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**1 - 2**of**2**### AMPLIFIED QUANTUM TRANSFORMS

, 2014

"... In this thesis we investigate two new Amplified Quantum Transforms. In par-ticular we create and analyze the Amplified Quantum Fourier Transform (Amplified-QFT) and the Amplified-Haar Wavelet Transform. The Amplified-QFT algorithm is used to solve the following problem: The Local Period Problem: Let ..."

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In this thesis we investigate two new Amplified Quantum Transforms. In par-ticular we create and analyze the Amplified Quantum Fourier Transform (Amplified-QFT) and the Amplified-Haar Wavelet Transform. The Amplified-QFT algorithm is used to solve the following problem: The Local Period Problem: Let L = {0, 1,..., N − 1} be a set of N labels and let A be a subset of M labels of period P, i.e. a subset of the form A = {j: j = s+ rP, r = 0, 1,...,M − 1} where P ≤ √N and M << N, and where M is assumed known. Given an oracle f: L → {0, 1} which is 1 on A and 0 elsewhere, find the local period P and the offset s. First, we provide a brief history of quantum mechanics and quantum comput-ing. Second, we examine the Amplified-QFT in detail and compare it against the Quantum Fourier Transform (QFT) and Quantum Hidden Subgroup (QHS) algo-rithms 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.