Quantum lower bounds for simulating fluid dynamics

Abstract

Developing quantum algorithms to simulate fluid dynamics has become an active area of research, as accelerating fluid simulations could have significant impact in both industry and fundamental science. While many approaches have been proposed for simulating fluid dynamics on quantum computers, it is largely unclear whether these algorithms will provide speedup over existing classical approaches. In this paper we give evidence that quantum computers cannot significantly outperform classical simulations of fluid dynamics in general. We study two models of fluids: the Korteweg-de Vries (KdV) equation, which models shallow water waves, and the incompressible Euler equations, which model ideal, inviscid fluids. We show that any quantum algorithm simulating the KdV equation or the Euler equations for time $T$ requires $Ω(T^2)$ and $e^{Ω(T)}$ copies of the initial state in the worst case, respectively. These lower bounds hold for the task of preparing the final state, and similar bounds hold for history state preparation. We prove the lower bound for the KdV equation by investigating divergence of solitons. For the Euler equations, we show that instabilities enable fast state discrimination.

Figures (3)

• Figure 1: Visual demonstration of the technique used for proving our lower bound. Given an unstable equilibrium (bottom row) and a small perturbation to it (top row), the two states will be driven apart from each other very quickly. Visualized here is the Rayleigh-Taylor instability, whereby a dense fluid resting on top of a less dense fluid (with gravity pointing downwards) becomes unstable to perturbations drazin2004hydrodynamicchandrasekhar2013hydrodynamic.
• Figure 2: As the nonlinear solution (green) cannot be analytically obtained, we rely on the linearized solution (dashed red) to prove the lower bound. By bounding the error from linearization (dashed blue), we can bound the inner product between the equilibrium and the nonlinear solution. However, if the linearization error bound is not sufficiently tight, as shown in the top row, we would not be able to show the desired lower bound for simulating the nonlinear dynamics. Thus, it is crucial to obtain a sufficiently tight bound, as shown in the bottom row, to prove the desired result.
• Figure 3: Numerical confirmation of the proof of \ref{['thm:euler-lower-bound']}. $\tilde{H}(t)$ denotes the bound on the inner product of the equilibrium with the nonlinear solution, as given in Eq. \ref{['eq:inner-product-bound']}. Figure \ref{['fig:min']} plots $\tilde{H}(t)$ as a function of simulation time $t$, verifying the existence of the local minimum. Figure \ref{['fig:eps-scaling']} plots the maximum value of $1-\tilde{H}(t)$ as a function of $1/\epsilon$, showing that the asymptotic scalings match Eq. \ref{['eq:asymptotics_euler']}. In both figures, $m=2$, $k=1$, and we arbitrarily set $\kappa = 10^{-6}$.

Theorems & Definitions (60)

• Lemma 1: bahouri2011fourier, Proposition 7.5
• Lemma 1
• Lemma 1
• Lemma 1
• Lemma 2: nielsen2010quantum, Theorem 9.2
• Lemma 3: Conservation of norm, KdV equation
• proof
• Lemma 4: Conservation of norm, Euler equations
• proof
• Lemma 5: Conservation of vorticity