Realization of -bit semiclassical quantum Fourier transform on IBM's quantum cloud computer

To overcome the difficulty of realizing large-scale quantum Fourier transform(QFT) within existing technology, this paper implements a resource-saving method(named t-bit semiclassical QFT over Z_(2~n)), which could realize large-scale QFT using an arbitrary-scale quantum register. By developing a feasible method to realize the control quantum gate Rk, we experimentally realize the 2-bit semiclassical QFT over Z_(2~3) on IBM's quantum cloud computer, which shows the feasibility of the method. Then, we compare the actual performance of 2-bit semiclassical QFT with standard QFT in the experiments.The squared statistical overlap experimental data shows that the fidelity of 2-bit semiclassical QFT is higher than that of standard QFT, which is mainly due to fewer two-qubit gates in the semiclassical QFT. Furthermore, based on the proposed method, N = 15 is successfully factorized by implementing Shor's algorithm.

Realization of quantum Fourier transform over Z_N

Since the difficulty in preparing the equal superposition state of amplitude is 1/√N, we construct a quantile transform of quantum Fourier transform （QFT） over ZN based on the elementary transforms, such as Hadamard transform and Pauli transform. The QFT over Z_N can then be realized by the quantile transform, and used to further design its quantum circuit and analyze the requirements for the quantum register and quantum gates. However, the transform needs considerable quantum computational resources and it is difficult to construct a high-dimensional quantum register. Hence, we investigate the design of t-bit quantile transform, and introduce the definition of t-bit semiclassical QFT over Z_N. According to probability amplitude, we prove that the transform can be used to realize QFT over ZN and further design its quantum circuit. For this transform, the requirements for the quantum register, the one-qubit gate, and two-qubit gate reduce obviously when compared with those for the QFT over Z_N.

Applications of quantum Fourier transform in photon-added coherent state

Quantum Fourier transform is realized by the Hadamard gate in a quantum computer, which can also be considered as a Hadamard transform. We introduce the Hadamard transformed photon-added coherent state （HTPACS）, which is obtained by letting the photon-added coherent state （PACS） across the quantum Hadamard gate, from this result. It is found that the HTPACS can be considered as a coordinate-momentum mutual exchanging followed by a squeezing transform of the PACS. In addition, the non-classical statistical properties of HTPACS, such as squeezing coefficient, Mandel parameter, etc., are also discussed.

Quantum Fourier Transform and Phase Estimation in Qudit System

量 Fourier 变换和量阶段评价是为许多量算法的关键部件，例如发现顺序， factoring，和等等。在这篇文章，量 Fourier 变换和阶段评价的一般过程为高维的大小写被调查。他们能在一台 qudit 量计算机在主程序跑被看作子程序，并且量电路被给。

Multiparty Quantum Secret Sharing Using Quantum Fourier Transform

一(n， n ) 多党的量秘密分享的阀值计划古典或量消息基于分离的量 Fourier 变换被建议。在我们的建议计划，仅当所有参加者在音乐会工作，秘密消息，被使用前面的量 Fourier 变换编码并且由使用颠倒译码，以如此的一个方法被切开并且分享，它能在他们之中被重建。而且，我们也讨论这个协议怎么必须小心地为改正错误或一个不诚实的参加者被设计。安全分析证明我们的计划是安全的。另外，这个计划有它与量计算完全兼容的一个优点并且更容易在分布式的量认识到安全计算。

Implementation of Quantum Fourier Transform and Its Applications via Quantum-Dot Spins and Microcavity

A scheme for implementing discrete quantum Fourier transform is proposed via quantum dots embeddedin a microcavity,and then some of its applications are investigated,i.e.,Deutsch-Jozsa.algorithm and Shot's quantumfactoring.In particular,the detailed process of implementing one-qubit Deutsch Jozsa algorithm and the factorization ofN=15 are given.The microcavity mode is only virtually excited in the whole interaction,so the effective decoherent hasslight effect on the current scheme.These schemes would be an important step to fabricate a solid quantum computer.

