A Grover-compatible manifold optimization algorithm for quantum search

Zhijian Lai; Dong An; Jiang Hu; Zaiwen Wen

Paper

A Grover-compatible manifold optimization algorithm for quantum search

Abstract

Grover's algorithm is a fundamental quantum algorithm that offers a quadratic speedup for the unstructured search problem by alternately applying physically implementable oracle and diffusion operators. In this paper, we reformulate the unstructured search as a maximization problem on the unitary manifold and solve it via the Riemannian gradient ascent (RGA) method. To overcome the difficulty that generic RGA updates do not, in general, correspond to physically implementable quantum operators, we introduce Grover-compatible retractions to restrict RGA updates to valid oracle and diffusion operators. Theoretically, we establish a local Riemannian

-Polyak-Łojasiewicz (PL) inequality with

, which yields a linear convergence rate of

toward the global solution. Here, the condition number

, where

denotes the Riemannian Lipschitz constant of the gradient. Taking into account both the geometry of the unitary manifold and the special structure of the cost function, we show that

for problem size

. Consequently, the resulting iteration complexity is

for attaining an

-accurate solution, which matches the quadratic speedup of

achieved by Grover's algorithm. These results demonstrate that an optimization-based viewpoint can offer fresh conceptual insights and lead to new advances in the design of quantum algorithms.