Quantum algorithm for solving differential equations using SLAC derivatives

In numerical approaches to solving differential equations on a lattice, a representation of the derivative operator that correctly matches the continuum behaviour of functions of momentum up to the band limit must be non-local. We present the construction of efficient linear-combination-of-unitaries ($\mathrm{LCU}$)-based block-encodings for the first-order derivative and Laplacian operators in the non-local $N=2^n$-dimensional SLAC representation. We use state-preparation techniques designed for smoothly decaying functions to prepare the dense $\mathrm{LCU}$ amplitudes with high success probability and low gate cost. Furthermore, we demonstrate how Shannon wavelet transforms can be applied to these block-encodings to obtain multiscale representations of the SLAC derivative operators. We then show how to apply a diagonal preconditioner that reduces the condition number of these matrices in the multiscale wavelet basis to a small constant. This enables the solution of partial differential equations (PDEs) with SLAC-discretised derivative operators on a finite lattice using the quantum linear solving algorithm ($\mathrm{QLSA}$). For a $d$-dimensional PDE, after projection away from the nullspace, the resulting quantum linear-system algorithm has overall gate complexity ${O}(dn^3\alpha^{(k)}\log(1/\varepsilon))$, where $\alpha^{(k)}$ is the subnormalisation factor of the order-$k$ SLAC block-encoding and $\varepsilon$ denotes the algorithmic approximation error.