[Paper Review] Two-Dimensional Quantum Search Algorithm

Arti Chamoli, Samina Masood|arXiv (Cornell University)|Dec 27, 2010

Quantum Computing Algorithms and Architecture1 references3 citations

TL;DR

This paper proposes a two-dimensional quantum search algorithm that maintains polynomial speedup without increasing computational time, leveraging quantum parallelism to process multi-dimensional queries efficiently. The key contribution is demonstrating that quantum search performance remains invariant under dimensionality expansion, preserving O(√N) complexity in 2D databases via amplitude amplification and Grover-like iteration.

ABSTRACT

Quantum mechanical search induces polynomial speed up in an unsorted database search process. In case of classical linear search the computational time increases with the dimensionality of the query. However, quantum parallelism, inherent to quantum systems, does not let multi-dimensional query processing affect the computational time of the quantum search algorithm. In this letter, a two-dimensional search process has been proposed. It has been shown that a two-dimensional search process can be accomplished without increasing the computational time due to implicit quantum parallelism

Motivation & Objective

• To investigate whether quantum search algorithms can maintain efficiency when extended to two-dimensional databases.
• To address the classical intuition that higher-dimensional queries increase computational cost.
• To demonstrate that quantum parallelism eliminates the need for increased time complexity in 2D search scenarios.
• To formalize a quantum algorithm that achieves two-dimensional search with the same asymptotic complexity as one-dimensional Grover search.

Proposed method

• The algorithm employs amplitude amplification techniques adapted for two-dimensional data structures.
• It uses a Grover-like iterative process to amplify the amplitude of the target state across a 2D array of size N×N.
• The search space is encoded in a 2D quantum register, with unitary operations designed to rotate the state toward the marked element.
• The oracle marks the target state in 2D by applying a phase flip conditional on both indices.
• The diffusion operator is generalized to operate uniformly over the 2D Hilbert space, maintaining symmetry and convergence.
• The algorithm ensures that the number of iterations remains O(√N), independent of dimensionality, due to implicit quantum parallelism.

Experimental results

Research questions

• RQ1 Can a quantum search algorithm maintain O(√N) complexity when applied to a two-dimensional database?
• RQ2 Does the inclusion of a second dimension in the search space increase the required number of queries compared to one-dimensional search?
• RQ3 How does quantum parallelism affect the scalability of search in multi-dimensional data structures?
• RQ4 What modifications to the standard Grover algorithm are necessary to support 2D indexing and state marking?
• RQ5 Is the computational cost of 2D search fundamentally different from classical 2D linear search?

Key findings

• The two-dimensional quantum search algorithm achieves the same asymptotic complexity O(√N) as the one-dimensional Grover search, despite the increased dimensionality.
• The number of required oracle queries remains proportional to √N, where N is the total number of elements in the 2D array.
• Quantum parallelism allows simultaneous evaluation of all 2D indices, preventing computational overhead from dimensionality.
• The algorithm maintains high success probability after O(√N) iterations, confirming robustness and scalability.
• The proposed method generalizes Grover's algorithm to higher dimensions without sacrificing efficiency or increasing query count.