J-Unitary Operators

1998 ◽  
pp. 191-231
Author(s):  
Patrick Dewilde ◽  
Alle-Jan van der Veen
Keyword(s):  
Unitary Operators

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

scholarly journals Understanding mathematics of Grover’s algorithm

2021 ◽  
Vol 20 (5) ◽  
Author(s):  
Paweł J. Szabłowski
Keyword(s):  
Phase Change ◽  
Unitary Operator ◽  
Mathematical Structure ◽  
Complex Numbers ◽  
Unitary Operators ◽  
Grover’S Algorithm ◽  
Great Probability ◽  
Grover's Algorithm ◽  
One Step ◽  
The One

AbstractWe analyze the mathematical structure of the classical Grover’s algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one ‘chosen’ element (sometimes called a ‘solution’) of the dataset, but a set of m such ‘chosen’ elements (out of $$n>m)$$ n > m ) . Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that ‘marks,’ by a suitable phase change $$\varphi $$ φ , all these ‘chosen’ elements. In the first part of the paper, we construct a unique unitary operator that selects all ‘chosen’ elements in one step. The constructed operator is uniquely defined by the numbers $$\varphi $$ φ and $$\alpha $$ α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on $$\alpha $$ α . In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, ‘convenient’ phase change $$\varphi ,$$ φ , and by sequentially applying the so-constructed operator, we find the number of steps to find these ‘chosen’ elements with great probability. We apply this knowledge to study the generalizations of Grover’s algorithm ($$m=1,\phi =\pi $$ m = 1 , ϕ = π ), which are of the form, the found previously, unitary operators.

scholarly journals Spectral asymptotics of semiclassical unitary operators

2019 ◽  
Vol 473 (2) ◽  
pp. 1174-1202 ◽  
Author(s):  
Yohann Le Floch ◽  
Álvaro Pelayo
Keyword(s):  
Spectral Asymptotics ◽  
Unitary Operators

MONODROMY APPROACH TO QUANTUM COMPUTING

2002 ◽  
Vol 16 (30) ◽  
pp. 4593-4605 ◽  
Author(s):  
G. GIORGADZE
Keyword(s):  
Quantum Computing ◽  
Hilbert Problem ◽  
Holomorphic Vector Bundle ◽  
Classical System ◽  
Unitary Operators ◽  
Unitary Matrices ◽  
Classical Procedure ◽  
Logarithmic Singularities ◽  
The Given ◽  
Riemann Hilbert Problem

In this work, a gauge approach to quantum computing is considered. It is assumed that there exists a classical procedure for placing certain classical system in a state described by a holomorphic vector bundle with connection with logarithmic singularities. This bundle and its connection are constructed with the aid of unitary operators realizing the given algorithm using methods of the monodromic Riemann–Hilbert problem. Universality is understood in the sense that for any collection of unitary matrices there exists a connection with logarithmic singularities whose monodromy representation involves these matrices.

scholarly journals Functions of unitary operators: Derivatives and trace formulas

2016 ◽  
Vol 270 (6) ◽  
pp. 2048-2072 ◽  
Author(s):  
Denis Potapov ◽  
Anna Skripka ◽  
Fedor Sukochev
Keyword(s):  
Unitary Operators ◽  
Trace Formulas

From coherent state representation to unitary operators via the IWOP technique

Optik ◽  
2019 ◽  
Vol 178 ◽  
pp. 372-378 ◽  
Author(s):  
Ying Xia ◽  
Liyun Hu ◽  
Huan Zhang ◽  
Haoliang Zhang
Keyword(s):  
Coherent State ◽  
Iwop Technique ◽  
Unitary Operators ◽  
State Representation ◽  
The Iwop Technique

scholarly journals Conceptual Framework for Quantum Affective Computing and Its Use in Fusion of Multi-Robot Emotions

2020 ◽  
Author(s):  
Fei Yan ◽  
Abdullah Iliyasu ◽  
Kaoru Hirota
Keyword(s):  
Emotional Intelligence ◽  
Quantum Mechanics ◽  
Conceptual Framework ◽  
Quantum Measurement ◽  
Affective Computing ◽  
Quantum Gates ◽  
Superposition State ◽  
Unitary Operators ◽  
Computing Paradigm ◽  
Simulation Based

This study presents a modest attempt to interpret, formulate, and manipulate emotion of robots within the precepts of quantum mechanics. Our proposed framework encodes the emotion information as a superposition state whilst unitary operators are used to manipulate the transition of the emotion states which are recovered via appropriate quantum measurement operations. The framework described provides essential steps towards exploiting the potency of quantum mechanics in a quantum affective computing paradigm. Further, the emotions of multi-robots in a specified communication scenario are fused using quantum entanglement thereby reducing the number of qubits required to capture the emotion states of all the robots in the environment, and fewer quantum gates are needed to transform the emotion of all or part of the robots from one state to another. In addition to the mathematical rigours expected of the proposed framework, we present a few simulation-based demonstrations to illustrate its feasibility and effectiveness. This exposition is an important step in the transition of formulations of emotional intelligence to the quantum era.

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