Detailed assessment of calculating drag force with quantum computers: Explicit time-evolution precludes exponential advantage for nonlinear differential equations

TL;DR

<3-5 sentence high-level summary> The paper assesses whether fault-tolerant quantum computers can meaningfully accelerate incompressible CFD, focusing on drag calculations for ship hull design using Carleman-linearized lattice Boltzmann methods coupled with quantum linear solvers and amplitude estimation. It provides detailed quantum resource estimates (QREs) for flow-past-sphere problem instances and several ship-hull designs, finding no exponential advantage: the QREs scale roughly as Re^{2.68}, driven by the intrinsic CFL-related time-step constraints of explicit time marching. The authors attribute the lack of exponential speedup to the polynomial relationship between spatial grid resolution and time steps, and they conclude that quantum utility for time-evolving nonlinear fluids is unlikely without new algorithmic breakthroughs or steady-state/implicit formulations. This work offers a rigorous benchmark framework for end-to-end quantum CFD utility assessment and highlights where future quantum advantages might realistically arise.</narrator>

Abstract

This study examines the potential for fault-tolerant quantum computers to provide utility in fluid dynamics simulations, with a focus on drag force calculations for ship hull design. We assess whether quantum algorithms can surpass classical computational limits by generating detailed quantum resource estimates (QREs) in terms of logical qubits and $T$-gate counts. Our analysis is based on a quantum algorithm leveraging Carleman linearization of the lattice Boltzmann method (LBM), which has been suggested to offer exponential speedup. We develop efficient block encodings for LBM matrices and a method for amplitude-encoding drag force. We apply the method to the simple case of fluid flow past a sphere across a range of Reynolds numbers ($\mathrm{Re}$). We estimate the required (logical qubits)$\times$($T$-gates), finding them to be prohibitively large, ranging from $10^{21}$ to $10^{39}$. While classical simulations scale as $O(\mathrm{Re}^3)$, our QREs exhibit a modest polynomial scaling of $O(\mathrm{Re}^{2.68})$, indicating no exponential quantum advantage. We attribute this limitation to an intrinsic power-law relationship between spatial grid resolution and time-stepping requirements that is a fundamental characteristic of explicit methods for evolving nonlinear differential equations. Thus, quantum computers are unlikely to provide utility in applications that require time-evolving fluids and other systems of nonlinear differential equations.

Figures (40)

• Figure 1: Top-level classical workflow (left path) vs. quantum workflow (right path). The workflows are well-suited to each platform.
• Figure 2: Free streaming. The population $f_i(\mathbf{x},t)$ (left node) becomes the population $f_i(\mathbf{x}+\mathbf{c}_i,t+1)$ (right node).
• Figure 3: Bounce back. The population $f_j(\mathbf{x} + \mathbf{c}_i,t)$ (flowing leftward from right node) becomes the population $f_i(\mathbf{x} +\mathbf{c}_i,t+1)$ (flowing rightward from right node). Note the change in velocity vector index $f_j \rightarrow f_i$, where $\mathbf{c}_i = -\mathbf{c}_j$.
• Figure 4: Overview of the quantum algorithmic pipeline workflow for extracting the drag coefficient. See figure \ref{['fig:drag_est_call_graph']} for details on how to break apart this diagram from a resource estimation perspective.
• Figure 5: Circuits used in QAE subroutine.

Theorems & Definitions (34)

• Definition 1.1: Application Instance
• Definition 1.2: Problem Instance
• Definition 1.3: Performance Metric
• Definition 1.4: Utility Threshold
• Definition 1.5: Utility Estimate
• Definition E.1: Block Encoding
• Definition E.2
• Lemma E.3
• Lemma E.4
• Theorem E.5