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arXiv:2406.06323

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

John Penuel, Amara Katabarwa, Johnson, Peter D., Kuklinski, Parker, Rempfer, Benjamin, Collin Farquhar, Yudong Cao, Michael C. Garrett

Published Jun 10, 2024• 4 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2406.06323

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Computational fluid dynamics (CFD) is a cornerstone of classical scientific computing, and there is growing interest in whether quantum computers can accelerate such simulations. The existing proposals for fault-tolerant quantum algorithms for CFD have almost exclusively been based on the Carleman embedding method, used to encode nonlinearities on a quantum computer. Here, we first identify several key challenges that must be addressed in order to establish quantum advantage using this method, including Carleman convergence, time-stepping requirements, condition-number scaling, and data extraction. Building on this analysis, we develop a novel algorithm for the incompressible lattice Boltzmann equation that addresses them, and provide its detailed analysis, including all potential sources of algorithmic complexity, as well as gate count estimates. We find that for an end-to-end problem, a modest quantum advantage may be preserved for selected observables in the high-error-tolerance regime. We establish a lower bound for the Reynolds number scaling of our quantum algorithm in dimension DDD at Kolmogorov microscale resolution with O(Re34(1+D2)×qM)O(\mathrm{Re}^{\frac{3}{4}(1+\frac{D}{2})} \times q_M)O(Re43​(1+2D​)×qM​), where qMq_MqM​ is a multiplicative overhead for data extraction with qM=O(Re38)q_M = O(\mathrm{Re}^{\frac{3}{8}})qM​=O(Re83​) for the drag force. This upper bounds the scaling improvement over classical algorithms by O(Re3D8)O(\mathrm{Re}^{\frac{3D}{8}})O(Re83D​). However, our numerical investigations suggest a lower speedup, with a scaling estimate of O(Re1.936×qM)O(\mathrm{Re}^{1.936} \times q_M)O(Re1.936×qM​) for D=2D=2D=2. Our results give robust evidence that small, but nontrivial, quantum advantages can be achieved in the context of CFD, and motivate the need for additional rigorous end-to-end quantum algorithm development.
Numerical simulation of turbulent fluid dynamics needs to either parametrize turbulence—which introduces large uncertainties—or explicitly resolve the smallest scales—which is prohibitively expensive. Here, we provide evidence through analytic bounds and numerical studies that a potential quantum speedup can be achieved to simulate fluid dynamics using quantum computing. Specifically, we provide a lattice Boltzmann formulation of fluid dynamics for which we give evidence that low-order Carleman linearization is much more accurate than previously believed for these systems. This is achieved via a combination of reformulating the Navier-Stokes nonlinearity ( u · ∇ u ) to lattice-Boltzmann nonlinearity ( u 2 ) and accurately linearizing the dynamical equations, which effectively trades nonlinearity for additional degrees of freedom that add negligible expense in the quantum solver. Based on this, we apply a quantum algorithm for simulating the Carleman-linearized lattice Boltzmann equation and provide evidence that its cost scales logarithmically with system size compared with polynomial scaling in the best known classical algorithms. In this paper, we suggest that a quantum advantage may exist for simulating fluid dynamics, paving the way for simulating nonlinear multiscale transport phenomena in a wide range of disciplines using quantum computing.
The implementation of quantum algorithms for the simulation of classical fluid dynamics poses a fundamental challenge due to the nonlinearity of the fluid equations. In this work, we provide a pedagogical introduction to quantum computing algorithms for simulating classical fluids, with a special focus on the Carleman-Lattice Boltzmann algorithm, which has captured significant attention in the last couple of years. While this algorithm demonstrates satisfactory convergence to analytical solutions for systems at low-to-moderate Reynolds numbers, it also shows an exponential depth of the corresponding quantum circuit. As a result much further analysis is needed to assess the availability of the Carleman-Lattice Boltzmann method on a quantum computer.
Quantum algorithms have been identified as a potential means to accelerate computational fluid dynamics (CFD) simulations, with the lattice Boltzmann method (LBM) being a promising candidate for realizing quantum speedups. Here, we extend the recent quantum algorithm for the incompressible LBM to account for realistic fluid dynamics setups by incorporating walls, inlets, outlets, and external forcing. We analyze the associated complexity cost and show that these modifications preserve the asymptotic scaling, and potential quantum advantage, of the original algorithm. Moreover, to support our theoretical analysis, we provide a classical numerical study illustrating the accuracy, complexity, and convergence of the algorithm for representative incompressible-flow cases, including the driven Taylor-Green vortex, the lid-driven cavity flow, and the flow past a cylinder. Our results provide a pathway to accurate quantum simulation of nonlinear fluid dynamics, and a framework for extending quantum LBM to more challenging flow configurations.
We present a quantum computing algorithm for fluid flows based on the Carleman-linearization of the Lattice Boltzmann (LB) method. First, we demonstrate the convergence of the classical Carleman procedure at moderate Reynolds numbers, namely, for Kolmogorov-like flows. Then we proceed to formulate the corresponding quantum algorithm, including the quantum circuit layout, and analyze its computational viability. We show that, at least for moderate Reynolds numbers between 10 and 100, the Carleman–LB procedure can be successfully truncated at second order, which is a very encouraging result. We also show that the quantum circuit implementing the single time-step collision operator has a fixed depth, regardless of the number of lattice sites. However, such depth is of the order of ten thousands quantum gates, meaning that quantum advantage over classical computing is not attainable today, but could be achieved in the near or mid-term future. The same goal for the multi-step version remains, however, an open topic for future research.
Recent advances in quantum hardware and quantum computing algorithms promise significant breakthroughs in computational capabilities. Quantum computers can achieve exponential improvements in speed vs classical computers by employing principles of quantum mechanics like superposition and entanglement. However, designing quantum algorithms to solve the nonlinear partial differential equations governing fluid dynamics is challenging due to the inherent linearity of quantum mechanics, which requires unitary transformation. In this study, we first address in detail several challenges that arise when trying to deal with nonlinearity using quantum algorithms and then propose a novel pure quantum algorithm for solving a nonlinear Burgers' equation. We employed multiple copies of the state vector to calculate the nonlinear term, which is necessary due to the no-cloning theorem. By reusing qubits from the previous time steps, we significantly reduced the number of qubits required for multi-step simulations, from exponential/quadratic scaling in earlier studies to linear scaling in time in the current study. We also employed various advanced quantum techniques, including block-encoding, quantum Hadamard product, and the linear combination of unitaries, to design a quantum circuit for the proposed quantum algorithm. The quantum circuit was executed on quantum simulators, and the obtained results demonstrated excellent agreement with those from classical simulations.