[Paper Review] Potential quantum advantage for simulation of fluid dynamics

Xiangyu Li, Xiaolong Yin+5 more|arXiv (Cornell University)|Mar 29, 2023

Lattice Boltzmann Simulation Studies19 citations

TL;DR

The paper argues that, by Carleman-linearizing the lattice Boltzmann form of Navier–Stokes, a quantum algorithm can potentially achieve exponential speedup, with evidence that low-order linearization is accurate and the cost scales logarithmically with system size.

ABSTRACT

Numerical simulation of turbulent fluid dynamics needs to either parameterize turbulence-which introduces large uncertainties-or explicitly resolve the smallest scales-which is prohibitively expensive. Here we provide evidence through analytic bounds and numerical studies that a potential quantum exponential speedup can be achieved to simulate the Navier-Stokes equations governing turbulence using quantum computing. Specifically, we provide a formulation of the lattice Boltzmann equation for which we give evidence that low-order Carleman linearization is much more accurate than previously believed for these systems and that for computationally interesting examples. This is achieved via a combination of reformulating the nonlinearity and accurately linearizing the dynamical equations, effectively trading nonlinearity for additional degrees of freedom that add negligible expense in the quantum solver. Based on this we apply a quantum algorithm for simulating the Carleman-linearized lattice Boltzmann equation and provide evidence that its cost scales logarithmically with system size, compared to polynomial scaling in the best known classical algorithms. This work suggests that an exponential quantum advantage may exist for simulating fluid dynamics, paving the way for simulating nonlinear multiscale transport phenomena in a wide range of disciplines using quantum computing.

Motivation & Objective

• Motivate the need to simulate turbulent fluid dynamics beyond classical turbulence modeling and direct NSE simulations.
• Propose a Boltzmann equation reformulation to reduce nonlinearity and enable quantum advantages.
• Introduce Carleman linearization to convert nonlinear LBE into a linear system.
• Show that the resulting quantum algorithm can achieve logarithmic scaling in system size under practical conditions.

Proposed method

• Use the lattice Boltzmann equation (LBE) as the mesoscopic formulation of fluid dynamics.
• Nonlinear terms are managed by expanding 1/ρ and rewriting fm in terms of a polynomial in fm.
• Apply Carleman linearization to map the nonlinear LBE to an infinite linear system via f, f[2], ..., f[k], and truncate at degree k.
• Construct n-point Carleman linearized LBE, with diagonality in x for efficient quantum treatment.
• Represent the linearized system with block-encoded matrices suitable for quantum linear systems algorithms.
• Analyze truncation errors and scalability, and discuss how the macroscopic quantities (ρ, ρu) are recovered from moments.

Experimental results

Research questions

• RQ1 Can low-order Carleman linearization provide accurate macroscopic fluid quantities (ρ, ρu) for weakly compressible flows (Ma ≪ 1)?
• RQ2 Does the Carleman-linearized lattice Boltzmann equation admit a quantum algorithm whose cost scales logarithmically with system size N?
• RQ3 What is the impact of truncating Carleman order k on accuracy and on the derived macroscopic observables?
• RQ4 Under what conditions can a quantum speedup be realized for simulating multiscale fluid dynamics via LBE compared to best classical methods?

Key findings

• Carleman truncation yields corrections to ρ and ρu that are suppressed by O(Ma^2), indicating controlled approximation at low Ma.
• The macroscopic fluid properties follow a hierarchy where higher-order corrections are suppressed by factors of O(Ma^2) relative to retained terms.
• The n-point Carleman LBE is linear and diagonally structured in Carleman degree, enabling efficient quantum root-of-solution access.
• A quantum algorithm for the Carleman-linearized LBE shows evidence that cost scales logarithmically with system size N, versus polynomial scaling classically.
• Overall, the results suggest the possibility of exponential quantum advantage for simulating nonlinear multiscale transport phenomena in fluid dynamics.
• The analysis is framed to indicate how turbulence simulation could be tractable on quantum computers using LBE and Carleman linearization.