QPE) provides provable performance when given a guiding state with non-trivial overlap with the ground state, we introduce a variational quantum algorithm with guiding states aiming towards predicting ground-state properties of quantum many-body systems. We then develop a proof technique—the linearization trick—that maps the training dynamics of the algorithm to those of a kernel model. This connection yields theoretical guarantees on both convergence and generalization for the VQA under the guiding state assumption. Our analysis shows that guiding states facilitate convergence, suppress finite-size error terms, and ensure stability across system dimensions. Finally, we validate our findings with numerical experiments on 2D random Heisenberg models.
The analysis of quantum state properties is of fundamental importance in quantum physics and serves as a primary motivation for the development of quantum computing. In particular, low-temperature and ground states are of significant interest, as they most prominently exhibit phenomena unique to quantum mechanics. However, extracting such properties typically requires costly numerical simulations or laboratory experiments, while simulation on quantum hardware remains limited by the capabilities of current devices. This thesis proposes two complementary approaches to address these challenges.The first contribution presents an algorithm for predicting ground-state properties across an entire family of quantum states within the same phase of matter, based on training data. Under slightly stronger assumptions, we improve upon the sample complexity of existing methods, prove the same rigorous guarantees for a deep neural network-based approach, and demonstrate its practical advantages through extensive numerical experiments.The second contribution addresses the challenge of preparing quantum states of interest using Variational Quantum Algorithms (VQAs), a framework based on the classical optimization of parametrized quantum circuits. We introduce multi-armed stochastic bandits into this setting, propose an instance-optimal algorithm for continuous single-parameter optimization, and and motivate further exploration of the framework we introduced with an array of experiments.
Solving the ground state and the ground-state properties of quantum many-body systems is generically a hard task for classical algorithms. For a family of Hamiltonians defined on an m -dimensional space of physical parameters, the ground state and its properties at an arbitrary parameter configuration can be predicted via a machine learning protocol up to a prescribed prediction error ɛ , provided that a sample set (of size N ) of the states can be efficiently prepared and measured. In a recent work [Huang , ], a rigorous guarantee for such a generalization was proved. Unfortunately, an exponential scaling for the provable sample complexity, N = m O ( 1 ɛ ) , was found to be universal for generic gapped Hamiltonians. This result applies to the situation where the dimension of the parameter space is large, while the scaling with the accuracy is not an urgent factor, not entering the realm of more precise learning and prediction. In this work, we consider an alternative relevant scenario, where the effective dimension m is a finite, not necessarily large constant, while the scaling with the prediction error becomes the central concern. By jointly preserving the fundamental properties of density matrices in the learning protocol and utilizing the continuity of quantum states in the parameter range of interest, we rigorously obtain a polynomial sample complexity for predicting quantum many-body states and their properties, with respect to the prediction error ɛ and the number of qubits, n , with N = poly ( ɛ − 1 , n , ln 1 δ ) , where poly denotes a polynomial function, and ( 1 − δ ) is the probability of success. Moreover, if restricted to learning local quantum-state properties, the number of samples can be further reduced to N = poly ( ɛ − 1 , ln n δ ) . Numerical demonstrations confirm our findings, and an alternative approach utilizing statistical learning theory with reproducing kernel Hilbert space achieves consistent results. The mere continuity assumption indicates that our results are not restricted to gapped Hamiltonian systems and properties within the same phase. Published by the American Physical Society 2024
Abstract 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.
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.
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 (VQAs) are constrained by a trade-off: deeper circuits can cover a larger reachable quantum states but suffer from barren plateaus, while shallow circuits remain trainable yet can have insufficient reachability to the target state. Here, we propose a general framework to address this challenge by enhancing the VQA performance with a designed input state constructed using a linear combination technique. This approach modifies the set of states reachable by the original circuit, enhancing accuracy while preserving efficiency. We provide a rigorous proof that such framework increases the performance of any given VQA ansatz, and demonstrate its broad applicability across different ansatz families. In ground state preparation for representative quantum many-body models, it achieves consistently higher accuracy than standard methods at the same gate budget. These results highlight input-state design as a powerful complement to circuit design for improving reachability of the target state within a fixed ansatz. Variational quantum algorithms face a practical trade-off between circuit depth and trainability, which limits the reachability of target quantum states. Here, the authors show that input-state design based on linear combinations systematically modifies the reachable-state set of a fixed ansatz, thereby enhancing target-state reachability and improving ground-state preparation accuracy across representative quantum many-body models without increasing gate cost.