Quantum Data Loading for Carleman Linearized Systems: Application to the Lattice-Boltzmann Equation

TL;DR

This paper introduces a quantum data loading strategy for Carleman linearized systems, enabling efficient simulation of nonlinear dynamical systems like the lattice Boltzmann equation on quantum computers.

Contribution

It develops a generalized linear combination of unitaries framework for Carleman linearized systems, with scalable complexity estimates for quantum simulation.

Findings

Number of terms in LCU scales as $\mathcal{O}(\alpha^2 Q^2)$, independent of discretization points.

T gate cost for block encoding scales as $\mathcal{O}(\alpha^3 Q^2 (\log_2 n)^2)$.

Circuit cost for variational solver scales with $N_s^2$ and $\mathcal{O}(\alpha (\log_2 Qn)^2)$.

Abstract

Herein, we introduce a strategy to decompose an arbitrary square matrix into a linear combination of non-unitaries (LCNU) where each non-unitary term is embedded into a unitary matrix. The result is a linear combination of unitaries (LCU) with an equal number of terms as the LCNU. Using this approach, we construct a generalized LCU framework for any Carleman linearized autonomous dynamical system with a polynomial nonlinearity. This framework is then used to construct an LCU for the 3-dimensional Carleman linearized lattice Boltzmann equation (LBE) in which the number of terms scales like $N_s \sim \mathcal{O}(\alpha^2 Q^2)$, where $\alpha$ is the Carleman truncation order and $Q$ is the number of discrete velocities from the LBE. Importantly, $N_s$ is completely independent of both the number of temporal and spatial discretization points. Lastly, we provide an estimate of our LCNU strategy's T gate cost in conjunction with (1) PREP and SELECT block encoding oracles, and (2) the variational quantum linear solver. In the former, the T cost scales like $\mathcal{O}(\alpha^3 Q^2 (\log_2 n)^2)$, where $n$ is the total number of spatial grid points across all dimensions. Next, the latter requires $N_s^2(\log_2 (2n_tn^\alpha)+1)$ circuits per iteration, with a worst case T gate cost of $\mathcal{O}(\alpha (\log_2 Qn)^2)$ among them. We, therefore, provide an efficient decomposition strategy useful for both fault-tolerant and variational approaches.