Amplified Quantum Transforms

TL;DR

This thesis introduces two new amplified quantum transforms, the Amplified-QFT and Amplified-Haar Wavelet Transform, demonstrating their advantages over existing algorithms in solving specific quantum problems.

Contribution

The paper presents the development and analysis of the Amplified-QFT and Amplified-Haar Wavelet Transform, showing their quadratic speedup and broader applicability in quantum algorithms.

Findings

Amplified-QFT is quadratically faster than QFT and QHS algorithms.

Amplified-Haar Wavelet Transform can solve the Local Constant or Balanced Signal Decision Problem.

An uncertainty relation for the Amplified-QFT algorithm is established.

Abstract

In this thesis we investigate two new Amplified Quantum Transforms. In particular we create and analyze the Amplified Quantum Fourier Transform (Amplified-QFT) and the Amplified-Haar Wavelet Transform. First, we provide a brief history of quantum mechanics and quantum computing. Second, we examine the Amplified-QFT in detail and compare it against the Quantum Fourier Transform (QFT) and Quantum Hidden Subgroup (QHS) algorithms for solving the Local Period Problem. We calculate the probabilities of success of each algorithm and show the Amplified-QFT is quadratically faster than the QFT and QHS algorithms. Third, we examine the Amplified-QFT algorithm for solving The Local Period Problem with an Error Stream. Fourth, we produce an uncertainty relation for the Amplified-QFT algorithm. Fifth, we show how the Amplified-Haar Wavelet Transform can solve the Local Constant or Balanced Signal Decision Problem which is a generalization of the Deutsch-Jozsa algorithm.