Quantum Lattice Boltzmann Method (QLBM)

• QLBM is a quantum algorithmic framework that leverages quantum parallelism and amplitude encoding to simulate nonlinear heat transfer and phase transitions.
• It integrates classical lattice Boltzmann collision-streaming dynamics with quantum circuits, using SWAP and MCRY gates to efficiently track moving phase interfaces.
• Simulation results on a D1Q3 lattice with 51 qubits and 17 nodes achieved RMS errors below 0.005, demonstrating accurate modeling of phase change dynamics.

The Quantum Lattice Boltzmann Method (QLBM) is a quantum algorithmic framework for simulating heat transfer with phase change, specifically designed to address the computational intensity and nonlinearity associated with evolving phase boundaries. The QLBM builds on the principles of the classical lattice Boltzmann method (LBM), incorporating quantum parallelism and quantum circuit encoding techniques to efficiently model the discontinuous and nonlinear nature of enthalpy-temperature relationships during phase transitions. By storing phase change information directly within the quantum circuit, the algorithm significantly reduces the frequency of classical-quantum communication, which is a known bottleneck in hybrid approaches. Presented results for a one-dimensional D1Q3 case using 17 lattice nodes and 51 qubits demonstrate root-mean-square (RMS) errors below 0.005 compared to analytical and classical LBM solutions, accurately tracking the evolution and movement of the solid-liquid interface through the phase transition process (Jawetz et al., 25 Sep 2025).

1. Foundational Principles and Quantum Encoding

QLBM exploits the inherently statistical and local nature of the LBM, wherein fictitious particle populations on a discrete lattice evolve according to prescribed collision and streaming steps. In the D1Q3 context, each node is associated with three velocity channels, and the distribution function for each channel at node $x_k$ and time $t$ is encoded as the amplitude of a qubit state:

$$|q_i(x_k, t)\rangle = \sqrt{1-f_i(x_k, t)}\,|0\rangle + \sqrt{f_i(x_k, t)}\,|1\rangle$$

This amplitude encoding corresponds to the probabilistic occupation of each velocity channel, an approach that aligns with the Born rule of quantum measurement and preserves the statistical fidelity of the physical model throughout the quantum circuit execution. The quantum circuit implements the collision operator as a unitary transformation $U_c$ (enforcing BGK-like relaxation), while the streaming operator is realized by SWAP gates, enabling efficient spatial propagation of populations without explicit data shifting in classical memory.

2. Interface Tracking and Domain Partitioning

Phase change problems introduce a discontinuity in the enthalpy–temperature relation, particularly at the melting temperature $T_\text{melt}$. The enthalpy $H$ is defined in a piecewise manner:

$$H = \begin{cases} c_p T, & T < T_\text{melt} \ c_p T + \mathcal{L}_\text{melt}, & T > T_\text{melt} \end{cases}$$

To efficiently deal with this discontinuity on a quantum computer, the QLBM partitions the computational lattice into separate solid and liquid registers. The algorithm tracks the node associated with the phase interface—referred to as the "melting node"—and assigns both the solid and liquid temperature fields to this node. The liquid fraction $\eta$ at the interface is also tracked within the quantum circuit, yielding a local enthalpy at the melting node:

$H = c_p T + \mathcal{L}_\text{melt} \eta$

The subgrid interface position is reconstructed from the local state as

$x_I = x + (\eta - 0.5)\Delta x$

This approach allows for a sharp, dynamically adaptive resolution of the phase front within an inherently discrete lattice architecture.

3. Quantum Circuit Implementation for Nonlinear Phase Behavior

At the heart of the quantum implementation is the embedding of nonlinear phase change information directly within the quantum state, sidestepping the need for frequent measurement and classical update cycles. This is realized using ancilla qubits and multi-controlled rotation (MCRY) circuits. At the melting node, the circuit considers all combinations of distribution functions $f_i$ to generate the correct local temperature and liquid fraction:

• The ancilla rotation angle is

$\theta = 2\sin^{-1}\left(\sqrt{S/3}\right)$

where $S$ denotes the sum over relevant control qubit states associated with phase fractions.

