Discrete Superconvergence Analysis for Quantum Magnus Algorithms of Unbounded Hamiltonian Simulation

• URL: http://arxiv.org/abs/2502.20255v1
• Date: Thu, 27 Feb 2025 16:43:28 GMT
• Title: Discrete Superconvergence Analysis for Quantum Magnus Algorithms of Unbounded Hamiltonian Simulation
• Authors: Yonah Borns-Weil, Di Fang, Jiaqi Zhang,
• Abstract summary: We provide the first superconvergence estimate in the fully discrete setting with a finite number of spatial discretization points $N$.<n>We establish a semiclassical framework by identifying two parameters through the discretization number and the time step size rescaled by the operator norm.
• Score: 3.5148549831413036
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Motivated by various applications, unbounded Hamiltonian simulation has recently garnered great attention. Quantum Magnus algorithms, designed to achieve commutator scaling for time-dependent Hamiltonian simulation, have been found to be particularly efficient for such applications. When applied to unbounded Hamiltonian simulation in the interaction picture, they exhibit an unexpected superconvergence phenomenon. However, existing proofs are limited to the spatially continuous setting and do not extend to discrete spatial discretizations. In this work, we provide the first superconvergence estimate in the fully discrete setting with a finite number of spatial discretization points $N$, and show that it holds with an error constant uniform in $N$. The proof is based on the two-parameter symbol class, which, to our knowledge, is applied for the first time in algorithm analysis. The key idea is to establish a semiclassical framework by identifying two parameters through the discretization number and the time step size rescaled by the operator norm, such that the semiclassical uniformity guarantees the uniformity of both. This approach may have broader applications in numerical analysis beyond the specific context of this work.

Related papers

• Exponential Quantum Speedup for Simulating Classical Lattice Dynamics [0.0]
  We introduce a rigorous quantum framework for simulating general harmonic lattice dynamics. We exploit well established quantum Hamiltonian simulation techniques. We demonstrate the applicability of the method across a broad class of lattice models.
  arXiv  Detail & Related papers  (2025-04-07T19:41:22Z)
• Quantum Simulation of Nonlinear Dynamical Systems Using Repeated Measurement [42.896772730859645]
  We present a quantum algorithm based on repeated measurement to solve initial-value problems for nonlinear ordinary differential equations. We apply this approach to the classic logistic and Lorenz systems in both integrable and chaotic regimes.
  arXiv  Detail & Related papers  (2024-10-04T18:06:12Z)
• 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)
• Time-dependent Hamiltonian Simulation via Magnus Expansion: Algorithm and Superconvergence [0.0]
  We introduce a new time-dependent Hamiltonian simulation algorithm based on the Magnus series expansion. We prove that the commutator in the second-order algorithm leads to a surprising fourth-order superconvergence.
  arXiv  Detail & Related papers  (2024-05-21T16:49:54Z)
• Quantum Algorithm for Solving the Advection Equation using Hamiltonian Simulation [0.0]
  One-dimensional advection can be simulated directly since the central finite difference operator for first-order derivatives is anti-Hermitian. A single copy of the initial quantum state is required and the circuit depth grows linearly with the required number of time steps.
  arXiv  Detail & Related papers  (2023-12-15T13:39:27Z)
• Simulating Markovian open quantum systems using higher-order series expansion [1.713291434132985]
  We present an efficient quantum algorithm for simulating the dynamics of Markovian open quantum systems. Our algorithm is conceptually cleaner, and it only uses simple quantum primitives without compressed encoding.
  arXiv  Detail & Related papers  (2022-12-05T06:02:50Z)
• Time Dependent Hamiltonian Simulation Using Discrete Clock Constructions [42.3779227963298]
  We provide a framework for encoding time dependent dynamics as time independent systems. First, we create a time dependent simulation algorithm based on performing qubitization on the augmented clock system. Second, we define a natural generalization of multiproduct formulas for time-ordered exponentials.
  arXiv  Detail & Related papers  (2022-03-21T21:29:22Z)
• Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schr\"odinger Equation [2.973326951020451]
  We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. For the simulation of the Schr"odinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate.
  arXiv  Detail & Related papers  (2021-11-04T18:50:36Z)
• Hybridized Methods for Quantum Simulation in the Interaction Picture [69.02115180674885]
  We provide a framework that allows different simulation methods to be hybridized and thereby improve performance for interaction picture simulations. Physical applications of these hybridized methods yield a gate complexity scaling as $log2 Lambda$ in the electric cutoff. For the general problem of Hamiltonian simulation subject to dynamical constraints, these methods yield a query complexity independent of the penalty parameter $lambda$ used to impose an energy cost.
  arXiv  Detail & Related papers  (2021-09-07T20:01:22Z)
• A Random Matrix Perspective on Random Tensors [40.89521598604993]
  We study the spectra of random matrices arising from contractions of a given random tensor. Our technique yields a hitherto unknown characterization of the local maximum of the ML problem. Our approach is versatile and can be extended to other models, such as asymmetric, non-Gaussian and higher-order ones.
  arXiv  Detail & Related papers  (2021-08-02T10:42:22Z)
• 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 algorithm for time-dependent Hamiltonian simulation by permutation expansion [6.338178373376447]
  We present a quantum algorithm for the dynamical simulation of time-dependent Hamiltonians. We demonstrate that the cost of the algorithm is independent of the Hamiltonian's frequencies.
  arXiv  Detail & Related papers  (2021-03-29T05:02:02Z)
• Efficient classical simulation of random shallow 2D quantum circuits [104.50546079040298]
  Random quantum circuits are commonly viewed as hard to simulate classically. We show that approximate simulation of typical instances is almost as hard as exact simulation. We also conjecture that sufficiently shallow random circuits are efficiently simulable more generally.
  arXiv  Detail & Related papers  (2019-12-31T19:00:00Z)

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.