A quantum-inspired approach to exploit turbulence structures

Gourianov, Nikita; Lubasch, Michael; Dolgov, Sergey; Berg, Q. Y. van den; Babaee, Hessam; Givi, Peyman; Kiffner, Martin; Jaksch, Dieter

doi:10.1038/s43588-021-00181-1

Cited by 43 publications

(39 citation statements)

References 46 publications

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“…Additionally, while it is unclear how well an MPS with interleaved ordering would perform for higher dimensional systems (Gourianov et al consider 3-D simulations using the grouped interwoven geometry, but they refrain from making strong claims about its performance [46]), we expect it to outperform an MPS with sequential ordering. For one, it is no longer possible to have an MPS with S3-like ordering in which dimensions are separated sequentially while also having tensors corresponding to fine grids be grouped together.…”

Section: Discussionmentioning

confidence: 88%

“…Alternatively, as mentioned by Refs. [29] and [46], one might consider representing the data using other tensor network ansatzë, such as tree tensor networks or 2-D tensor networks (PEPS) [50,51].…”

Section: Discussionmentioning

confidence: 99%

“…Alternatively, the data can be mapped such that tensors corresponding to similar grid resolutions but different dimensions are adjacent to each other (IF). This interleaved ordering is used by Gourianov et al [46], but they also contract tensors across different dimensions together so that now d = 2 K (IG). These different orderings are shown in the top row of Fig.…”

Section: A State Compressibilitymentioning

confidence: 99%

“…To the best of our knowledge, quantized tensor trains have only been discussed in a limited number of contexts including solving the Fokker-Planck equation [30], the chemical master equation [41,42], and finite element solvers of elliptic multi-scale problems [43][44][45]. More relevant is the recent work by Gourianov et al, in which they demonstrate the efficiency of MPS methods for simulating Navier-Stokes turbulence in two and three dimensions [46]. Still, the physics of the Navier-Stokes equation (a fluid equation) is fundamentally different from that of the more precise Vlasov equation (a kinetic equation) discussed here.…”

Section: Introductionmentioning

confidence: 99%

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A quantum-inspired method for solving the Vlasov-Poisson equations

Ye,

Loureiro

2022

Preprint

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Kinetic simulations of collisionless (or weakly collisional) plasmas using the Vlasov equation are often infeasible due to high resolution requirements and the exponential scaling of computational cost with respect to dimension. Recently, it has been proposed that matrix product state (MPS) methods, a quantum-inspired but classical algorithm, can be used to solve partial differential equations with exponential speed-up, provided that the solution can be compressed and efficiently represented as an MPS within some tolerable error threshold. In this work, we explore the practicality of MPS methods for solving the Vlasov-Poisson equations in 1D1V, and find that important features of linear and nonlinear dynamics, such as damping or growth rates and saturation amplitudes, can be captured while compressing the solution significantly. Furthermore, by comparing the performance of different mappings of the distribution functions onto the MPS, we develop an intuition of the MPS representation and its behavior in the context of solving the Vlasov-Poisson equations, which will be useful for extending these methods to higher dimensional problems.

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Section: Discussionmentioning

confidence: 88%

Section: Discussionmentioning

confidence: 99%

Section: A State Compressibilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A quantum-inspired method for solving the Vlasov-Poisson equations

Ye,

Loureiro

2022

Preprint

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“…It is clear that the present thermal convection flow model is still very low-dimensional and thus far away from convective turbulence. Our efforts should be considered as one first step to model real fluid flows on a quantum computer, a possible route beside other directions, such as quantum embeddings of nonlinear dynamical systems by the Koopman operator framework [60] or variational quantum algorithms for the direct solution of the equations of motion [61], see also ref. [66] for further directions such as lattice Boltzmann methods.…”

Section: Discussionmentioning

confidence: 99%

Hybrid quantum-classical reservoir computing of thermal convection flow

Pfeffer¹,

Heyder²,

Schumacher³

2022

Preprint

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We simulate the nonlinear chaotic dynamics of Lorenz-type models for a classical two-dimensional thermal convection flow with 3 and 8 degrees of freedom by a hybrid quantum-classical reservoir computing model. The high-dimensional quantum reservoir dynamics is established by universal quantum gates that rotate and entangle the individual qubits of the tensor product quantum state. A comparison of the quantum reservoir computing model with its classical counterpart shows that the same prediction and reconstruction capabilities of classical reservoirs with thousands of perceptrons can be obtained by a few strongly entangled qubits. We demonstrate that the mean squared error between model output and ground truth in the test phase of the quantum reservoir computing algorithm increases when the reservoir is decomposed into separable subsets of qubits. Furthermore, the quantum reservoir computing model is implemented on a real noisy IBM quantum computer for up to 7 qubits. Our work thus opens the door to model the dynamics of classical complex systems in a high-dimensional phase space effectively with an algorithm that requires a small number of qubits.

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Variational quantum solutions to the advection–diffusion equation for applications in fluid dynamics

Demirdjian

Gunlycke

Reynolds

et al. 2022

Quantum Inf Process

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Constraints in power consumption and computational power limit the skill of operational numerical weather prediction by classical computing methods. Quantum computing could potentially address both of these challenges. Herein, we present one method to perform fluid dynamics calculations that takes advantage of quantum computing. This hybrid quantum-classical method, which combines several algorithms, scales logarithmically with the dimension of the vector space and quadratically with the number of nonzero terms in the linear combination of unitary operators that specifies the linear operator describing the system of interest. As a demonstration, we apply our method to solve the advection-diffusion equation for a small system using IBM quantum computers. We find that reliable solutions of the equation can be obtained on even the noisy quantum computers available today. This and other methods that exploit quantum computers could replace some of our traditional methods in numerical weather prediction as quantum hardware continues to improve. KeywordsEarth and atmospheric sciences • Partial differential equations • Linear system of equations • Fluid mechanics DISTRIBUTION A. Approved for public release: distribution unlimited.

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A quantum-inspired approach to exploit turbulence structures

Cited by 43 publications

References 46 publications

A quantum-inspired method for solving the Vlasov-Poisson equations

A quantum-inspired method for solving the Vlasov-Poisson equations

Hybrid quantum-classical reservoir computing of thermal convection flow

Variational quantum solutions to the advection–diffusion equation for applications in fluid dynamics

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