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ASME Journal of Heat and Mass Transfer

Modeling Heat Conduction With an Internal Heat Source Using a Quantum Lattice Boltzmann Method

Yijin Mao, Yuwen Zhang

Published Nov 10, 2025• 1 citation
Publisher

Abstract

Abstract This work presents a fully quantum implementation of the Quantum Lattice Boltzmann Method (QLBM) for simulating one-dimensional heat conduction with internal heat generation. Building on the classical Lattice Boltzmann framework, QLBM reformulates the collision and streaming steps as unitary operations on quantum circuits, enabling site-wise parallel updates on near-term quantum hardware. In this formulation, classical distribution functions are encoded into quantum amplitudes, and the evolution of the system is performed entirely through quantum gates, including a collision operator and a streaming step realized as a qubit permutation. The model is validated by simulating a canonical diffusion problem with a localized heat source, and results are benchmarked against both classical LBM and an analytical solution obtained via ODE integration. The impact of quantum sampling noise is systematically assessed by varying the number of measurement shots, revealing that higher shot counts significantly improve fidelity to classical results. Root-mean-square error (RMSE) analyses confirm that the fully quantum model accurately captures the spatiotemporal behavior of the system, demonstrating the viability of QLBM for modeling nonunitary physics in thermally driven systems.

DOI
10.1115/1.4070381

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Abstract This work presents a fully quantum implementation of the Quantum Lattice Boltzmann Method (QLBM) for simulating one-dimensional heat conduction with internal heat generation. Building on the classical Lattice Boltzmann framework, QLBM reformulates the collision and streaming steps as unitary operations on quantum circuits, enabling site-wise parallel updates on near-term quantum hardware. In this formulation, classical distribution functions are encoded into quantum amplitudes, and the evolution of the system is performed entirely through quantum gates, including a collision operator and a streaming step realized as a qubit permutation. The model is validated by simulating a canonical diffusion problem with a localized heat source, and results are benchmarked against both classical LBM and an analytical solution obtained via ODE integration. The impact of quantum sampling noise is systematically assessed by varying the number of measurement shots, revealing that higher shot counts significantly improve fidelity to classical results. Root-mean-square error (RMSE) analyses confirm that the fully quantum model accurately captures the spatiotemporal behavior of the system, demonstrating the viability of QLBM for modeling nonunitary physics in thermally driven systems.
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Abstract Heat transfer involving phase change is computationally intensive due to moving phase boundaries, nonlinear computations, and time step restrictions. This paper presents a quantum lattice Boltzmann method (QLBM) for simulating heat transfer with phase change. The approach leverages the statistical nature of the lattice Boltzmann method (LBM) while addressing the challenges of discontinuous phase transitions in quantum computing. The method implements an interface-tracking strategy that partitions the problem into separate solid and liquid domains, enabling the algorithm to handle the discontinuity in the enthalpy–temperature relationship. We store phase change information in the quantum circuit to reduce information exchange between classical and quantum hardware, a bottleneck in many quantum applications. Results from the implementation agree with both classical LBM and analytical solutions, demonstrating QLBM as an effective approach for analyzing thermal systems with phase transitions. Simulations using 17 lattice nodes with 53 qubits demonstrate temperature root-mean-square errors of order 0.01 when compared against classical solutions. The method accurately tracks interface movement during phase transition.
The Quantum Lattice Boltzmann Method (QLBM) is one of the most promising approaches for realizing the potential of quantum computing in simulating computational fluid dynamics. Many recent works mostly focus on classical simulation, and rely on full state tomography. Several key algorithmic issues like observable readout, data encoding, and impractical circuit depth remain unsolved. As a result, these are not directly realizable on any quantum hardware. We present a series of novel algorithmic advances which allow us to implement the QLBM algorithm, for the first time, on a quantum computer. Hardware results for the time evolution of a 2D Gaussian initial density distribution subject to a uniform advection-diffusion field are presented. Furthermore, 3D simulation results are presented for particular non-uniform advection fields, devised so as to avoid the problem of diminishing probability of success due to repeated post-selection operations required for multiple timesteps. We demonstrate the evolution of an initial quantum state governed by the advection-diffusion equation, accounting for the iterative nature of the explicit QLBM algorithm. A tensor network encoding scheme is used to represent the initial condition supplied to the advection-diffusion equation, significantly reducing the two-qubit gate count affording a shorter circuit depth. Further reductions are made in the collision and streaming operators. Collectively, these advances give a path to realizing more practical, 2D and 3D QLBM applications with non-trivial velocity fields on quantum hardware.
