quantum-lbm-surrogate

name	quantum-lbm-surrogate
description	Hybrid quantum-classical surrogate model for Lattice Boltzmann Method (LBM) collision dynamics. Uses parameterized quantum circuits with data re-uploading to implement partial Fourier series, recovering complete BGK collision dynamics across full physically admissible range of relaxation without retraining. Use when building quantum surrogates for PDE solvers, fluid dynamics, or any physics simulation requiring non-unitary operation approximation.
version	1.0.0
tags	["quantum","surrogate-model","LBM","fluid-dynamics","VQC","PDE","non-unitary","data-reuploading","physics-simulation"]
source	arXiv:2606.31351
authors	["Lukas C. Birk","David M. Wawrzyniak","Josef M. Winter","Steffen J. Schmidt","Thomas Indinger","Christian F. Janßen","Nikolaus A. Adams"]
published	2026-06-30T00:00:00.000Z
category	quantum-computing
trigger_words	["quantum surrogate LBM","quantum fluid dynamics","quantum collision operator","quantum PDE solver","VQC expressibility","data re-uploading quantum","quantum BGK collision"]

Quantum LBM Surrogate Model

Core Insight

Pure quantum solvers struggle with non-unitary operations. A hybrid approach uses a quantum machine learning surrogate to approximate non-linear collision dynamics of the Lattice Boltzmann Method (LBM), effectively offloading the non-unitary operations that challenge pure quantum solvers.

Architecture

1. Parameterized Quantum Circuit (VQC) Surrogate

• Expressivity source: Parameterized quantum circuits implement partial Fourier series
• Data re-uploading: Extends the spectrum of representable frequencies
• Complete BGK recovery: Surrogate recovers complete Bhatnagar-Gross-Krook collision dynamics across the full physically admissible range of relaxation parameters WITHOUT retraining

2. Why This Works

• LBM collision operator is non-unitary (cannot be directly implemented on quantum hardware)
• VQC with data re-uploading can approximate arbitrary functions via Fourier series
• The surrogate learns the mapping: input distribution → post-collision distribution
• Once trained, it works across ALL physically valid relaxation parameters

Implementation Pattern

# Conceptual workflow
1. Generate training data from classical BGK collision operator
   - Sample across full range of relaxation parameters (τ ∈ [0.5, 2.0])
   - Include diverse flow configurations

2. Design VQC with data re-uploading
   - Encode input distribution states
   - Apply parameterized gates with trainable angles
   - Re-upload data multiple times to extend frequency spectrum
   - Measure output distribution

3. Train VQC to minimize surrogate error
   - Loss: ||f_post_collision - VQC_output||²
   - Use gradient-based or gradient-free optimization

4. Validate on benchmark problems
   - Taylor-Green vortex (energy dissipation)
   - Double shear layer (shear-driven instabilities)

5. Assess VQC metrics
   - Expressibility → surrogate accuracy
   - Entanglement capability → representational power
   - Effective dimension → generalization capacity

Key Findings

VQC Metric Relevance