# Quantum algorithm based on the $\varepsilon$-random linear disequations for the continuous hidden shift problem

## TL;DR

This paper introduces a quantum algorithm for the continuous hidden shift problem on real vector spaces, extending previous discrete group results, and demonstrates its polynomial-time solvability.

## Contribution

It defines the continuous hidden shift problem and the $ ext{ extsterling}$-random linear disequations, and presents a quantum algorithm solving the problem efficiently.

## Key findings

- Quantum algorithm solves the continuous hidden shift problem in polynomial time.
- Extension of hidden shift problem to $ ext{ extsterling}$-random linear disequations.
- Provides a new framework for quantum algorithms on continuous domains.

## Abstract

There have been several research works on the hidden shift problem, quantum algorithms for the problem, and their applications. However, all the results have focused on discrete groups with discrete oracle functions. In this paper, we define the continuous hidden shift problem on $\mathbb{R}^n$ with a continuous oracle function as an extension of the hidden shift problem, and also define the $\varepsilon$-random linear disequations which is a generalization of the random linear disequations. By employing the newly defined concepts, we show that there exists a quantum computational algorithm which solves this problem in time polynomial in $n$.