Quantum algorithm for the advection–diffusion equation simulated with the lattice Boltzmann method

Budinski, Ljubomir

doi:10.1007/s11128-021-02996-3

Cited by 34 publications

(16 citation statements)

References 39 publications

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“…What is even more surprising is that other disciplines away from quantum physics, yet heavily relying on numerical calculus (fluid mechanics, finance, etc) already applied quantum algorithms to their own cost-intensive problems. For instance, several works in fluid mechanics field used quantum subroutines to solve both the lattice Boltzmann (Mezzacapo et al, 2015;Todorova and Steijl, 2020;Budinski, 2021a) and the Navier-Stokes (Steijl and Barakos, 2018;Gaitan, 2020;Budinski, 2021b;Gaitan, 2021) equations. The hope is that structual mechanics will also explore the use of quantum algorithms to support expensive simulations, such as those involving material nonlinearities and large structural deformations.…”

Section: Discussionmentioning

confidence: 99%

Review and perspectives in quantum computing for partial differential equations in structural mechanics

et al. 2022

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Structural mechanics is commonly modeled by (systems of) partial differential equations (PDEs). Except for very simple cases where analytical solutions exist, the use of numerical methods is required to find approximate solutions. However, for many problems of practical interest, the computational cost of classical numerical solvers running on classical, that is, silicon-based computer hardware, becomes prohibitive. Quantum computing, though still in its infancy, holds the promise of enabling a new generation of algorithms that can execute the most cost-demanding parts of PDE solvers up to exponentially faster than classical methods, at least theoretically. Also, increasing research and availability of quantum computing hardware spurs the hope of scientists and engineers to start using quantum computers for solving PDE problems much faster than classically possible. This work reviews the contributions that deal with the application of quantum algorithms to solve PDEs in structural mechanics. The aim is not only to discuss the theoretical possibility and extent of advantage for a given PDE, boundary conditions and input/output to the solver, but also to examine the hardware requirements of the methods proposed in literature.

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Section: Discussionmentioning

confidence: 99%

Review and perspectives in quantum computing for partial differential equations in structural mechanics

et al. 2022

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“…The Navier-Stokes equations are a set of non-linear partial differential equations that describes the motion of fluids across continuum length scales. There are several studies aimed at applying quantum algorithms to computational fluid dynamics (see review 49 ), ranging from reduction of partial differential equations to ordinary differential equations 50 and quantum solutions of sub-steps of the classical algorithm 51 , 52 to the quantum Lattice Boltzmann scheme 53 .…”

Section: Applications To the Navier–stokes Equationsmentioning

confidence: 99%

Variational quantum evolution equation solver

Leong

Ewe

Koh

2022

Sci Rep

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Variational quantum algorithms offer a promising new paradigm for solving partial differential equations on near-term quantum computers. Here, we propose a variational quantum algorithm for solving a general evolution equation through implicit time-stepping of the Laplacian operator. The use of encoded source states informed by preceding solution vectors results in faster convergence compared to random re-initialization. Through statevector simulations of the heat equation, we demonstrate how the time complexity of our algorithm scales with the Ansatz volume for gradient estimation and how the time-to-solution scales with the diffusion parameter. Our proposed algorithm extends economically to higher-order time-stepping schemes, such as the Crank–Nicolson method. 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, and demonstrate its validity by proof-of-concept results.

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“…It is plausible that empirical advantages with these methods may be achievable in the near term given their hybrid quantum-classical structure. Finally, the past few years have also seen progress in quantum algorithms for solving classical differential equations, either for general cases [295,296,377,378,379] or specific applications, like the finite element method (FEM) [380,381] or Navier-Stokes [382,383]. Importantly, among these quantum algorithms are ones for solving the more difficult cases of non-homogeneous and nonlinear PDEs [296,379].…”

Section: Simulating Classical Physicsmentioning

confidence: 99%

Biology and medicine in the landscape of quantum advantages

Cordier¹,

Sawaya²,

Guerreschi³

et al. 2021

Preprint

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Quantum computing holds significant potential for applications in biology and medicine, spanning from the simulation of biomolecules to machine learning approaches for subtyping cancers on the basis of clinical features. This potential is encapsulated by the concept of a quantum advantage, which is typically contingent on a reduction in the consumption of a computational resource, such as time, space, or data. Here, we distill the concept of a quantum advantage into a simple framework that we hope will aid researchers in biology and medicine pursuing the development of quantum applications. We then apply this framework to a wide variety of computational problems relevant to these domains in an effort to i) assess the potential of quantum advantages in specific application areas and ii) identify gaps that may be addressed with novel quantum approaches. Bearing in mind the rapid pace of change in the fields of quantum computing and classical algorithms, we aim to provide an extensive survey of applications in biology and medicine that may lead to practical quantum advantages.

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Quantum algorithm for the advection–diffusion equation simulated with the lattice Boltzmann method

Cited by 34 publications

References 39 publications

Review and perspectives in quantum computing for partial differential equations in structural mechanics

Review and perspectives in quantum computing for partial differential equations in structural mechanics

Variational quantum evolution equation solver

Biology and medicine in the landscape of quantum advantages

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