Herein, we present a polylogarithmic decomposition method to load the matrix from the linearized one-dimensional Burgers' equation onto a quantum computer. First, we use the Carleman linearization method to map the nonlinear Burgers' equation into an infinite linear system of equations, which is subsequently truncated to order $\alpha$. This new finite linear system is then embedded into a larger system of equations with the key property that its matrix can be decomposed into a linear combination of $\mathcal{O}({log}_2{n}_t+{\alpha }^2{log}_2{n}_x)$ terms for $n_t$ time steps and $n_x$ spatial grid points. While the terms in this linear combination are not unitary, each can be implemented using a simple block-encoding procedure. A numerical simulation is performed by combining our approach with the variational quantuam linear solver, demonstrating that accurate solutions are possible. Finally, a resource estimate shows that the upper bound of the Clifford- and $T$-gate counts scale like $\mathcal{O}\left[\alpha {\left({log}_2{n}_x\right)}^2\right]$ and $\mathcal{O}\left[{\left({log}_2{n}_x\right)}^2\right]$, respectively. This is therefore the first explicit polylogarithmic data loading method with respect to $n_x$ and $n_t$ for a Carleman linearized system.

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• Quantum algorithms
• Quantum algorithms & computation
• Quantum computation
• Quantum information processing
• Quantum information theory
• Quantum simulation
• Dynamical systems
• Differential equations