Simple Scheme for Realizing the General Conditional Phase Shift Gate and a Simulation of Quantum Fourier Transform in Circuit QED

We propose a theoretical scheme for realizing the general conditional phase shift gate of charge qubitssituated in a high-Q superconducting transmission line resonator.The phase shifting angle can be tuned from 0 to 2n bysimply adjusting the qubit-resonator detuning and the interaction time.Based on this gate proposal,we give a detailedprocedure to implement the three-qubit quantum Fourier transform with circuit quantum electrodynamics (QED).Acareful analysis of the decoherence sources shows that the algorithm can be achieved with a high fidelity using currentcircuit QED techniques.

An Image Encryption Method Based on Quantum Fourier Transformation

The image security problem is an important area in information security, and image encryption plays a vital role in this day. To protect the image encryption from the attack of quantum algorithm appeared recently, an image encryption method based on quantum Fourier transformation is proposed here. First, the image encryption and Fourier transformation are discussed here, then a encryption function is proposed. Second, a quantum Fourier transformation is introduced to quantum encryption, and the full step of quantum encryption is given as well. Third, the security of the proposed quantum encryption if analyzed, and some propositions are also presented. Lastly, some conclusions are indicated and some possible directions are also listed.

t-bit semiclassical quantum Fourier transform

Because of the difficulty of building a high-dimensional quantum register,this paper presents an implementation of the high-dimensional quantum Fourier transform(QFT)based on a low-dimensional quantum register.First,we define the t-bit semi- classical quantum Fourier transform.In terms of probability amplitude,we prove that the transform can realize quantum Fourier transformation,illustrate that the requirement for the two-qubit gate reduces obviously,and further design a quantum circuit of the transform.Combining the classical fixed-window method and the implementation of Shor's quantum factorization algorithm,we then redesign a circuit for Shor's algorithm,whose required computation resource is approximately equal to that of Parker's.The requirement for elementary quantum gates for Parker's algorithm is 3 O (logN),and the quantum register for our circuit re- quires t-1 more dimensions than Parker's.However,our circuit is t2 times as fast as Parker's,where t is the width of the window.

Quantum entangled fractional Fourier transform based on the IWOP technique

In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform.The core function of the coordinate-momentum exchange operators in the addition law of fractional Fourier transform was analyzed too.In this paper,the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators(IWOP)are used to establish the entanglement fractional Fourier transform theory to the extent of quantum.A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.

The Fourier Transform and Principles of Quantum Mechanics

From the hypotheses that the position-representation of a physical state is the Fourier transform of its momentum-representation and that the timerepresentation is the inverse Fourier transform of its energy-representation, we are able to obtain the Planck relation E = hν , the de Broglie relation p = h /λ , the Dirac fundamental commutation relation, the Schr?dinger equations, the Heisenberg uncertainty principle in quantum mechanics, and the annihilation/creation of a photon from excitation/de-excitation of an atom following Bohr.

Convolution Theorem of Fractional Fourier Transformation Derived by Representation Transformation in Quantum Mechancis

Based on our previous paper (Commun.Theor.Phys.39 (2003) 417) we derive the convolution theoremof fractional Fourier transformation in the context of quantum mechanics,which seems a convenient and neat way.Generalization of this method to the complex fractional Fourier transformation case is also possible.

The Fourier slice transformation of the Wigner operator and the quantum tomogram of the density operator

Using the Weyl quantization scheme and based on the Fourier slice transformation （FST） of the Wigner operator, we construct a new expansion formula of the density operator p, with the expansion coefficient being the FST of p＇s classical Weyl correspondence, and the latter the Fourier transformation of p＇s quantum tomogram. The coordinate momentum intermediate representation is used as the Radon transformation of the Wigner operator.

Quantum Discrete Fourier Transform in an Ion Trap System

<正> We propose two schemes for the implementation of quantum discrete Fourier transform in the ion trap sys-tem.In each scheme we design a tunable two-qubit phase gate as the main ingredient.The experimental implementationof the schemes would be an important step toward complex quantum computation in the ion trap system.

