On power sums of complex numbers whose sum is 0

1995 ◽  
Vol 66 (1-2) ◽  
pp. 51-60
Author(s):  
G. Harcos
Keyword(s):  
Complex Numbers ◽  
Power Sums

scholarly journals Complex numbers with vanishing power sums

1957 ◽  
Vol 33 (1) ◽  
pp. 10-12 ◽  
Author(s):  
Saburô Uchiyama
Keyword(s):  
Complex Numbers ◽  
Power Sums

Remark on power-sums of complex numbers

1967 ◽  
Vol 18 (3-4) ◽  
pp. 279-281 ◽  
Author(s):  
J. Ławrynowicz
Keyword(s):  
Complex Numbers ◽  
Power Sums

On some problems of P. Turán concerning power sums of complex numbers

1992 ◽  
Vol 59 (1-2) ◽  
pp. 11-24 ◽  
Author(s):  
L. Erdős
Keyword(s):  
Complex Numbers ◽  
Power Sums

Complex numbers

2006 ◽  
Author(s):  
Stephen C. Roy
Keyword(s):  
Complex Numbers

On higher order Fibonacci hyper complex numbers

2021 ◽  
Vol 148 ◽  
pp. 111044
Author(s):  
Can Kızılateş ◽  
Tiekoro Kone
Keyword(s):  
Higher Order ◽  
Complex Numbers

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.

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