Approximate Quantum Fourier Transform and Quantum Algorithm for Phase Estimation

2015 ◽  
pp. 391-405 ◽  
Author(s):  
Alexander N. Prokopenya
Keyword(s):  
Fourier Transform ◽  
Quantum Algorithm ◽  
Phase Estimation ◽  
Quantum Fourier Transform

scholarly journals QUANTUM PHASE ESTIMATION WITH AN ARBITRARY NUMBER OF QUBITS

2013 ◽  
Vol 11 (01) ◽  
pp. 1350008
Author(s):  
CHEN-FU CHIANG
Keyword(s):  
Fourier Transform ◽  
Quantum Computing ◽  
Diophantine Approximation ◽  
Great Difficulty ◽  
Quantum Computers ◽  
Phase Estimation ◽  
Quantum Fourier Transform ◽  
Factorization Algorithm ◽  
Simultaneous Diophantine Approximation ◽  
Parallel Circuits

Due to the great difficulty in scalability, quantum computers are limited in the number of qubits during the early stages of the quantum computing regime. In addition to the required qubits for storing the corresponding eigenvector, suppose we have additional k qubits available. Given such a constraint k, we propose an approach for the phase estimation for an eigenphase of exactly n-bit precision. This approach adopts the standard recursive circuit for quantum Fourier transform (QFT) in [R. Cleve and J. Watrous, Fast parallel circuits for quantum fourier transform, Proc. 41st Annual Symp. on Foundations of Computer Science (2000), pp. 526–536.] and adopts classical bits to implement such a task. Our algorithm has the complexity of O(n log k), instead of O(n2) in the conventional QFT, in terms of the total invocation of rotation gates. We also design a scheme to implement the factorization algorithm by using k available qubits via either the continued fractions approach or the simultaneous Diophantine approximation.

scholarly journals Which role does multiphoton interference play in small phase estimation in quantum Fourier transform interferometers?

2017 ◽  
Vol 15 (08) ◽  
pp. 1740020 ◽  
Author(s):  
Olaf Zimmermann ◽  
Vincenzo Tamma
Keyword(s):  
Fourier Transform ◽  
Constant Factor ◽  
Point Of View ◽  
Experimental Point ◽  
Phase Estimation ◽  
Single Photons ◽  
Phase Sensitivity ◽  
Quantum Fourier Transform ◽  
Small Phase ◽  
Number Formula

Recently, quantum Fourier transform interferometers have been demonstrated to allow a quantum metrological enhancement in phase sensitivity for a small number [Formula: see text] of identical input single photons [J. P. Olson, K. R. Motes, P. M. Birchall, N. M. Studer, M. LaBorde, T. Moulder, P. P. Rohde and J. P. Dowling, Phys. Rev. A 96 (2017) 013810; K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson and P. P. Rohde, Phys. Rev. Lett. 114 (2015) 170802; O. Zimmermann, Bachelor Thesis (Ulm University, 2015) arXiv: 1710.03805.]. However, multiphoton distinguishability at the detectors can play an important role from an experimental point of view [V. Tamma and S. Laibacher, Phys. Rev. Lett. 114 (2015) 243601.]. This raises a fundamental question: How is the phase sensitivity affected when the photons are completely distinguishable at the detectors and therefore do not interfere? In other words, which role does multiphoton interference play in these schemes? Here, we show that for small phase values, the phase sensitivity achievable in the proposed schemes with indistinguishable photons is enhanced only by a constant factor with respect to the case of completely distinguishable photons at the detectors. Interestingly, this enhancement arises from the interference of only a polynomial number (in [Formula: see text]) of the total [Formula: see text] multiphoton path amplitudes in the [Formula: see text]-port interferometer. These results are independent of the number [Formula: see text] of single photons and of the phase weight factors employed at each interferometer channel.

scholarly journals An efficient high dimensional quantum Schur transform

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 122 ◽  
Author(s):  
Hari Krovi
Keyword(s):  
Fourier Transform ◽  
Representation Theory ◽  
Tensor Product ◽  
Unitary Operator ◽  
Group Representation ◽  
Quantum Algorithm ◽  
Unitary Groups ◽  
High Dimensional ◽  
Quantum Fourier Transform ◽  
Dual Algorithm

