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Communications in Computational Physics

Prospective Merger Between Car-Parrinello and Lattice Boltzmann Methods for Quantum Many-Body Simulations

Sauro Succi A, Silvia Palpacelli

Published Jan 1, 2011• 1 citation
Publisher

Abstract

Formal analogies between the Car-Parrinello (CP) ab-initio molecular dynamics for quantum many-body systems, and the Lattice Boltzmann (LB) method for classical and quantum fluids, are pointed out. A theoretical scenario, whereby the quantum LB would be coupled to the CP framework to speed-up many-body quantum simulations, is also discussed, together with accompanying considerations on the computational efficiency of the prospective CP-LB scheme.

DOI
10.4208/cicp.141009.091110s

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Formal analogies between the Car-Parrinello (CP) ab-initio molecular dynamics for quantum many-body systems, and the Lattice Boltzmann (LB) method for classical and quantum fluids, are pointed out. A theoretical scenario, whereby the quantum LB would be coupled to the CP framework to speed-up many-body quantum simulations, is also discussed, together with accompanying considerations on the computational efficiency of the prospective CP-LB scheme.
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Ab-initio molecular dynamics (AIMD) is a powerful tool to simulate physical movements of molecules for investigating properties of materials. While AIMD is successful in some applications, circumventing its high computational costs is imperative to perform large-scale and long-time simulations. In recent days, near-term quantum computers have attracted much attentions as a possible solution to alleviate the challenge. Specifically, Kuroiwa et al. proposed a new AIMD method called quantum Car-Parrinello molecular dynamics (QCPMD), which exploits the Car-Parrinello method and Langevin formulation to realize cost-efficient simulations at the equilibrium state, using near-term quantum devices. In this work, we build on the proposed QCPMD method and introduce the classical shadow technique to further improve resource efficiency of the simulations. More precisely, classical shadows are used to estimate the forces of all nuclei simultaneously, implying this approach is more effective as the number of molecules increases. We numerically study the performance of our scheme on the H2\text{H}_2H2​ molecule and show that QCPMD with classical shadows can simulate the equilibrium state. Our results will give some insights into efficient AIMD simulations on currently-available quantum computers.
Quantum computing offers a promising pathway to accelerate fluid dynamics simulations. In this work, we propose a hybrid quantum-classical lattice Boltzmann method (LBM) for solving the advection–diffusion equation, as a step toward an efficient and general quantum linear solver framework for fluid flows. Quantum circuits handle the computationally intensive collision and streaming steps, exploiting quantum parallelism and removing parameter constraints, while boundary corrections are managed classically to avoid unnecessary overhead. Various boundary conditions are incorporated within the quantum framework, enhancing the flexibility and applicability of the method. Numerical experiments with Gaussian hill initial conditions under periodic boundaries and zero initial conditions under mixed Dirichlet–Neumann boundaries, each in one and two dimensions, demonstrate the precision of the proposed approach. The computational complexity is analyzed in detail. The collision and streaming steps achieve a lower complexity than their classical LBM counterparts, although overall performance is still limited by the input and output costs. This work represents a critical step toward the practical quantum simulation of complex fluid flows.
Abstract The Lattice Boltzmann concepts and applications described so far refer to classical, i.e., non-quantum physics. However, the LB formalism is not restricted to classical Newtonian mechanics and indeed an LB formulation of quantum mechanics, going by the name of quantum LB (QLB) has been in existence for more than two decades. Even though it would far-fetched to say that QLB represents a mainstream, in the recent years it has captured some revived interest, mostly on account of recent developments in quantum-computing research. This chapter provides an account of the QLB formulation: stay tuned, LBE goes quantum!
Lattice Boltzmann (LB) methods have proven to be powerful tools for simulating classical physical systems such as waves, diffusion and fluid dynamics. For quantum mechanics in one dimension there are two main schemes: one proposed by Meyer, where the Schrödinger equation is interpreted as a diffusion equation with imaginary diffusion constant; the other, proposed by Succi, is based on the Dirac equation, which in the mesoscopic limit reproduces the non-relativistic quantum behaviour in a similar way to lattice Boltzmann models for fluids. In the present work we compare these models by testing the quantum time evolution of a free Gaussian wavepacket. It is found to be in close correspondence with the results of the two models. Time oscillations are found both for the mean and variance of the wavepacket. Also we found their dependences from factors not present in the predicted values. These elements could provide clues for developing a good adaptation of lattice Boltzmann method for the quantum world.
Quantum computing holds transformative potential for simulating nonlinear physical systems, such as fluid turbulence. However, mapping nonlinear dynamics to the linear operations required by quantum hardware remains a fundamental challenge. Here, we bridge this gap by introducing a novel node-level ensemble description of a lattice gas, which enables the simulation of nonlinear fluid dynamics on quantum computers. This approach combines the advantages of the lattice Boltzmann method with low-dimensional representation (computational cost) and lattice gas cellular automata with linear collision treatment (quantum compatibility). Building on this framework, we propose a quantum lattice Boltzmann method that relies on linear operations with medium dimensionality, offering the potential for quantum speedup. We validated the algorithm through simulations of vortex-pair merging and decaying turbulence on up to 16.8 million computational grid points. The results demonstrate remarkable agreement with direct numerical simulation, effectively capturing the essential nonlinear mechanisms of fluid dynamics. This work potentially advances the development of quantum algorithms for other nonlinear problems across various transport phenomena in engineering.