Quantum Fourier analysis for multivariate functions and applications to a class of Schr\"odinger-type partial differential equations

• URL: http://arxiv.org/abs/2104.02668v3
• Date: Wed, 2 Feb 2022 13:27:24 GMT
• Title: Quantum Fourier analysis for multivariate functions and applications to a class of Schr\"odinger-type partial differential equations
• Authors: Paula Garc\'ia-Molina, Javier Rodr\'iguez-Mediavilla and Juan Jos\'e Garc\'ia-Ripoll
• Abstract summary: We create a variational hybrid quantum algorithm to solve static, Schr"odinger-type, Hamiltonian partial differential equations. We use this algorithm to benchmark the performance of the representation techniques. We obtain low infidelities of order $10-4-10-5$ using only three to four qubits, demonstrating the high compression of information in a quantum computer.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this work, we develop a highly efficient representation of functions and differential operators based on Fourier analysis. Using this representation, we create a variational hybrid quantum algorithm to solve static, Schr\"odinger-type, Hamiltonian partial differential equations (PDEs), using space-efficient variational circuits, including the symmetries of the problem, and global and gradient-based optimizers. We use this algorithm to benchmark the performance of the representation techniques by means of the computation of the ground state in three PDEs, i.e., the one-dimensional quantum harmonic oscillator, and the transmon and flux qubits, studying how they would perform in ideal and near-term quantum computers. With the Fourier methods developed here, we obtain low infidelities of order $10^{-4}-10^{-5}$ using only three to four qubits, demonstrating the high compression of information in a quantum computer. Practical fidelities are limited by the noise and the errors of the evaluation of the cost function in real computers, but they can also be improved through error mitigation techniques.

Related papers

• DimOL: Dimensional Awareness as A New 'Dimension' in Operator Learning [63.5925701087252]
  We introduce DimOL (Dimension-aware Operator Learning), drawing insights from dimensional analysis. To implement DimOL, we propose the ProdLayer, which can be seamlessly integrated into FNO-based and Transformer-based PDE solvers. Empirically, DimOL models achieve up to 48% performance gain within the PDE datasets.
  arXiv  Detail & Related papers  (2024-10-08T10:48:50Z)
• Fourier Neural Operators for Learning Dynamics in Quantum Spin Systems [77.88054335119074]
  We use FNOs to model the evolution of random quantum spin systems. We apply FNOs to a compact set of Hamiltonian observables instead of the entire $2n$ quantum wavefunction.
  arXiv  Detail & Related papers  (2024-09-05T07:18:09Z)
• 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)
• Quantum error mitigation for Fourier moment computation [49.1574468325115]
  This paper focuses on the computation of Fourier moments within the context of a nuclear effective field theory on superconducting quantum hardware. The study integrates echo verification and noise renormalization into Hadamard tests using control reversal gates. The analysis, conducted using noise models, reveals a significant reduction in noise strength by two orders of magnitude.
  arXiv  Detail & Related papers  (2024-01-23T19:10:24Z)
• Quantum Fourier Networks for Solving Parametric PDEs [4.409836695738518]
  Recently, a deep learning architecture called Fourier Neural Operator (FNO) proved to be capable of learning solutions of given PDE families for any initial conditions as input. We propose quantum algorithms inspired by the classical FNO, which result in time complexity logarithmic in the number of evaluations.
  arXiv  Detail & Related papers  (2023-06-27T12:21:02Z)
• Gibbs Sampling of Continuous Potentials on a Quantum Computer [0.0]
  We build a quantum algorithm for Gibbs sampling from periodic real-valued functions. Our algorithm makes zeroeth order queries to a quantum oracle of the function.
  arXiv  Detail & Related papers  (2022-10-14T20:56:44Z)
• Variational Quantum Continuous Optimization: a Cornerstone of Quantum Mathematical Analysis [0.0]
  We show how universal quantum computers can handle mathematical analysis calculations for functions with continuous domains. The basic building block of our approach is a variational quantum circuit where each qubit encodes up to three continuous variables. By combining this encoding with quantum state tomography, a variational quantum circuit of $n$ qubits can optimize functions of up to $3n$ continuous variables.
  arXiv  Detail & Related papers  (2022-10-06T18:00:04Z)
• Numerical Simulations of Noisy Quantum Circuits for Computational Chemistry [51.827942608832025]
  Near-term quantum computers can calculate the ground-state properties of small molecules. We show how the structure of the computational ansatz as well as the errors induced by device noise affect the calculation.
  arXiv  Detail & Related papers  (2021-12-31T16:33:10Z)
• Bosonic field digitization for quantum computers [62.997667081978825]
  We address the representation of lattice bosonic fields in a discretized field amplitude basis. We develop methods to predict error scaling and present efficient qubit implementation strategies.
  arXiv  Detail & Related papers  (2021-08-24T15:30:04Z)
• Solving nonlinear differential equations with differentiable quantum circuits [21.24186888129542]
  We propose a quantum algorithm to solve systems of nonlinear differential equations. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits. We show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space.
  arXiv  Detail & Related papers  (2020-11-20T13:21:11Z)
• Variational Monte Carlo calculations of $\mathbf{A\leq 4}$ nuclei with an artificial neural-network correlator ansatz [62.997667081978825]
  We introduce a neural-network quantum state ansatz to model the ground-state wave function of light nuclei. We compute the binding energies and point-nucleon densities of $Aleq 4$ nuclei as emerging from a leading-order pionless effective field theory Hamiltonian.
  arXiv  Detail & Related papers  (2020-07-28T14:52:28Z)

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.