Quantum Search without Global Diffusion

TL;DR

This paper demonstrates that quantum search can retain its quadratic speedup without a global diffusion operator by using local operations and recursive constructions, reducing circuit depth.

Contribution

It introduces a recursive method that preserves quantum search efficiency using only local operators, challenging the necessity of a global diffusion step.

Findings

Retains O(√N) complexity with local operators when partitions have at least log log N qubits.

Reduces circuit depth by up to 96% on 18-qubit problems compared to Grover's algorithm.

Depth reductions persist even when the oracle circuit is significantly deeper than the diffusion operator.

Abstract

Quantum search is among the most important algorithms in quantum computing. At its core is quantum amplitude amplification, a technique that achieves a quadratic speedup over classical search by combining two global reflections: the oracle, which marks the target, and the diffusion operator, which reflects about the initial state. We show that this speedup can be preserved when the oracle is the only global operator, with all other operations acting locally on non-overlapping partitions of the search register. We present a recursive construction that, when the initial and target states both decompose as tensor products over these chosen partitions, admits an exact closed-form solution for the algorithm's dynamics. This is enabled by an intriguing degeneracy in the principal angles between successive reflections, which collapse to just two distinct values governed by a single recursively defined angle. Applied to unstructured search, a problem that naturally satisfies the tensor decomposition, the approach retains the $O(\sqrt{N})$ oracle complexity of Grover search when each partition contains at least $\log_2(\log_2 N)$ qubits. On an 18-qubit search problem, partitioning into two stages reduces the non-oracle circuit depth by as much as 51%-96% relative to Grover, requiring up to 9% additional oracle calls. For larger problem sizes this oracle overhead rapidly diminishes, and valuable depth reductions persist when the oracle circuit is substantially deeper than the diffusion operator. More broadly, these results show that a global diffusion operator is not necessary to achieve the quadratic speedup in quantum search, offering a new perspective on this foundational algorithm. Moreover, the scalar reduction at the heart of our analysis inspires and motivates new directions and innovations in quantum algorithm design and evaluation.