The Schur transform is a unitary operator that block diagonalizes the action of the symmetric and unitary groups on an n fold tensor product V⊗n of a vector space V of dimension d. Bacon, Chuang and Harrow [5] gave a quantum algorithm for this transform that is polynomial in n, d and log⁡ϵ−1, where ϵ is the precision. In a footnote in Harrow's thesis [18], a brief description of how to make the algorithm of [5] polynomial in log⁡d is given using the unitary group representation theory (however, this has not been explained in detail anywhere). In this article, we present a quantum algorithm for the Schur transform that is polynomial in n, log⁡d and log⁡ϵ−1 using a different approach. Specifically, we build this transform using the representation theory of the symmetric group and in this sense our technique can be considered a ''dual" algorithm to [5]. A novel feature of our algorithm is that we construct the quantum Fourier transform over the so called permutation modules, which could have other applications.

Quantum Fourier Transform and Phase Estimation in Qudit System

2011 ◽  
Vol 55 (5) ◽  
pp. 790-794 ◽  
Author(s):  
Ye Cao ◽  
Shi-Guo Peng ◽  
Chao Zheng ◽  
Gui-Lu Long
Keyword(s):  
Fourier Transform ◽  
Phase Estimation ◽  
Quantum Fourier Transform

scholarly journals Digital-analog quantum algorithm for the quantum Fourier transform

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Ana Martin ◽  
Lucas Lamata ◽  
Enrique Solano ◽  
Mikel Sanz
Keyword(s):  
Fourier Transform ◽  
Quantum Algorithm ◽  
Quantum Fourier Transform

scholarly journals EXACT QUANTUM FOURIER TRANSFORMS AND DISCRETE LOGARITHM ALGORITHMS

2004 ◽  
Vol 02 (01) ◽  
pp. 91-100 ◽  
Author(s):  
MICHELE MOSCA ◽  
CHRISTOF ZALKA
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Quantum Algorithms ◽  
Discrete Logarithm ◽  
Quantum Algorithm ◽  
Quantum Fourier Transform ◽  
The Family ◽  
The Fourier Transform ◽  
Quantum Fourier Transforms ◽  
Amplitude Amplification

We show how the Quantum Fast Fourier Transform (QFFT) can be made exact for arbitrary orders (first showing it for large primes). Most quantum algorithms only need a good approximation of the quantum Fourier transform of order 2n to succeed with high probability, and this QFFT can in fact be done exactly. Kitaev1 showed how to approximate the Fourier transform for any order. Here we show how his construction can be made exact by using the technique known as "amplitude amplification". Although unlikely to be of any practical use, this construction allows one to make Shor's discrete logarithm quantum algorithm exact. Thus we have the first example of an exact non black box fast quantum algorithm, thereby giving more evidence that "quantum" need not be probabilistic. We also show that in a certain sense the family of circuits for the exact QFFT is uniform. Namely, the parameters of the gates can be approximated efficiently.

Optical simulator of the quantum Fourier transform

2016 ◽  
Vol 114 (2) ◽  
pp. 20004 ◽  
Author(s):  
Y. S. Nam ◽  
R. Blümel
Keyword(s):  
Fourier Transform ◽  
Quantum Fourier Transform

A watermark strategy for quantum images based on quantum fourier transform

2012 ◽  
Vol 12 (2) ◽  
pp. 793-803 ◽  
Author(s):  
Wei-Wei Zhang ◽  
Fei Gao ◽  
Bin Liu ◽  
Qiao-Yan Wen ◽  
Hui Chen
Keyword(s):  
Fourier Transform ◽  
Quantum Fourier Transform ◽  
Quantum Images

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

2015 ◽  
Vol 13 (07) ◽  
pp. 1550059 ◽  
Author(s):  
Shruti Dogra ◽  
Arvind Dorai ◽  
Kavita Dorai
Keyword(s):  
Fourier Transform ◽  
Quantum Algorithms ◽  
Quantum Computers ◽  
Quantum Fourier Transform ◽  
Specific Implementation ◽  
Circuit Decomposition

The quantum Fourier transform (QFT) is a key ingredient of several quantum algorithms and a qudit-specific implementation of the QFT is hence an important step toward the realization of qudit-based quantum computers. This work develops a circuit decomposition of the QFT for hybrid qudits based on generalized Hadamard and generalized controlled-phase gates, which can be implemented using selective rotations in NMR. We experimentally implement the hybrid qudit QFT on an NMR quantum emulator, which uses four qubits to emulate a single qutrit coupled to two qubits.

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