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Towards Quantum Machine Learning of Lattice Boltzmann Collision Operators for Fluid Dynamic Simulations

Wael Itani, Katepalli R. Sreenivasan

TL;DR

This work explores learning a unitary operator to approximate the lattice Boltzmann collision operator, leveraging a modified amplitude encoding to avoid renormalization and embedding LB symmetries directly into the quantum circuit. By training symmetry-aware collision ansatze against datasets generated from LB flows and evaluating on the lid-driven cavity, the authors show that nonlinear collision dynamics can be approximated within a limited velocity range and that symmetry hard-wiring markedly improves training efficiency and conservation. A detailed complexity and error analysis reveals qubit and gate requirements, as well as bounds on applicable Reynolds and Mach numbers, highlighting both the promise and current limits of quantum ML for nonlinear fluid simulations. The study suggests that quantum ML can serve as a nonlinear corrector for linear LB components and lays groundwork for future encoding strategies (e.g., autoencoders) and more scalable quantum approaches to nonlinear transport phenomena. Overall, the results provide critical insights into how quantum circuit design, data encoding, and symmetry considerations intersect to enable (and constrain) quantum simulations of fluid dynamics at the mesoscale.

Abstract

We attempt the use of a unitary operator to approximate the lattice Boltzmann collision operator. We use a modified amplitude encoding to bypass the renormalization that would have required classical processing at every step (thus eroding any quantum advantage to be had). We describe the hard-wiring of the lattice Boltzmann symmetries into the quantum circuit and show that, for the specific case of the cavity flow, approximating the nonlinear system is limited to low velocities. These findings may help us understand better the possibilities of nonlinear simulations on a quantum computer, and also pave the way for a discussion on how quantum machine learning might be harnessed to address more complex problems.

Towards Quantum Machine Learning of Lattice Boltzmann Collision Operators for Fluid Dynamic Simulations

TL;DR

This work explores learning a unitary operator to approximate the lattice Boltzmann collision operator, leveraging a modified amplitude encoding to avoid renormalization and embedding LB symmetries directly into the quantum circuit. By training symmetry-aware collision ansatze against datasets generated from LB flows and evaluating on the lid-driven cavity, the authors show that nonlinear collision dynamics can be approximated within a limited velocity range and that symmetry hard-wiring markedly improves training efficiency and conservation. A detailed complexity and error analysis reveals qubit and gate requirements, as well as bounds on applicable Reynolds and Mach numbers, highlighting both the promise and current limits of quantum ML for nonlinear fluid simulations. The study suggests that quantum ML can serve as a nonlinear corrector for linear LB components and lays groundwork for future encoding strategies (e.g., autoencoders) and more scalable quantum approaches to nonlinear transport phenomena. Overall, the results provide critical insights into how quantum circuit design, data encoding, and symmetry considerations intersect to enable (and constrain) quantum simulations of fluid dynamics at the mesoscale.

Abstract

We attempt the use of a unitary operator to approximate the lattice Boltzmann collision operator. We use a modified amplitude encoding to bypass the renormalization that would have required classical processing at every step (thus eroding any quantum advantage to be had). We describe the hard-wiring of the lattice Boltzmann symmetries into the quantum circuit and show that, for the specific case of the cavity flow, approximating the nonlinear system is limited to low velocities. These findings may help us understand better the possibilities of nonlinear simulations on a quantum computer, and also pave the way for a discussion on how quantum machine learning might be harnessed to address more complex problems.
Paper Structure (56 sections, 34 equations, 13 figures, 8 tables)

This paper contains 56 sections, 34 equations, 13 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Lattice Boltzmann Method
  3. Overview
  4. Formulation
  5. Streaming
  6. Collision
  7. Equilibrium Distribution for a Single-Phase Fluid
  8. Prior Attempts at Quantum Algorithms for Lattice Boltzmann
  9. Brief Overview of Quantum Machine Learning
  10. Methods
  11. Dataset Generation
  12. Lattice Basis Vector Encoding
  13. Modified Amplitude Encoding of Discrete Densities
  14. Neural Network Training and Architecture
  15. Training
  16. ...and 41 more sections

Figures (13)

  • Figure 1: Different lattice configurations in three, two and one dimensions
  • Figure 2: Main lines of research for the approaches on a quantum algorithm for the lattice Boltzmann method.
  • Figure 3: Overview of the quantum circuit used for training, including the lattice basis vector, value, and lattice site registers, as well as the ancilla qubits.
  • Figure 4: A quantum circuit showing the minimum viable entangling layer
  • Figure 5: A quantum circuit showing an entangling layer with periodic CNOTs
  • ...and 8 more figures