A quantum computation problem is discussed in this paper.Many new features that make quantum computation superior to classical computation can be attributed to quantum coherence effect, which depends on the phase of quantum coherent state.Quantum Fourier transform algorithm, the most commonly used algorithm, is introduced.And one of its most important applications, phase estimation of quantum state based on quantum Fourier transform, is presented in details.The flow of phase estimation algorithm and the quantum circuit model are shown.And the error of the output phase value, as well as the probability of measurement, is analysed.The probability distribution of the measuring result of phase value is presented and the computational efficiency is discussed.The researchis focused on how to determine the optimum number of qubit which satisfy the accuracy of phase estimation.
A quantum computation problem is discussed in this paper.Many new features that make quantum computation superior to classical computation can be attributed to quantum coherence effect, which depends on the phase of quantum coherent state.Quantum Fourier transform algorithm, the most commonly used algorithm, is introduced.And one of its most important applications, phase estimation of quantum state based on quantum Fourier transform, is presented in details.The flow of phase estimation algorithm and the quantum circuit model are shown.And the error of the output phase value, as well as the probability of measurement, is analysed.The probability distribution of the measuring result of phase value is presented and the computational efficiency is discussed.The researchis focused on how to determine the optimum number of qubit which satisfy the accuracy of phase estimation.
A quantum computation problem is discussed in this paper.Many new features that make quantum computation superior to classical computation can be attributed to quantum coherence effect, which depends on the phase of quantum coherent state.Quantum Fourier transform algorithm, the most commonly used algorithm, is introduced.And one of its most important applications, phase estimation of quantum state based on quantum Fourier transform, is presented in details.The flow of phase estimation algorithm and the quantum circuit model are shown.And the error of the output phase value, as well as the probability of measurement, is analysed.The probability distribution of the measuring result of phase value is presented and the computational efficiency is discussed.
The Quantum Fourier Transform (QFT)andQuantumPhaseEstimation (QPE)are twofundamental algorithmic frameworks in quantum computation, and they provide the mathematical basis of many exponential-speedup quan tum algorithms, such as Shor’s factoring algorithm and quantum simulation techniques. In this work we present a detailed theoretical and computational study of the Quantum Fourier Transform and its integration in the Quan tum Phase Estimation. The discussion begins with the linear algebraic structure of the Hilbert spaces, the unitary transformations, and the complex vector representations that enable quantum interference and superposition. The QFT is studied as the unitary transform of computational basis states in the amplitudes in the frequency domain by phase rotations and tensor-product decompositions. The efficient circuit implementation in terms of Hadamard and controlled phase gates of this transform is discussed. The advantage of the transform in terms of polynomial time complexity over the classical discrete Fourier transform in some quantum scenarios is discussed.The QFT is implemented and the Quantum Phase Estimation algorithm is proposed to obtain the eigenphase information of the unitary operators. In the report the mathematical representation of QPE is derived by using the controlled unitary evolution, the inverse Fourier transforms and the measurement theory in finite dimensional Hilbert spaces. We discuss the probabilistic interpretation of phase extraction in detail and the convergence properties of the bi nary phase approximation. We also discuss the relation of QPE to eigenvalue problems in quantum mechanics and its applications to Hamiltonian simulation, quantum chemistry, cryptography, and hidden subgroup prob lems.Implementation aspects of these algorithms are studied using quantum circuit representations and oracle based computation models. We also discuss error sensitivity, qubit precision requirements, decoherence limita tions and practical constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware. The work synthesizes the mathematical foundations, computational complexity analysis and physical realizations of the QFT and the QPE, demonstrating how the Fourier-based quantum transformations provide the essential mechanism underlying modern quantum algorithmic speedups and scalable quantum information processing.
