  We present several improvements to the recently developed ground state
preparation algorithm based on the Quantum Eigenvalue Transformation for
Unitary Matrices (QETU), apply this algorithm to a lattice formulation of U(1)
gauge theory in 2+1D, as well as propose a novel application of QETU, a highly
efficient preparation of Gaussian distributions.
  The QETU technique has been originally proposed as an algorithm for
nearly-optimal ground state preparation and ground state energy estimation on
early fault-tolerant devices. It uses the time-evolution input model, which can
potentially overcome the large overall prefactor in the asymptotic gate cost
arising in similar algorithms based on the Hamiltonian input model. We present
modifications to the original QETU algorithm that significantly reduce the cost
for the cases of both exact and Trotterized implementation of the time
evolution circuit. We use QETU to prepare the ground state of a U(1) lattice
gauge theory in 2 spatial dimensions, explore the dependence of computational
resources on the desired precision and system parameters, and discuss the
applicability of our results to general lattice gauge theories. We also
demonstrate how the QETU technique can be utilized for preparing Gaussian
distributions and wave packets in a way which outperforms existing algorithms
for as little as $n_q \gtrsim 2-5$ qubits.
