Quantum algorithms for the fractional Poisson equation via rational approximation

• URL: http://arxiv.org/abs/2604.00603v1
• Date: Wed, 01 Apr 2026 08:08:33 GMT
• Title: Quantum algorithms for the fractional Poisson equation via rational approximation
• Abstract summary: We present a quantum algorithm for solving the fractional equation ((-)s u = f) with (s in (0,1)) on bounded domains.<n>The proposed approach combines rational approximation techniques with quantum linear system solvers to achieve exponential quantum advantage.
• Score: 20.497126318288906
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: This paper presents a quantum algorithm for solving the fractional Poisson equation \((-Δ)^s u = f\) with \(s \in (0,1)\) on bounded domains. The proposed approach combines rational approximation techniques with quantum linear system solvers to achieve exponential quantum advantage. The rational approximation represents the inverse fractional Laplacian as a weighted sum of standard resolvents, transforming the original nonlocal problem into a collection of shifted integer-order partial differential equations. These equations are consolidated into a single large linear system through a modified right-hand side construction that simplifies the quantum implementation. To enable practical implementation, we develop explicit quantum circuits via the Schrödingerization technique, which converts the non-unitary dynamics of the linear system into a higher-dimensional Schrödinger-type equation, allowing the use of standard Hamiltonian simulation. The circuit construction leverages the decomposition of shift operators to realize the discrete Laplacian and employs controlled operations to implement the select oracle. Under finite difference discretization, we provide detailed algorithmic procedures utilizing block-encoding techniques for the coefficient matrices. A comprehensive complexity analysis demonstrates that the quantum algorithm achieves a dependence on the inverse mesh size \(h^{-1}\) that is independent of the spatial dimension \(d\), in stark contrast to classical methods which suffer from exponential growth in high dimensions. This establishes an exponential quantum advantage for high-dimensional fractional problems, effectively overcoming the curse of dimensionality that limits classical approaches.

