Quantum Fourier Transform and Phase Estimation in Qudit System

Abstract

The quantum Fourier transform and quantum phase estimation are the key components for many quantum algorithms, such as order-finding, factoring, and etc. In this article, the general procedure of quantum Fourier transform and phase estimation are investigated for high dimensional case. They can be seen as subroutines in a main program run in a qudit quantum computer, and the quantum circuits are given.

Qudits and High-Dimensional Quantum Computing

“…QFT, as the heart of many quantum algorithms, can also be performed in a qudit system [145,165]. In an N-dimensional system represented with n d-dimensional qudits, the QFT , F(d, N) into a new basis set [26]…”

“…Although the qudit system's advantages in various applications and potentials for future development are substantial, this system receives less attention than the conventional qubitbased quantum computing, and a comprehensive review of the qudit-based models and technologies is needed. This review article provides an overview of qudit-based quantum computing covering a variety of topics ranging from circuit building [39,61,71,89,133]; algorithm designs [2,17,26,62,79,119,121]; to experimental methods [2,11,48,60,62,91,99,106]. In this article, high-dimensional generalizations of many widely used quantum gates are presented and the universality of the qudit gates is shown.…”

“…In this article, high-dimensional generalizations of many widely used quantum gates are presented and the universality of the qudit gates is shown. Qudit versions of three major classes of quantum algorithms-algorithms for the oracles decision problems (e.g., the Deutsch-Jozsa algorithm [121], algorithms for the hidden non-abelian subgroup problems (e.g., the phaseestimation algorithms (PEAs) [26] and the quantum search algorithm (e.g., Grover's algorithm [79]-are discussed and the comparison of the qudit designs vs. the qubit designs is analyzed. Finally, we introduce various physical platforms that can implement qudit computation and compare their performances with their qubit counterparts.…”

Implementation of the quantum Fourier transform on a hybrid qubit–qutrit NMR quantum emulator

“…An efficient decomposition of the QFT for qubits achieves an exponential speedup over the classical fast Fourier transform [30], and has been experimentally implemented by several groups on different physical systems [31][32][33][34][35]. Decompositions of the QFT for qudit systems have been worked out by several groups [36][37][38][39].…”

t-bit semiclassical quantum Fourier transform

“…In 1996, Grover [2] presented a quantum searching algorithm that reduces the computational complexity of current exhaustive search attacks from O(2 n ) to O(2 n/2 ). To overcome this problem, scholars [3][4][5][6][7][8][9][10][11] in China and abroad have since intensively investigated quantum computation and quantum cryptology. Although the correctness of quantum computation and Shor's algorithm has been proved to hold true [12], it is still difficult to break 2048-bit RSA or 191-bit ECC in the case of the quantum computer for the reason that the kilobit quantum computer does not exist.…”