The Quantum Lattice Boltzmann Method (QLBM) has emerged as one of the most promising quantum computing approaches for the numerical simulation of problems in computational fluid dynamics (CFD). The dynamics is formulated in terms of mesoscopic particle distribution functions governed by a discrete Boltzmann transport equation, comprising local streaming and collision operations. In this work, the resulting macroscopic behavior corresponds to the advection-diffusion equation, which we adopt as a canonical model problem for transport phenomena. Building upon recent progress in QLBM implementations, we advance towards more realistic problem settings that better reflect conventional CFD requirements. We address, for the first time, transport under the action of non uniform velocity fields on quantum hardware. We implement our demonstration using IonQ's trapped-ion systems including Forte generation systems and a 64-qubit Barium development system similar to the forthcoming IonQ Tempo line. We identify the density readout and subsequent reloading of the fluid density as a potential bottleneck of the current algorithm and discuss several approaches to mitigate this bottleneck. We identify the use of MPS shadow tomography as a promising method to efficiently scale the readout to large system with complex density distributions. Lastly, we introduce and simulate a novel method to implement wall boundaries for advection-diffusion in QLBM, and discuss the prospects of scaling to higher-complexity problems.
This article introduces a novel framework for developing quantum algorithms for the Lattice Boltzmann Method (LBM) applied to the advection-diffusion equation. We formulate the collision-streaming evolution of the LBM as a compact time-marching scheme and rigorously establish its stability under low Mach number conditions. This unified formulation eliminates the need for classical measurement at each time step, enabling a systematic and fully quantum implementation. Building upon this representation, we investigate two distinct quantum algorithmic approaches. The first is a time-marching quantum algorithm realized through sequential evolution operators, for which we provide a detailed implementation-including block-encoding and dilating unitarization-along with a full complexity analysis. The second employs a quantum linear systems algorithm, which encodes the entire time evolution into a single global linear system. We demonstrate that both methods achieve comparable asymptotic time complexities. The proposed algorithms are validated through numerical simulations of benchmark problems in one and two dimensions. This work provides a systematic pathway that avoids full-state measurement and reinitialization at every time step for the quantum simulation of advection-diffusion processes via the lattice Boltzmann paradigm.
The Lattice Boltzmann method (LBM) is a well-established mesoscopic approach for simulating fluid dynamics by evolving particle distribution functions on discrete lattices. While the LBM is highly parallelizable on classical hardware, its translation to quantum algorithms is impeded by the collision process, which is intrinsically nonlinear and irreversible. Several existing quantum formulations implement this process through repeated quantum tomography and state preparation at every timestep, leading to significant overheads. We introduce a quantum LBM based on a denoising-type collision operator that avoids tomography-based updates. The collision dynamics are reformulated as an orthogonal projection onto the linearized manifold of equilibrium distributions around a reference state. This geometric approach filters non-equilibrium components while preserving lattice symmetries and approximating nonlinear terms needed to recover hydrodynamic behavior. A complete pipeline is presented with efficient gate-level realizations, incorporating encoding of distributions, collision, streaming, boundary conditions, and measurement of physical quantities such as hydrodynamic forces. In addition, we outline an approach for implementing projector-based operators deterministically without postselection, paving the way to fully coherent multi-timestep LBM simulations. Numerical experiments for advection-diffusion and flow problems demonstrate that the method reproduces macroscopic behaviors with high accuracy, with performance depending on the choice of reference state.