Unitary matrix decompositions for optimal and modular linear optics architectures

2021 ◽  
Vol 54 (4) ◽  
pp. 045301
Author(s):  
Shreya P Kumar ◽  
Ish Dhand
Keyword(s):  
Unitary Matrix ◽  
Linear Optics ◽  
Matrix Decompositions

scholarly journals Central Splitting of A2 Discrete Fourier–Weyl Transforms

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1828 ◽  
Author(s):  
Jiří Hrivnák ◽  
Mariia Myronova ◽  
Jiří Patera
Keyword(s):  
Reflection Group ◽  
Unitary Matrix ◽  
Weyl Transform ◽  
Matrix Decompositions ◽  
Weight Lattice ◽  
Affine Weyl Group ◽  
Weyl Transforms ◽  
Original Weight ◽  
Unitary Transform ◽  
Point Set

Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2 constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A2 is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.

scholarly journals DECOMPOSITION OF UNITARY MATRICES AND QUANTUM GATES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350015 ◽  
Author(s):  
CHI-KWONG LI ◽  
REBECCA ROBERTS ◽  
XIAOYAN YIN
Keyword(s):  
General Scheme ◽  
Quantum Gate ◽  
Quantum Gates ◽  
Unitary Matrix ◽  
Additional Structure ◽  
Unitary Matrices ◽  
Decomposition Scheme ◽  
Single Qubit ◽  
Matlab Program

A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d - 1 classes, where each class is isomorphic to the group of 2 × 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2n-1(2n-1) fully controlled single-qubit gates chosen from 2n-1 classes, where the quantum gates in each class share the same n - 1 control qubits. Moreover, it is shown that one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.

scholarly journals Uniqueness Analysis of Non-Unitary Matrix Joint Diagonalization

2013 ◽  
Vol 61 (7) ◽  
pp. 1786-1796 ◽  
Author(s):  
Martin Kleinsteuber ◽  
Hao Shen
Keyword(s):  
Unitary Matrix ◽  
Joint Diagonalization

Implementation of a single-photon fully quantum router with cavity QED and linear optics

2021 ◽  
Vol 53 (1) ◽  
Author(s):  
Cong Cao ◽  
Yu-Hong Han ◽  
Xin Yi ◽  
Pan-Pan Yin ◽  
Xiu-Yu Zhang ◽  
...  
Keyword(s):  
Cavity Qed ◽  
Single Photon ◽  
Linear Optics ◽  
Quantum Router

scholarly journals Multiple phases in a generalized Gross-Witten-Wadia matrix model

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Jorge G. Russo ◽  
Miguel Tierz
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Characteristic Polynomial ◽  
Matrix Model ◽  
Phase Structure ◽  
Unitary Matrix ◽  
Two Dimensional ◽  
Toeplitz Determinants ◽  
Dimensional Parameter ◽  
Planar Limit

Abstract We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large N results are obtained by using Szegö theorem with a Fisher-Hartwig singularity. In the large N (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space.

Polarimetry of soil and vegetation in the visible II: Mueller matrix decompositions

2021 ◽  
pp. 107622
Author(s):  
Sergey N. Savenkov ◽  
Alexander A. Kohkanovsky ◽  
Evgen A. Oberemok ◽  
Ivan S. Kolomiets ◽  
Alexander S. Klimov
Keyword(s):  
Mueller Matrix ◽  
Matrix Decompositions

scholarly journals A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1134
Author(s):  
Kenta Higuchi ◽  
Takashi Komatsu ◽  
Norio Konno ◽  
Hisashi Morioka ◽  
Etsuo Segawa
Keyword(s):  
Unit Circle ◽  
Stationary State ◽  
Discrete Time ◽  
Quantum Walk ◽  
Time Limit ◽  
The Other ◽  
Unitary Matrix ◽  
Initial State ◽  
Time Quantum ◽  
Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

scholarly journals Linear optics control of sideband instability for improved free-electron laser spectral brightness

2020 ◽  
Vol 23 (11) ◽  
Author(s):  
G. Perosa ◽  
E. M. Allaria ◽  
L. Badano ◽  
N. Bruchon ◽  
P. Cinquegrana ◽  
...  
Keyword(s):  
Free Electron ◽  
Free Electron Laser ◽  
Spectral Brightness ◽  
Linear Optics

scholarly journals A Unified Mathematical Formalism for First to Third Order Dielectric Response of Matter: Application to Surface-Specific Two-Colour Vibrational Optical Spectroscopy

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 153 ◽  
Author(s):  
Christophe Humbert ◽  
Thomas Noblet
Keyword(s):  
Dielectric Response ◽  
Structure And Properties ◽  
Linear Optics ◽  
Third Order ◽  
Mathematical Functions ◽  
Sum Frequency ◽  
Non Linear Optics ◽  
Non Linear ◽  
Light Excitation

To take advantage of the singular properties of matter, as well as to characterize it, we need to interact with it. The role of optical spectroscopies is to enable us to demonstrate the existence of physical objects by observing their response to light excitation. The ability of spectroscopy to reveal the structure and properties of matter then relies on mathematical functions called optical (or dielectric) response functions. Technically, these are tensor Green’s functions, and not scalar functions. The complexity of this tensor formalism sometimes leads to confusion within some articles and books. Here, we do clarify this formalism by introducing the physical foundations of linear and non-linear spectroscopies as simple and rigorous as possible. We dwell on both the mathematical and experimental aspects, examining extinction, infrared, Raman and sum-frequency generation spectroscopies. In this review, we thus give a personal presentation with the aim of offering the reader a coherent vision of linear and non-linear optics, and to remove the ambiguities that we have encountered in reference books and articles.

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