From String Detection to Orthogonal Vector Problem

TL;DR

This paper explores quantum search algorithms for string detection and orthogonal vector problems, extending existing methods and proposing new circuits for efficient computation under various conditions.

Contribution

It extends the 3-qubit String Detection Problem to 4-qubits with multiple winners, analyzes the Orthogonal Vector Problem, and proposes a short-depth circuit for orthogonal pair calculation.

Findings

Original GSA framework lacks numerical stability for OVP

Modified GSA can stabilize measurements under certain conditions

Proposed constant runtime circuit efficiently finds orthogonal pairs

Abstract

Considering Grover's Search Algorithm (GSA) with the standard diffuser stage applied, we revisit the $3$-qubit unique String Detection Problem (SDP) and extend the algorithm to $4$-qubit SDP with multiple winners. We then investigate unstructured search problems with non-uniform distributions and define the Orthogonal Vector Problem (OVP) under quantum settings. Although no numerically stable results is reached under the original GSA framework, we provide intuition behind our implementation and further observations on OVP. We further perform a special case analysis under the modified GSA framework which aims to stabilize the final measurement under arbitrary initial distribution. Based on the result of the analysis, we generalize the initial condition under which neither the original framework nor the modification works. Instead of utilizing GSA, we also propose a short-depth circuit that can calculate the orthogonal pair for a given vector represented as a binary string with constant runtime.