Related papers

• Analytical Series Expansion for Efficient Gradient Evaluation in Multi-Qubit Optimal Control [42.97938764966747]
  We present a unifying framework for gradient-based quantum optimal control.<n>For the case of unitary propagators, we derive a series expansion involving time-independent commutators and time-dependent coefficients.<n>The method is particularly suited for simulating optimal control tasks in quantum systems with local interactions.
  arXiv  Detail & Related papers  (2026-07-29T12:50:52Z)
• Explicit Block Encodings of Discrete Laplacians with Mixed Boundary Conditions [45.88028371034407]
  Block encoding provides the standard method for accessing matrix data within quantum circuits.<n>We present a unified framework for efficiently block encoding finite-difference discretizations of the Laplacian.
  arXiv  Detail & Related papers  (2026-03-12T19:35:16Z)
• Quantum simulation of Liouville equation in geometrical optics with partial transmission and reflection via Schrödingerization [42.22857110018992]
  This paper investigates quantum simulation algorithms for the Liouville equation in geometrical optics with partial transmission and reflection at sharp interfaces.<n>By means of a warped phase transformation in one higher dimension, the Schrdingerization method converts linear partial differential equations into a system of Schrdinger-type equations with unitary evolution.<n>We present detailed constructions of the resulting quantum algorithms and show through complexity analysis that the proposed methods achieve quantum advantage in the precision parameter $$ over their classical counterparts.
  arXiv  Detail & Related papers  (2026-03-12T14:47:16Z)
• Block encoding of sparse matrices with a periodic diagonal structure [67.45502291821956]
  We provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure.<n>Various applications for the presented methodology are discussed in the context of solving differential problems.
  arXiv  Detail & Related papers  (2026-02-11T07:24:33Z)
• Quantum Random Feature Method for Solving Partial Differential Equations [36.58357595906332]
  Quantum computing holds promise for scientific computing due to its potential for exponential speedups over classical methods.<n>In this work, we introduce a quantum random method (QRFM) that leverages advantages from both numerical analysis and neural analysis.
  arXiv  Detail & Related papers  (2025-10-09T08:42:09Z)
• Quantum Algorithms for Solving Generalized Linear Systems via Momentum Accelerated Gradient and Schrödingerization [34.04094622012203]
  We propose a quantum algorithm that combines the momentum accelerated gradient method with Schr"odingerization.<n>The algorithm achieves speedup over its classical counterpark in solving linear systems.<n>It can overcome the practical limitations of existing non-Schr"odingerization-based quantum linear system algorithms.
  arXiv  Detail & Related papers  (2025-09-20T08:40:53Z)
• Efficient Quantum Access Model for Sparse Structured Matrices using Linear Combination of Things [0.6138671548064355]
  We present a novel framework for Linear Combination of Unitaries (LCU)-style decomposition tailored to structured sparse matrices.<n>LCU is a foundational primitive in both variational and fault-tolerant quantum algorithms.<n>We introduce the Sigma basis, a compact set of simple, non-unitary operators that can better capture sparsity and structure.
  arXiv  Detail & Related papers  (2025-07-04T17:05:07Z)
• On the Schrödingerization method for linear non-unitary dynamics with optimal dependence on matrix queries [42.104910612491885]
  The Schr"odingerization method converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schr"odinger-type equations with unitary evolution.<n>The original proposal used a particular initial function in the auxiliary space that did not achieve optimal scaling in precision.<n>Here we show that, by choosing smoother initial functions in auxiliary space, Schr"odingerization textitcan in fact achieve near optimal and even optimal scaling in matrix queries.
  arXiv  Detail & Related papers  (2025-05-01T07:46:50Z)
• High order schemes for solving partial differential equations on a quantum computer [0.0]
  We show that higher-order methods can reduce the number of qubits necessary for discretization, similar to the classical case.<n>This result has important consequences for the practical application of quantum algorithms based on Hamiltonian evolution.
  arXiv  Detail & Related papers  (2024-12-26T14:21:59Z)
• Double-Logarithmic Depth Block-Encodings of Simple Finite Difference Method's Matrices [0.0]
  Solving differential equations is one of the most computationally expensive problems in classical computing. Despite recent progress made in the field of quantum computing and quantum algorithms, its end-to-end application towards practical realization still remains unattainable.
  arXiv  Detail & Related papers  (2024-10-07T17:44:30Z)
• A probabilistic imaginary-time evolution quantum algorithm for advection-diffusion equation: Explicit gate-level implementation and comparisons to quantum linear system algorithms [0.0]
  We propose a quantum algorithm for solving the advection-diffusion-reaction equation.<n>Our algorithm achieves an exponential speedup regarding the matrix size at the cost of a worse dependence on the error bound.
  arXiv  Detail & Related papers  (2024-09-27T08:56:21Z)
• Boundary Treatment for Variational Quantum Simulations of Partial Differential Equations on Quantum Computers [1.6318838452579472]
  The paper presents a variational quantum algorithm to solve initial-boundary value problems described by partial differential equations. The approach uses classical/quantum hardware that is well suited for quantum computers of the current noisy intermediate-scale quantum era.
  arXiv  Detail & Related papers  (2024-02-28T18:19:33Z)
• GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls [77.34726150561087]
  The GRadient Ascent Pulse Engineering (GRAPE) method is widely used for optimization in quantum control. We adopt GRAPE method for optimizing objective functionals for open quantum systems driven by both coherent and incoherent controls. The efficiency of the algorithm is demonstrated through numerical simulations for the state-to-state transition problem.
  arXiv  Detail & Related papers  (2023-07-17T13:37:18Z)
• Automatic and effective discovery of quantum kernels [41.61572387137452]
  Quantum computing can empower machine learning models by enabling kernel machines to leverage quantum kernels for representing similarity measures between data.<n>We present an approach to this problem, which employs optimization techniques, similar to those used in neural architecture search and AutoML.<n>The results obtained by testing our approach on a high-energy physics problem demonstrate that, in the best-case scenario, we can either match or improve testing accuracy with respect to the manual design approach.
  arXiv  Detail & Related papers  (2022-09-22T16:42:14Z)

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.