Quantum Fourier transform (QFT) is a key function to realize quantum computers. A QFT followed by measurement was demonstrated on a simple circuit based on fiber-optics. The QFT was shown to be robust against imperfections in the rotation gate. Error probability was estimated to be 0.01 per qubit, which corresponded to error-free operation on 100 qubits. The error probability can be further reduced by taking the majority of the accumulated results. The reduction of error probability resulted in a successful QFT demonstration on 1024 qubits. 1
Quantum Fourier transform (QFT) is a key function to realize quantum computers. A QFT followed by measurement was demonstrated on a simple circuit based on fiber-optics. The QFT was shown to be robust against imperfections in the rotation gate. Error probability was estimated to be 0.01 per qubit, which corresponded to error-free operation on 100 qubits. The error probability can be further reduced by taking the majority of the accumulated results. The reduction of error probability resulted in a successful QFT demonstration on 1024 qubits.
Among a number of candidates, photons have advantages for implementing qubits: very weak coupling to the environment, the existing single photon measurement technique, and so on. Moreover, commercially available fiber-optic devices enable us to construct quantum circuits that consist of one-qubit operations (including classically controlled gates). Fiber optics resolves the mode matching problems in conventional optics and provides mechanically stable optical circuits. A quantum Fourier transform (QFT) followed by measurement was demonstrated with a simple circuit based on fiber optics. The circuit was shown to be robust against imperfections in the rotation gate. The error probability was estimated to be 0.01 per qubit, which corresponded to error-free operation for 100 qubits. The error probability can be further reduced to achieve successful QFT of 1024 qubits by taking the majority of the accumulated results. As is well known, QFT is a key function in quantum computations such as the final part of Shor’s factorization algorithm. The present QFT circuit, in combination with controlled unitary gates, would make possible practical quantum computers. Possible schemes of realizing quantum computers in this line are explored.
The quantum Fourier transform (QFT) is one of the most widely used quantum algorithms, ranging from its primary role in finding the periodicity hidden in a quantum state to its use in constructing a quantum adder. Testing how the QFT performs under more realistic conditions, we find that the QFT, when used for period finding, shows extraordinary robustness with respect to static gate defects. For instance, replacing all rotation angles $\ensuremath{\pi}/{2}^{j}$ of the controlled rotation gates in the QFT circuit by $\ensuremath{\pi}(1+r)/{2}^{j}$, where
r is a uniformly distributed random variable taking values in the range $[\ensuremath{-}1,1]$, effectively resulting in a QFT with random gates, the QFT performs well above the expected random result. However, it is important to keep the
2j terms in the denominators of the rotation angles, resulting in random, but hierarchically random, gates. Relaxing this hierarchical structure of the QFT circuit, we find that the performance of the QFT deteriorates significantly. This observation indicates that the hierarchical structure of the quantum circuit of the QFT is more important for the observed robustness in performance than the precise actions of individual gates. In addition to the specific example of the QFT circuit studied here, this observation also corroborates our experience with more general and more complex quantum circuits. Thus, backed by our detailed numerical and analytical results, we may condense the results of our research into the following general principle: The topology of a quantum circuit is more important than the precise actions of its gates.
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.