Solving the Nonlinear Vlasov Equation on a Quantum Computer

• URL: http://arxiv.org/abs/2411.19310v1
• Date: Thu, 28 Nov 2024 18:32:30 GMT
• Title: Solving the Nonlinear Vlasov Equation on a Quantum Computer
• Authors: Tamás Á Vaszary, Animesh Datta, Thomas Goffrey, Brian Appelbe,
• Abstract summary: We present a mapping of the nonlinear, electrostatic Vlasov equation with Krook type collision operators, discretized on a (1 + 1) dimensional grid. We show that the quantum algorithm is guaranteed to converge only when the plasma parameters take unphysical values.
• Score: 0.0
• License:
• Abstract: We present a mapping of the nonlinear, electrostatic Vlasov equation with Krook type collision operators, discretized on a (1 + 1) dimensional grid, onto a recent Carleman linearization based quantum algorithm for solving ordinary differential equations (ODEs) with quadratic nonlinearities. We show that the quantum algorithm is guaranteed to converge only when the plasma parameters take unphysical values. This is due to the high level of dissipation in the ODE system required for convergence, that far exceeds the physical dissipation effect provided by the Krook operator. Additionally, we derive upper bounds for the query- and gate complexities of the quantum algorithm in the limit of large grid sizes. We conclude that these are polynomially larger than the time complexity of the corresponding classical algorithms. We find that this is mostly due to the dimension, sparsity and norm of the Carleman linearized evolution matrix.

Related papers

• Quantum and classical algorithms for nonlinear unitary dynamics [0.5729426778193399]
  We present a quantum algorithm for a non-linear differential equation of the form $fracd|urangledt. We also introduce a classical algorithm based on the Euler method allowing comparably scaling to the quantum algorithm in a restricted case.
  arXiv  Detail & Related papers  (2024-07-10T14:08:58Z)
• A Catalyst Framework for the Quantum Linear System Problem via the Proximal Point Algorithm [9.804179673817574]
  We propose a new quantum algorithm for the quantum linear system problem (QLSP) inspired by the classical proximal point algorithm (PPA) Our proposed method can be viewed as a meta-algorithm that allows inverting a modified matrix via an existing texttimattQLSP_solver. By carefully choosing the step size $eta$, the proposed algorithm can effectively precondition the linear system to mitigate the dependence on condition numbers that hindered the applicability of previous approaches.
  arXiv  Detail & Related papers  (2024-06-19T23:15:35Z)
• An Efficient Quantum Algorithm for Linear System Problem in Tensor Format [4.264200809234798]
  We propose a quantum algorithm based on the recent advances on adiabatic-inspired QLSA. We rigorously show that the total complexity of our implementation is polylogarithmic in the dimension.
  arXiv  Detail & Related papers  (2024-03-28T20:37:32Z)
• A hybrid quantum-classical algorithm for multichannel quantum scattering of atoms and molecules [62.997667081978825]
  We propose a hybrid quantum-classical algorithm for solving the Schr"odinger equation for atomic and molecular collisions. The algorithm is based on the $S$-matrix version of the Kohn variational principle, which computes the fundamental scattering $S$-matrix. We show how the algorithm could be scaled up to simulate collisions of large polyatomic molecules.
  arXiv  Detail & Related papers  (2023-04-12T18:10:47Z)
• A quantum algorithm for the linear Vlasov equation with collisions [0.0]
  We present a quantum algorithm that simulates the linearized Vlasov equation with and without collisions. We show that a quadratic speedup in system size is attainable.
  arXiv  Detail & Related papers  (2023-03-06T19:19:30Z)
• Dynamical chaos in nonlinear Schr\"odinger models with subquadratic power nonlinearity [137.6408511310322]
  We deal with a class of nonlinear Schr"odinger lattices with random potential and subquadratic power nonlinearity. We show that the spreading process is subdiffusive and has complex microscopic organization. The limit of quadratic power nonlinearity is also discussed and shown to result in a delocalization border.
  arXiv  Detail & Related papers  (2023-01-20T16:45:36Z)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• A theory of quantum differential equation solvers: limitations and fast-forwarding [19.080267236745623]
  We show that quantum algorithms suffer from computational overheads due to two types of non-quantumness'' We then show that ODEs in the absence of both types of non-quantumness'' are equivalent to quantum dynamics. We show how to fast-forward quantum algorithms for solving special classes of ODEs which leads to improved efficiency.
  arXiv  Detail & Related papers  (2022-11-09T22:50:14Z)
• Designing Kerr Interactions for Quantum Information Processing via Counterrotating Terms of Asymmetric Josephson-Junction Loops [68.8204255655161]
  static cavity nonlinearities typically limit the performance of bosonic quantum error-correcting codes. Treating the nonlinearity as a perturbation, we derive effective Hamiltonians using the Schrieffer-Wolff transformation. Results show that a cubic interaction allows to increase the effective rates of both linear and nonlinear operations.
  arXiv  Detail & Related papers  (2021-07-14T15:11:05Z)
• Fixed Depth Hamiltonian Simulation via Cartan Decomposition [59.20417091220753]
  We present a constructive algorithm for generating quantum circuits with time-independent depth. We highlight our algorithm for special classes of models, including Anderson localization in one dimensional transverse field XY model. In addition to providing exact circuits for a broad set of spin and fermionic models, our algorithm provides broad analytic and numerical insight into optimal Hamiltonian simulations.
  arXiv  Detail & Related papers  (2021-04-01T19:06:00Z)
• Quantum-Inspired Algorithms from Randomized Numerical Linear Algebra [53.46106569419296]
  We create classical (non-quantum) dynamic data structures supporting queries for recommender systems and least-squares regression. We argue that the previous quantum-inspired algorithms for these problems are doing leverage or ridge-leverage score sampling in disguise.
  arXiv  Detail & Related papers  (2020-11-09T01:13:07Z)

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.