The foundational value of quantum computing for classical fluids

TL;DR

The paper investigates whether quantum algorithms for classical fluids can reveal new patterns of quantum information flow beyond the many-body Schrödinger equation, addressing foundational questions about macroscopic limits. It proposes a concrete framework—Carleman embedding combined with Block Encoding (BECLB)—to linearize nonlinear fluid dynamics and embed dissipation within a unitary quantum evolution. A frank assessment of scaling and probabilistic constraints shows that current ancilla resources and low per-step success rates necessitate telescopic, multistep strategies and possible multiscale coarse-graining to compete with classical methods or NBSE, especially for large problems like numerical weather forecasting. While the approach carries significant foundational interest, practical quantum advantage hinges on breakthroughs in telescope-like quantum time marching and synergistic multiscale techniques, currently an active area of exploration.

Abstract

Quantum algorithms for classical physics problems expose new patterns of quantum information flow as compared to the many-body Schrödinger equation. As a result, besides their potential practical applications, they also offer a valuable theoretical and computational framework to elucidate the foundations of quantum mechanics, particularly the validity of the many-body Schrödinger equation in the limit of large number of particles, on the order of the Avogadro number. This idea is illustrated by means of a concrete example, the Block-Encoded Carleman embedding of the Lattice Boltzmann formulation of fluid dynamics (CLB).

Figures (7)

• Figure 1: The converging (blue solid line) and the diverging (red solid line) solutions of the logistic equation, obtained by setting the initial condition smaller, $x_0 = 0.5$, and larger, $x_0=2$, than $1/R$ respectively, with $R=1.5$ (black dashed horizontal line). The gray vertical line marks the time singularity for the unstable solution $t^*$.
• Figure 2: The analytical solution of the logistic equation in the stable regime (blue solid line) is compared with the solutions of the Carleman system of equations with increasing truncation order $k_{max}$, which yields a better approximation of the solution. We set $x_0=0.5$ and $R=1.5$, hence $r=3/4$.
• Figure 3: The accuracy $\epsilon_k(t)$ calculated at $t=2$ for given $k_{max}=1$ (red dashed line), $k_{max}=2$ (green dotted line) and $k_{max}=3$ (purple solid line). The initial condition is set to $x_0=0.5$, therefore the stable region is $0<R<2$.
• Figure 4: The minimum value $k$ of the truncation order $k_{max}$ to achieve accuracy $\epsilon$ for given nonlinearity $R$. $\epsilon$ is calculated at time $t=2$ and represented in log scale, while the initial condition is set to $x_0=0.5$ and defines the stable region $0<R<2$. The isolines for $k$ are drawn in white.
• Figure 5: The quantum circuit for block encoding a nonunitary operation into the unitary operator B.E. The success of the algorithm is conditioned on measuring all the $q_a$ ancilla qubits in the state $|0\rangle$. This happens with a probability $p \sim 2^{-2q_a}$, with little dependence upon the number of system qubits $q_s.$