Quantum-Classical Algorithm for an Instantaneous Spectral Analysis of Signals:A Complement to Fourier Theory

A quantum time-dependent spectrum analysis, or simply, quantum spectral analysis (QSA) is presented in this work, and it’s based on Schr&#246;dinger’s equation. In the classical world, it is named frequency in time (FIT), which is used here as a complement of the traditional frequency-dependent spectral analysis based on Fourier theory. Besides, FIT is a metric which assesses the impact of the flanks of a signal on its frequency spectrum, not taken into account by Fourier theory and lets alone in real time. Even more, and unlike all derived tools from Fourier Theory (i.e., continuous, discrete, fast, short-time, fractional and quantum Fourier Transform, as well as, Gabor) FIT has the following advantages, among others: 1) compact support with excellent energy output treatment, 2) low computational cost, O(N) for signals and O(N2) for images, 3) it does not have phase uncertainties (i.e., indeterminate phase for a magnitude = 0) as in the case of Discrete and Fast Fourier Transform (DFT, FFT, respectively). Finally, we can apply QSA to a quantum signal, that is, to a qubit stream in order to analyze it spectrally.

Speeding up implementation for Shor's factorization quantum algorithm

In this paper, based on the implementation of semiclassical quantum Fourier transform, we first propose the concept of generation vector of ternary binary representation, construct the generation function's truth table, prove that the generation vector of ternary binary representation is one kind of k 's NAF representation and further find that its number of nonzero is not more than [(「logk」+1) /2]. Then we redesign a quantum circuit for Shor's algorithm, whose computation resource is approximately equal to that of Parker (Their requirements of elementary quantum gate are both O (「logN」3), and our circuit requires 2 qubits more than Parker's). However, our circuit is twice as fast as Parker's.

Realization of quantum permutation algorithm in high dimensional Hilbert space

Quantum algorithms provide a more efficient way to solve computational tasks than classical algorithms. We experimentally realize quantum permutation algorithm using light＇s orbital angular momentum degree of freedom. By exploiting the spatial mode of photons, our scheme provides a more elegant way to understand the principle of quantum permutation algorithm and shows that the high dimension characteristic of light＇s orbital angular momentum may be useful in quantum algorithms. Our scheme can be extended to higher dimension by introducing more spatial modes and it paves the way to trace the source of quantum speedup.

Studying the Two New Convolutions of Fractional Fourier Transform by Using Dirac Notation

Based on quantum mechanical representation and operator theory,this paper restates the two new convolutions of fractional Fourier transform(FrFT)by making full use of the conversion relationship between two mutual conjugates:coordinate representation and momentum representation.This paper gives full play to the efficiency of Dirac notation and proves the convolutions of fractional Fourier transform from the perspective of quantum optics,a field that has been developing rapidly.These two new convolution methods have potential value in signal processing.

Terahertz Generation Using Implanted InGaAs Photomixers and Multi-wavelength Quantum Dot Lasers

We report on a study of terahertz(THz) generation using implanted In Ga As photomixers and multi-wavelength quantum dot lasers. We carry out In Ga As materials growth, optical characterization, device design and fabrication, and photomixing experiments. This approach is capable of generating a comb of electromagnetic radiation from microwave to terahertz. For shortening photomixer carrier lifetime, we employ proton implantation into an epitaxial layer of lattice matched In Ga As grown on InP. Under a 1.55 μm multimode In GaAs/In GaAsP quantum dot laser excitation, a frequency comb with a constant frequency spacing of 50 GHz generated on the photomixer is measured, which corresponds to the beats of the laser longitudinal modes. The measurement is performed with a Fourier transform infrared spectrometer. This approach affords a convenient method to achieve a broadband multi-peak coherent THz source.

The quantum spectra analysis of the circular billiards in wells

We use a recently defined quantum spectral function and apply the method of closed-orbit theory to the 2D circular billiard system. The quantum spectra contain rich information of all classical orbits connecting two arbitrary points in the well. We study the correspondence between quantum spectra and classical orbits in the circular, 1/2 circular and 1/4 circular wells using the analytic and numerical methods. We find that the peak positions in the Fourier- transformed quantum spectra match accurately with the lengths of the classical orbits. These examples show evidently that semi-classlcal method provides a bridge between quantum and classical mechanics.