A notable aspect is the storage and update of the phase history through the quantum circuit itself, minimizing data transfers and maximizing quantum coherence time across timesteps.

4. Validation, Error Analysis, and Simulation Results

Simulations were executed for a one-dimensional bar with 17 spatial nodes and 51 qubits using a Qiskit Aer MPS simulator. The temperature field, interface location $x_I$, and liquid fraction, as computed by QLBM, accord with both the classical LBM and the analytical Stefan solution $x_I(t) = 2\lambda\sqrt{\alpha t}$ for solid-liquid interface progression. Measured root-mean-square errors for both temperature and interface position remain below 0.005 throughout the simulation, indicating high numerical fidelity.

The following error metric is used:

$$E_\text{RMS} = \sqrt{(1/M)\sum_{k=1}^M \left(q^\text{classical(k)} - q^\text{quantum(k)}\right)^2}$$

where $M$ is the number of lattice sites. This low error attests to the ability of QLBM to stably handle strong nonlinearities and discontinuities associated with the enthalpy jump during melting.

5. Algorithmic Workflow and Quantum Resource Management

QLBM advances the simulation using the following key operations each time step:

1. Mapping of distribution functions to quantum amplitudes using basis rotations for each $f_i$.
2. Application of a unitary collision operator $U_c$, structured as a direct sum of rotations in subspaces, such that $U_c = \oplus_j V_j$ with $V_j = \alpha J + \beta (I - J)$ and parameters (e.g., $\beta = e^{i\theta}$) chosen for unitarity and isotropy.
3. Propagation of populations using sequences of SWAP gates along the lattice, implementing quantum streaming.
4. At the interface, MCRY circuits encode phase information in the ancillary qubits, tracking $\eta$ and $T$ within the quantum state.
5. Macroscopic observables (temperature profile, interface position, liquid fraction) are efficiently reconstructed from the quantum register, avoiding repeated cycle measurement and classical intervention.

For the tested $17$-node lattice ($51$ qubits), all algorithmic steps are performed using quantum routines, and the error introduced by quantum encoding and discretization remains negligible with respect to analytical and classical benchmarks.

6. Relevance and Extension to Advanced Applications

Although the paper focuses on one-dimensional heat transfer with phase change, the outlined QLBM is applicable to more complex, higher-dimensional systems. The interface-tracking and quantum-encoded phase history strategies generalize to multidimensional lattices, paving the way for:

• Simulating higher-dimensional phase change phenomena in energy storage materials or additive manufacturing.
• Modeling multiphase flows or flows with complex moving boundaries in advanced thermal management systems.
• Extending to fully coupled convection–diffusion–phase change models (D2Q9 or D3Q19 lattice schemes).
• Exploring quantum simulation in material science applications with rapid or spatially localized phase changes.

This approach efficiently leverages quantum hardware strengths—parallelism, amplitude encoding, and circuit-based phase memory—for problems where nonlinearities and interfacial dynamics are critical, while minimizing classical-quantum data transfer bottlenecks.

7. Key Mathematical Formulations

Table: Main Equations Used in QLBM for Phase Change

Description	Equation
Enthalpy formulation	$$\dfrac{1}{c_p} \dfrac{\partial H}{\partial t} = \alpha \nabla^2 T$$
Enthalpy–temperature relation	$$H = \begin{cases} c_p T, & T < T_\text{melt} \ c_p T + \mathcal{L}_\text{melt}, & T > T_\text{melt} \end{cases}$$
Interface tracking	$x_I = x + (\eta - 0.5)\Delta x$
Amplitude encoding	$$|q_i(x_k,t)\rangle = \sqrt{1 - f_i(x_k,t)}\,|0\rangle + \sqrt{f_i(x_k,t)}\,|1\rangle$$
Ancilla rotation (MCRY)	$\theta = 2 \sin^{-1}\left(\sqrt{S/3}\right)$
RMS error metric	$$E_\text{RMS} = \sqrt{\frac{1}{M}\sum_k (q_\text{classical}^{(k)} - q_\text{quantum}^{(k)})^2}$$

Each mathematical relation is directly realized in the quantum circuit, ensuring the QLBM captures the essential physical features of phase change processes for heat transfer (Jawetz et al., 25 Sep 2025).