Efficient quantum algorithm for weighted partial sums and numerical integration

This paper presents a quantum algorithm for efficiently computing partial sums and specific weighted partial sums of quantum state amplitudes. Computation of partial sums has important applications, including numerical integration, cumulative probability distributions, and probabilistic modeling. The proposed quantum algorithm uses a custom unitary construction to achieve the desired partial sums with gate complexity and circuit depth of $O(\log_2 M)$, where $M$ represents the number of terms in the partial sum. For cases where $M$ is a power of two, the unitary construction is straightforward; however, for arbitrary $M$, we develop an efficient quantum algorithm to create the required unitary matrix. Computational examples for evaluation certain partial sums and numerical integration based on our proposed algorithm are provided. We also extend the algorithm to evaluate partial sums of even or odd components and more complex weighted sums over specified intervals.