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Fully Quantum Lattice Gas Automata Building Blocks for Computational Basis State Encodings

Călin A. Georgescu, Merel A. Schalkers, Matthias Möller

TL;DR

This work advances Quantum Lattice Gas Automata by introducing a cohesive set of fully quantum building blocks that operate under a Space-Time data encoding with computational-basis qubits. It details expressive pointwise and scalable volumetric initializations, scalable boundary-condition implementations (including edge-case handling and staircase geometries), and generalized collision operators that apply to arbitrary $D_dQ_q$ discretizations via permutation, redistribution, and Coll$(k,N)$ constructions. Measurement and momentum-exchange techniques are adapted to the Space-Time encoding, enabling mass, density, pressure, and force extraction directly from quantum states, with open-source implementations provided. Collectively, these building blocks enable end-to-end QLGA pipelines with complexity analyses, offering practical paths toward quantum-accelerated CFD simulations and establishing reusable primitives for broader QLGA research.

Abstract

Lattice Gas Automata (LGA) is a classical method for simulating physical phenomena, including Computational Fluid Dynamics (CFD). Quantum LGA (QLGA) is the family of methods that implement LGA schemes on quantum computers. In recent years, QLGA has garnered attention from researchers thanks to its potential of efficiently modeling CFD processes by either reducing memory requirements or providing simultaneous representations of exponentially many LGA states. In this work, we introduce novel building blocks for QLGA algorithms that rely on computational basis state encodings. We address every step of the algorithm, from initial conditions to measurement, and provide detailed complexity analyses that account for all discretization choices of the system under simulation. We introduce multiple ways of instantiating initial conditions, efficient boundary condition implementations for novel geometrical patterns, a novel collision operator that models less restricted interactions than previous implementations, and quantum circuits that extract quantities of interest out of the quantum state. For each building block, we provide intuitive examples and open-source implementations of the underlying quantum circuits.

Fully Quantum Lattice Gas Automata Building Blocks for Computational Basis State Encodings

TL;DR

This work advances Quantum Lattice Gas Automata by introducing a cohesive set of fully quantum building blocks that operate under a Space-Time data encoding with computational-basis qubits. It details expressive pointwise and scalable volumetric initializations, scalable boundary-condition implementations (including edge-case handling and staircase geometries), and generalized collision operators that apply to arbitrary discretizations via permutation, redistribution, and Coll constructions. Measurement and momentum-exchange techniques are adapted to the Space-Time encoding, enabling mass, density, pressure, and force extraction directly from quantum states, with open-source implementations provided. Collectively, these building blocks enable end-to-end QLGA pipelines with complexity analyses, offering practical paths toward quantum-accelerated CFD simulations and establishing reusable primitives for broader QLGA research.

Abstract

Lattice Gas Automata (LGA) is a classical method for simulating physical phenomena, including Computational Fluid Dynamics (CFD). Quantum LGA (QLGA) is the family of methods that implement LGA schemes on quantum computers. In recent years, QLGA has garnered attention from researchers thanks to its potential of efficiently modeling CFD processes by either reducing memory requirements or providing simultaneous representations of exponentially many LGA states. In this work, we introduce novel building blocks for QLGA algorithms that rely on computational basis state encodings. We address every step of the algorithm, from initial conditions to measurement, and provide detailed complexity analyses that account for all discretization choices of the system under simulation. We introduce multiple ways of instantiating initial conditions, efficient boundary condition implementations for novel geometrical patterns, a novel collision operator that models less restricted interactions than previous implementations, and quantum circuits that extract quantities of interest out of the quantum state. For each building block, we provide intuitive examples and open-source implementations of the underlying quantum circuits.

Paper Structure

This paper contains 40 sections, 46 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Classical and Quantum Lattice Gas Automata
  3. The Space-Time Data Encoding and Running Examples
  4. Example -- $\mathrm{D}_{1}\mathrm{Q}_{2}$, $\mathrm{D}_{2}\mathrm{Q}_{4}$, and $\mathrm{D}_{3}\mathrm{Q}_{6}$.
  5. Complexity analysis assumptions.
  6. Initial Conditions
  7. Pointwise Initialization
  8. Example -- $\mathrm{D}_{2}\mathrm{Q}_{4}$ pointwise initialization.
  9. Complexity Analysis.
  10. Volumetric Initialization
  11. Example -- $\mathrm{D}_{1}\mathrm{Q}_{2}$ volumetric initialization.
  12. Edge cases -- $\mathrm{D}_{2}\mathrm{Q}_{4}$ initialization.
  13. Complexity analysis.
  14. Streaming and Boundary Conditions
  15. Pointwise Imposition of Boundary Conditions
  16. ...and 25 more sections

Figures (22)

  • Figure 1: Overview of the LGA collision and streaming steps.
  • Figure 2: QLGA algorithm overview for $N_t$ time steps.
  • Figure 3: Space-Time data structure and quantum register for the $\mathrm{D}_{1}\mathrm{Q}_{2}$ discretization.
  • Figure 4: Space-Time data structure and quantum register for the $\mathrm{D}_{2}\mathrm{Q}_{4}$ discretization.
  • Figure 5: $\mathrm{D}_{2}\mathrm{Q}_{4}$ pointwise initialization of the gridpoint at location $(1, 5)$ and velocity profile $\ket{1100}$ for one time step.
  • ...and 17 more figures