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

• URL: http://arxiv.org/abs/2605.00302v1
• Date: Fri, 01 May 2026 00:10:50 GMT
• Title: Quantum Data Loading for Carleman Linearized Systems: Application to the Lattice-Boltzmann Equation
• Abstract summary: We decompose an arbitrary square matrix into a linear combination of non-unitaries (LCNU) where each non-unitary term is embedded into a unitary matrix.<n>We provide an estimate of our LCNU strategy's T gate cost in conjunction with (1) encoding block oracles, and (2) the variational quantum linear solver.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $N_s \sim \mathcal{O}(α^2 Q^2)$, where $α$ 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 $\mathcal{O}(α^3 Q^2 (\log_2 n)^2)$, where $n$ is the total number of spatial grid points across all dimensions. Next, the latter requires $N_s^2(\log_2 (2n_tn^α)+1)$ circuits per iteration, with a worst case T gate cost of $\mathcal{O}(α(\log_2 Qn)^2)$ among them. We, therefore, provide an efficient decomposition strategy useful for both fault-tolerant and variational approaches.

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