We give a rigorous and self-contained analysis of the Grover-Rudolph quantum state-preparation algorithm, which encodes a probability distribution $\{p_k\}$ as an $n$-qubit amplitude state $\sum _k\sqrt{p_k}|k⟩$ via a hierarchy of controlled ${\mathrm{R}}_y$ rotations determined by a dyadic refinement of the target. We formalize the dyadic probability tree, derive the trigonometric factorization of conditional masses, and prove by induction that the circuit prepares exactly the desired measurement law. We further prove that perturbing each rotation angle by at most $\eta$ changes the output distribution by at most $min(1,n\eta )$ in total variation, and combine this with a Hoeffding concentration bound to obtain an explicit design rule: $b\ge {\log }_2(2n\pi /\epsilon )$ bits and $S\ge 2^{n+1}\log (2/\delta )/{\epsilon }^2$ shots suffice to achieve accuracy $\epsilon$ with confidence $1-\delta$. As a circuit-theoretic complement, we provide an ancilla-free transpilation of each stage into $\{{\mathrm{R}}_y(\cdot ),X,\mathrm{C}\mathrm{N}\mathrm{O}\mathrm{T}\}$ via Gray-code ladders and a Walsh-Hadamard angle transform.

Algorithms that use tensor decompositions are widely used due to how well they perfor with large amounts of data. Among them, we find the algorithms that search for the solution of a linear system in separated form, where the greedy rank-one update method stands out, to be the starting point of the famous proper generalized decomposition family. When the matrices of these systems have a particular structure, called a Laplacian-like matrix which is related to the aspect of the Laplacian operator, the convergence of the previous method is faster and more accurate. The main goal of this paper is to provide a procedure that explicitly gives, for a given square matrix, its best approximation to the set of Laplacian-like matrices. Clearly, if the residue of this approximation is zero, we will be able to solve, by using the greedy rank-one update algorithm, the associated linear system at a lower computational cost. As a particular example, we prove that the discretization of a general partial differential equation of the second order without mixed derivatives can be written as a linear system with a Laplacian-type matrix. Finally, some numerical examples based on partial differential equations are given.