Quantum computing can provide speedups in solving many problems as the evolution of a quantum system is described by a unitary operator in an exponentially large Hilbert space. Such unitary operators change the phase of their eigenstates and make quantum algorithms fundamentally different from their classical counterparts. Based on this unique principle of quantum computing, we develop an algorithmic toolbox, quantum phase processing, that can directly apply arbitrary trigonometric transformations to eigenphases of a unitary operator. The quantum phase processing circuit is constructed simply, consisting of single-qubit rotations and controlled unitaries, typically using only one ancilla qubit. Besides the capability of phase transformation, quantum phase processing in particular can extract the eigeninformation of quantum systems by simply measuring the ancilla qubit, making it naturally compatible with indirect measurement. Quantum phase processing complements another powerful framework known as quantum singular value transformation and leads to more intuitive and efficient quantum algorithms for solving problems that are particularly phase related. As a notable application, we propose a quantum phase estimation algorithm without quantum Fourier transform, which requires the fewest ancilla qubits and matches the best performance so far. We further exploit the power of our method by investigating a plethora of applications in Hamiltonian simulation, entanglement spectroscopy, and quantum entropy estimation, demonstrating improvements or optimality for almost all cases.
Quantum computing is the field that studies computation using quantum mechanical systems, exploiting properties such as superposition or entanglement. The focus of this project will be on quantum phase estimation algorithms. The formalism commonly used to represent quantum algorithms as quantum circuits composed by qubits and quantum gates will be presented. Then, we will be able to explain the quantum Fourier transform (that is exponentially faster than its classical analogue) and three quantum phase estimation algorithms: Kitaev’s phase estimation (KPEA), the phase estimation algorithm based on QFT (PEA) and the iterative quantum phase estimation algorithm based on a semiclassical Fourier transform (IPEA). We will use the Qiskit framework in order to run simulations of such algorithms, and even run them in real IBM Q Experience quantum computers (specifically, we will use the 5 qubit computer ibmq_essex). Statistical post-processing techniques will be essential in order to extract a phase estimation from the results obtained. We will use two post-processing techniques: the majority and the average rule. The performance of these methods applied to the results of several simulations and executions in ibmq_essex will be studied. At last, we will talk about the possible applications of phase estimation algorithms. Phase estimation is used as a subroutine in the field of quantum simulation, allowing us to obtain several magnitudes of a quantum system such as the energy levels. We will present an efficient quantum algorithm for factoring called Shor’s algorithm. The cost of factorization with a classical computer increases exponentially with the size of the input, so this is a example of a task that cannot be solved efficiently on a classical computer, but can be solved efficiently by a quantum computer. Phase estimation is also used in the field of quantum metrology. Specifically, an N-V center can be used to measure magnetic fields with high sensitivity and spatial resolution operating at room temperature thanks to a measuring protocol based on phase estimation.
In this chapter, we will study the quantum Fourier transform and its application in different quantum algorithms. Problems such as factoring an integer into prime numbers or period finding are computationally intractable problems for a classical computer because of the exponentially large number of operations involved. Integer factoring and period finding can be efficiently solved using the quantum phase estimation algorithm that is heavily based on the quantum Fourier transform. Alternately, since quantum phase estimation aims to find the eigenvalue corresponding to an eigenvector of a unitary operator, it is backbone of important algorithms in optimization such as the HHL algorithm (named for Hassim, Harrow, and Lloyd), which serves as the matrix inversion routine in quantum computing. We start this chapter by revising our concepts of the Fourier transform and its discrete counterpart, the discrete Fourier transform , and then move on to the exciting domain of the quantum Fourier transform and the quantum phase estimation algorithm. We follow this up with a discussion and implementation of the few quantum Fourier transform–related algorithms such as factoring a number and period finding. At the end of the chapter, we briefly introduce the basics of group theory with an attempt to explain the hidden subgroup problem and how it relates to several of the Fourier transform–based algorithms.
In this paper, we introduce quantum fourier transform as a key ingredient for many useful algorithms. These algorithms make a solution for problems which is considered to be intractable problems on a classical computer. Quantum Fourier transform is propounded as a key for quantum phase estimation algorithm. In this paper our aim is the implementation of period-finding algorithm.Quantum computer solves this problem, exponentially faster than classical one. Quantum phase estimation algorithm is the key for the period-finding problem .Therefore, by means of simulating quantum Fourier transform, we are able to implement the period-finding algorithm. In this paper, the simulation of quantum Fourier transform is carried out by Matlab software.