A Quantum Hamiltonian Simulation Framework for Enabling Time-Resolved Flow Dynamics

We adopt the lattice Boltzmann method (LBM) as the underlying fluid model due to its local update structure, which makes it well-suited for quantum implementation and has been actively studied as a candidate for quantum-assisted flow simulation. Unlike conventional CFD methods based on differential operators, LBM naturally decomposes the dynamics into local interactions, offering a favorable structure for quantum encoding. To overcome the limitations of conventional approaches for constructing quantum-assisted flow computation methods based on Hamiltonian simulation, we first reformulate the governing equations and then cast them into a Hamiltonian simulation framework, thereby enabling the treatment of unsteady flow solutions within a quantum computational framework. Details of this reformulation will be discussed in the conference presentation. 

Since the resulting system remains nonlinear, we apply Carleman linearization [1] to lift the nonlinear system into an infinite-dimensional linear system, which is then truncated to obtain a finite-dimensional approximation. The resulting truncated linear ordinary differential equation inherently includes truncation errors associated with this approximation. To enable quantum computation, the resulting linear system is further embedded into a Hermitian form suitable for Hamiltonian simulation, for example through Hermitian embedding or related transformations. This linear system is then solved using Hamiltonian simulation, and the desired physical quantities are extracted through measurement. This procedure provides a systematic pathway from nonlinear fluid equations to quantum-computable representations. 

The proposed quantum-assisted framework is validated through implementation on classical hardware (CPU/GPU) using quantum simulation techniques that emulate Hamiltonian evolution. Due to current hardware limitations, the demonstration focuses on small-scale systems that fit within available memory resources. As a representative test case, we consider a two-dimensional channel flow problem.

The obtained results are quantitatively compared with those from conventional LBM simulations, evaluating consistency in velocity profiles and temporal evolution. In particular, the relative error of the velocity field and convergence behavior are examined to assess both accuracy and convergence characteristics. We further perform a numerical stability analysis, as numerical instabilities may arise from the reformulated linear system, its inherent structural properties, and the Carleman linearization. These effects are systematically analyzed to clarify their influence on convergence behavior.

Furthermore, we investigate the impact of key algorithmic design choices on computational performance and quantum suitability. These include the choice of governing formulation (fully discrete LBM versus differential-form FDLBM), collision models (BGK versus MRT), and linearization strategies (Carleman, Koopman–von Neumann, and machine-learning-based approaches). By evaluating these alternatives within a unified framework, we identify configurations that offer favorable stability, accuracy, and compatibility with quantum implementation. 

In this work, we focus on capabilities that are difficult to achieve within conventional time-marching frameworks. In particular, the proposed formulation enables direct access to full temporal evolution through Hamiltonian simulation, addressing a key limitation of standard quantum time-evolution algorithms that primarily provide final-time information. 

For small- to medium-scale problems, no computational speedup is expected compared to conventional methods, as the encoding involves degrees of freedom over both space and time. However, the proposed approach may offer a distinct advantage in terms of error behavior. While quantum implementations of time-marching-based simulation approaches accumulate numerical errors as well as algorithmic errors associated with quantum simulation during physical time evolution, the proposed approach may mitigate this accumulation through its reformulated treatment of unsteady systems. Details of the error behavior will also be discussed in the presentation.

Based on the demonstrated small-scale simulations, we estimate resource requirements for larger systems, including qubit counts and circuit depth, and analyze their scaling with respect to spatial and temporal resolution. These analyses provide insight into the feasibility of extending the proposed framework beyond classical limits and help identify regimes in which quantum-assisted simulations may become practically advantageous.