Quantum state preparation for multivariate functions

A fundamental step of any quantum algorithm is the preparation of qubit registers in a suitable initial state. Often qubit registers represent a discretization of continuous variables and the initial state is defined by a multivariate function. We develop protocols for preparing quantum states whose amplitudes encode multivariate functions by linearly combining block-encodings of Fourier and Chebyshev basis functions. Without relying on arithmetic circuits, quantum Fourier transforms, or multivariate quantum signal processing, our algorithms are simpler and more effective than previous proposals. We analyze requirements both asymptotically and pragmatically in terms of near/medium-term resources. Numerically, we prepare bivariate Students t-distributions, 2D Ricker wavelets and electron wavefunctions in a 3D Coulomb potential, which are initial states with potential applications in finance, physics and chemistry simulations. Finally, we prepare bivariate Gaussian distributions on the Quantinuum H2-1 trapped-ion quantum processor using 24 qubits and up to 237 two-qubit gates.