Trotter-based quantum algorithm for solving transport equations with exponentially fewer time-steps

• URL: http://arxiv.org/abs/2508.15691v2
• Date: Mon, 25 Aug 2025 12:00:54 GMT
• Title: Trotter-based quantum algorithm for solving transport equations with exponentially fewer time-steps
• Authors: Julien Zylberman, Thibault Fredon, Nuno F. Loureiro, Fabrice Debbasch,
• Abstract summary: We present a quantum numerical scheme based on three steps: quantum state preparation, evolution, and measurement.<n>We introduce novel vector-norm analysis and prove that the number of time-steps can be reduced by a factor exponential in the number of qubits.<n>We also present efficient quantum circuits and numerical simulations that confirm the predicted vector-norm scaling.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: The extent to which quantum computers can simulate physical phenomena and solve the partial differential equations (PDEs) that govern them remains a central open question. In this work, one of the most fundamental PDEs is addressed: the multidimensional transport equation with space- and time-dependent coefficients. We present a quantum numerical scheme based on three steps: quantum state preparation, evolution, and measurement of relevant observables. The evolution step combines a high-order centered finite difference with a time-splitting scheme based on product formula approximations, also known as Trotterization. We introduce novel vector-norm analysis and prove that the number of time-steps can be reduced by a factor exponential in the number of qubits compared to previously established operator-norm analysis, thereby significantly lowering the projected computational resources. We also present efficient quantum circuits and numerical simulations that confirm the predicted vector-norm scaling. We report results on real quantum hardware for the one-dimensional convection equation, and solve a non-linear ordinary differential equation via its associated Liouville equation, a particular case of transport equations. This work provides a practical framework for efficiently simulating transport phenomena on quantum computers, with potential applications in plasma physics, molecular gas dynamics and non-linear dynamical systems, including chaotic systems.

Related papers

• Quantum Hamiltonian simulation of linearised Euler equations in complex geometries [0.0]
  Quantum computing promises exponential improvements in solving systems of partial differential equations.<n>One approach is via mapping classical CFD problems to a quantum Hamiltonian evolution.<n>Here we extend this method to more complex and practically relevant cases.
  arXiv  Detail & Related papers  (2025-10-20T18:00:16Z)
• Quantum Algorithm for Subcellular Multiscale Reaction-Diffusion Systems [1.4090780535978131]
  A key challenge lies in the "curse of dimensionality"<n>We introduce a quantum algorithmic framework designed to overcome these challenges.<n>This framework opens the door to simulations of biologically relevant subcellular processes.
  arXiv  Detail & Related papers  (2025-09-25T02:04:41Z)
• Variational Quantum Linear Solver for Simulating Quantum Transport in Nanoscale Semiconductor Devices [4.258785028734343]
  We develop simulation methods that enable the application of the variational quantum linear solver (VQLS) to simulate quantum transport in nanoscale semiconductor devices.<n>We propose new forms of cost functions to solve complex and non-symmetric linear systems with faster computing speed.
  arXiv  Detail & Related papers  (2025-09-06T01:53:32Z)
• Quantum Speedup in Dissecting Roots and Solving Nonlinear Algebraic Equations [0.0]
  It is shown that quantum computer can detect the existence of root of a function almost exponentially more efficient than the classical counterpart.<n>Various applications and implications are discussed, including a quantum algorithm for solving dense linear systems.
  arXiv  Detail & Related papers  (2025-03-09T13:27:11Z)
• Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits [63.733312560668274]
  Given a quantum circuit containing d tunable RZ gates and G-d Clifford gates, can a learner perform purely classical inference to efficiently predict its linear properties? We prove that the sample complexity scaling linearly in d is necessary and sufficient to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. We devise a kernel-based learning model capable of trading off prediction error and computational complexity, transitioning from exponential to scaling in many practical settings.
  arXiv  Detail & Related papers  (2024-08-22T08:21:28Z)
• Parallel Quantum Computing Simulations via Quantum Accelerator Platform Virtualization [44.99833362998488]
  We present a model for parallelizing simulation of quantum circuit executions. The model can take advantage of its backend-agnostic features, enabling parallel quantum circuit execution over any target backend.
  arXiv  Detail & Related papers  (2024-06-05T17:16:07Z)
• Solving reaction dynamics with quantum computing algorithms [42.408991654684876]
  We study quantum algorithms for response functions, relevant for describing different reactions governed by linear response.<n>We focus on nuclear-physics applications and consider a qubit-efficient mapping on the lattice, which can efficiently represent the large volumes required for realistic scattering simulations.
  arXiv  Detail & Related papers  (2024-03-30T00:21:46Z)
• Ab-initio variational wave functions for the time-dependent many-electron Schrödinger equation [41.94295877935867]
  We introduce a variational approach for fermionic time-dependent wave functions, surpassing mean-field approximations. We use time-dependent Jastrow factors and backflow transformations, which are enhanced through neural networks parameterizations. The results showcase the ability of our variational approach to accurately capture the time evolution, providing insight into the quantum dynamics of interacting electronic systems.
  arXiv  Detail & Related papers  (2024-03-12T09:37:22Z)
• Quantum computing of reacting flows via Hamiltonian simulation [13.377719901871027]
  We develop the quantum spectral and finite difference methods for simulating reacting flows in periodic and general conditions. The present quantum computing algorithms offer a one-shot'' solution for a given time without temporal discretization.
  arXiv  Detail & Related papers  (2023-12-13T04:31:49Z)
• 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)
• Universality of critical dynamics with finite entanglement [68.8204255655161]
  We study how low-energy dynamics of quantum systems near criticality are modified by finite entanglement. Our result establishes the precise role played by entanglement in time-dependent critical phenomena.
  arXiv  Detail & Related papers  (2023-01-23T19:23:54Z)
• 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)
• Quantum Model-Discovery [19.90246111091863]
  Quantum algorithms for solving differential equations have shown a provable advantage in the fault-tolerant quantum computing regime. We extend the applicability of near-term quantum computers to more general scientific machine learning tasks. Our results show a promising path to Quantum Model Discovery (QMoD) on the interface between classical and quantum machine learning approaches.
  arXiv  Detail & Related papers  (2021-11-11T18:45:52Z)
• Information Scrambling in Computationally Complex Quantum Circuits [56.22772134614514]
  We experimentally investigate the dynamics of quantum scrambling on a 53-qubit quantum processor. We show that while operator spreading is captured by an efficient classical model, operator entanglement requires exponentially scaled computational resources to simulate.
  arXiv  Detail & Related papers  (2021-01-21T22:18:49Z)

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.