A Grover-compatible manifold optimization algorithm for quantum search

TL;DR

This paper reformulates Grover's quantum search algorithm as a manifold optimization problem, introducing a Riemannian gradient ascent method with Grover-compatible updates, achieving the quadratic speedup characteristic of Grover's algorithm.

Contribution

It presents a novel optimization framework for quantum search on the unitary manifold, with convergence guarantees and compatibility with physical quantum operators.

Findings

Achieves $O(\sqrt{N})$ iteration complexity matching Grover's speedup.

Establishes a local Riemannian er-Polyak-{}ojasiewicz inequality with er=1/2.

Demonstrates the effectiveness of manifold optimization in quantum algorithm design.

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 $\mu$-Polyak-{\L}ojasiewicz (PL) inequality with $\mu = \tfrac{1}{2}$, which yields a linear convergence rate of $1 - \kappa^{-1}$ toward the global solution. Here, the condition number $\kappa = L_{\mathrm{Rie}} / \mu$, where $L_{\mathrm{Rie}}$ 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 $L_{\mathrm{Rie}} = O(\sqrt{N})$ for problem size $N = 2^n$. Consequently, the resulting iteration complexity is $O(\sqrt{N} \log(1/\varepsilon))$ for attaining an $\varepsilon$-accurate solution, which matches the quadratic speedup of $O(\sqrt{N})$ 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.