Quantum Differential Equation Solvers with Low State Preparation Cost: Eliminating the Time Dependence in Dissipative Equations

• URL: http://arxiv.org/abs/2508.15170v1
• Date: Thu, 21 Aug 2025 02:17:21 GMT
• Title: Quantum Differential Equation Solvers with Low State Preparation Cost: Eliminating the Time Dependence in Dissipative Equations
• Authors: Gengzhi Yang, Akwum Onwunta, Dong An,
• Abstract summary: We propose efficient quantum algorithms for simulating linear dissipative differential equations.<n>The key idea of our algorithms is to perform the simulation only over an effective time period when the dynamics has not significantly dissipated yet.
• Score: 2.5677613431426978
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Linear dissipative differential equation is a fundamental model for a large number of physical systems, such as quantum dynamics with non-Hermitian Hamiltonian, open quantum system dynamics, diffusion process and damped system. In this work, we propose efficient quantum algorithms for simulating linear dissipative differential equations. The key idea of our algorithms is to perform the simulation only over an effective time period when the dynamics has not significantly dissipated yet, rather than over the entire physical evolution period. We conduct detailed analysis on the complexity of our algorithms and show that, while maintaining low state preparation cost, our algorithms can completely eliminate the time dependence. This is a more than exponential improvement compared to the previous state-of-the-art quantum algorithms.

Related papers

• Self-Supervised Coarsening of Unstructured Grid with Automatic Differentiation [55.88862563823878]
  In this work, we present an original algorithm to coarsen an unstructured grid based on the concepts of differentiable physics.<n>We demonstrate performance of the algorithm on two PDEs: a linear equation which governs slightly compressible fluid flow in porous media and the wave equation.<n>Our results show that in the considered scenarios, we reduced the number of grid points up to 10 times while preserving the modeled variable dynamics in the points of interest.
  arXiv  Detail & Related papers  (2025-07-24T11:02:13Z)
• Quantum simulation of a noisy classical nonlinear dynamics [2.6874004806796528]
  We consider the problem of simulating dynamics of classical nonlinear dissipative systems with $Ngg 1$ of freedom.<n>To make the problem tractable for quantum computers, we add a weak Gaussian noise to the equation of motion and the initial state.<n>For any constant nonzero noise rate, the runtime quantum scales to $log(N)$, evolution time, inverse error tolerance, and the relative strength of nonlinearity and dissipation.
  arXiv  Detail & Related papers  (2025-07-08T17:25:40Z)
• A time-marching quantum algorithm for simulation of the nonlinear Lorenz dynamics [0.0]
  We develop a quantum algorithm that implements the time evolution of a second order time-discretized version of the Lorenz model.<n> Notably, we showcase that it accurately captures the structural characteristics of the Lorenz system.
  arXiv  Detail & Related papers  (2025-06-26T15:08:00Z)
• Time-dependent Neural Galerkin Method for Quantum Dynamics [42.81677042059531]
  We introduce a classical computational method for quantum dynamics that relies on a global-in-time variational principle.<n>Our scheme computes the entire state trajectory over a finite time window by minimizing a loss function that enforces the Schr"odinger's equation.<n>We showcase the method by simulating global quantum quenches in the paradigmatic Transverse-Field Ising model in both 1D and 2D.
  arXiv  Detail & Related papers  (2024-12-16T13:48:54Z)
• 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)
• Solving Fractional Differential Equations on a Quantum Computer: A Variational Approach [0.1492582382799606]
  We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation cost scales economically with the number of time steps.
  arXiv  Detail & Related papers  (2024-06-13T02:27:16Z)
• Hamiltonian simulation for hyperbolic partial differential equations by scalable quantum circuits [1.6268784011387605]
  This paper presents a method that enables us to explicitly implement the quantum circuit for Hamiltonian simulation. We show that the space and time complexities of the constructed circuit are exponentially smaller than those of conventional classical algorithms.
  arXiv  Detail & Related papers  (2024-02-28T15:17:41Z)
• The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts [0.0]
  We give the first non-asymptotic computation of the cost of encoding the solution to general linear ordinary differential equations into quantum states. We show that the stability properties of a large class of classical dynamics allow their fast-forwarding. We find that the history state can always be output with complexity $O(T1/2)$ for any stable linear system.
  arXiv  Detail & Related papers  (2023-09-14T17:25:43Z)
• Variational Quantum Evolution Equation Solver [0.0]
  Variational quantum algorithms offer a promising new paradigm for solving partial differential equations on near-term quantum computers. We propose a variational quantum algorithm for solving a general evolution equation through implicit time-stepping of the Laplacian operator. We present a semi-implicit scheme for solving systems of evolution equations with non-linear terms, such as the reaction-diffusion and the incompressible Navier-Stokes equations.
  arXiv  Detail & Related papers  (2022-04-06T16:02:11Z)
• 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)
• 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)
• 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)
• Fast and differentiable simulation of driven quantum systems [58.720142291102135]
  We introduce a semi-analytic method based on the Dyson expansion that allows us to time-evolve driven quantum systems much faster than standard numerical methods. We show results of the optimization of a two-qubit gate using transmon qubits in the circuit QED architecture.
  arXiv  Detail & Related papers  (2020-12-16T21:43:38Z)

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.