Breaking the Curse of Dimensionality in Quantum PDE Solvers via Gevrey Regularity

• URL: http://arxiv.org/abs/2608.07893v1
• Date: Sat, 08 Aug 2026 03:40:25 GMT
• Title: Breaking the Curse of Dimensionality in Quantum PDE Solvers via Gevrey Regularity
• Abstract summary: We use the Gevrey hierarchy to stratifies the space between smooth and analytic functions.<n>We show how our Fourier methods do so efficiently at varying target precisions on a quantum computer.<n>As an application, we introduce a hierarchy of many-body quantum simulation pipelines.
• Score: 4.219859693008517
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We connect different degrees of smoothness of real-valued periodic functions to the cost of preparing their high-precision Fourier-basis amplitude encodings as quantum states. Our central observation is that the Gevrey hierarchy, which stratifies the space between smooth and analytic functions, provides a natural class for high-precision quantum algorithms. We then specialize to solving general linear partial differential equations (PDEs), showing how our Fourier methods do so efficiently at varying target precisions on a quantum computer. This also demonstrates how our framework enables passage from query-complexity results to explicit elementary gate counts. As an application, we introduce a hierarchy of many-body quantum simulation pipelines that harness these high-precision algorithms to probe the linear response of atomistic systems in first quantization. Each level of the hierarchy unlocks a further polynomial-degree quantum speedup, yielding a gradual improvement in simulation efficiency as quantum computers scale.

Related papers

• Balancing Expressivity and Learnability in Quantum Kernel Bandit Optimization [50.880384999888044]
  We develop misspecified GP bandit algorithms and derive regret bounds that characterize the trade-off between approximation error and information gain.<n>Our methods outperform full quantum kernels in sample efficiency, while substantially reducing computational overhead.
  arXiv  Detail & Related papers  (2026-07-01T15:38:00Z)
• Quantum Orthogonal Separable Physics-Informed Neural Networks [0.0]
  This paper introduces Quantum Orthogonal Separable Physics-Informed Neural Networks (QO-SPINNs), a novel architecture for solving Partial Differential Equations.<n>We leverage a quantum algorithm for accelerating matrix multiplication within each layer, achieving a $mathcal O(dlog d/2)$ complexity.<n>We provide a robust and efficient framework for uncertainty quantification (UQ) which, to our knowledge, is the first UQ method specifically designed for Separable PINNs.
  arXiv  Detail & Related papers  (2025-11-16T14:15:19Z)
• Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits [62.46800898243033]
  Recent progress in quantum learning theory prompts a question: can linear properties of a large-qubit circuit be efficiently learned from measurement data generated by varying classical inputs?<n>We prove that the sample complexity scaling linearly in $d$ is required to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d.<n>We propose a kernel-based method leveraging classical shadows and truncated trigonometric expansions, enabling a controllable trade-off between prediction accuracy and computational overhead.
  arXiv  Detail & Related papers  (2024-08-22T08:21:28Z)
• Unveiling quantum phase transitions from traps in variational quantum algorithms [0.0]
  We introduce a hybrid algorithm that combines quantum optimization with classical machine learning.<n>We use LASSO for identifying conventional phase transitions and the Transformer model for topological transitions.<n>We validated the method with numerical simulations and real-hardware experiments on Rigetti's Ankaa 9Q-1 quantum computer.
  arXiv  Detail & Related papers  (2024-05-14T09:01:41Z)
• Quantum Subroutine for Variance Estimation: Algorithmic Design and Applications [80.04533958880862]
  Quantum computing sets the foundation for new ways of designing algorithms. New challenges arise concerning which field quantum speedup can be achieved. Looking for the design of quantum subroutines that are more efficient than their classical counterpart poses solid pillars to new powerful quantum algorithms.
  arXiv  Detail & Related papers  (2024-02-26T09:32:07Z)
• Scalable Quantum Computation of Highly Excited Eigenstates with Spectral Transforms [0.76146285961466]
  We use the HHL algorithm to prepare excited interior eigenstates of physical Hamiltonians in a variational and targeted manner. This is enabled by the efficient computation of the expectation values of inverse Hamiltonians on quantum computers. We detail implementations of this scheme for both fault-tolerant and near-term quantum computers.
  arXiv  Detail & Related papers  (2023-02-13T19:01:02Z)
• The Quantum Path Kernel: a Generalized Quantum Neural Tangent Kernel for Deep Quantum Machine Learning [52.77024349608834]
  Building a quantum analog of classical deep neural networks represents a fundamental challenge in quantum computing. Key issue is how to address the inherent non-linearity of classical deep learning. We introduce the Quantum Path Kernel, a formulation of quantum machine learning capable of replicating those aspects of deep machine learning.
  arXiv  Detail & Related papers  (2022-12-22T16:06:24Z)
• A self-consistent field approach for the variational quantum eigensolver: orbital optimization goes adaptive [52.77024349608834]
  We present a self consistent field approach (SCF) within the Adaptive Derivative-Assembled Problem-Assembled Ansatz Variational Eigensolver (ADAPTVQE) This framework is used for efficient quantum simulations of chemical systems on nearterm quantum computers.
  arXiv  Detail & Related papers  (2022-12-21T23:15:17Z)
• Improved iterative quantum algorithm for ground-state preparation [4.921552273745794]
  We propose an improved iterative quantum algorithm to prepare the ground state of a Hamiltonian system. Our approach has advantages including the higher success probability at each iteration, the measurement precision-independent sampling complexity, the lower gate complexity, and only quantum resources are required when the ancillary state is well prepared.
  arXiv  Detail & Related papers  (2022-10-16T05:57:43Z)
• Quantum Phase Processing and its Applications in Estimating Phase and Entropies [10.8525801756287]
  "quantum phase processing" can directly apply arbitrary trigonometric transformations to eigenphases of a unitary operator. Quantum phase processing can extract the eigen-information of quantum systems by simply measuring the ancilla qubit. We propose a new quantum phase estimation algorithm without quantum Fourier transform, which requires the fewest ancilla qubits and matches the best performance so far.
  arXiv  Detail & Related papers  (2022-09-28T17:41:19Z)
• Fundamental limitations on optimization in variational quantum algorithms [7.165356904023871]
  A leading paradigm to establish such near-term quantum applications is variational quantum algorithms (VQAs) We prove that for a broad class of such random circuits, the variation range of the cost function vanishes exponentially in the number of qubits with a high probability. This result can unify the restrictions on gradient-based and gradient-free optimizations in a natural manner and reveal extra harsh constraints on the training landscapes of VQAs.
  arXiv  Detail & Related papers  (2022-05-10T17:14:57Z)
• Quantum algorithms for quantum dynamics: A performance study on the spin-boson model [68.8204255655161]
  Quantum algorithms for quantum dynamics simulations are traditionally based on implementing a Trotter-approximation of the time-evolution operator. variational quantum algorithms have become an indispensable alternative, enabling small-scale simulations on present-day hardware. We show that, despite providing a clear reduction of quantum gate cost, the variational method in its current implementation is unlikely to lead to a quantum advantage.
  arXiv  Detail & Related papers  (2021-08-09T18:00:05Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.