Grover’s Algorithm and Its Generalization

2018 ◽  
pp. 41-67
Author(s):  
Renato Portugal
Keyword(s):  
Grover’S Algorithm ◽  
Grover's Algorithm

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 A framework for reducing the overhead of the quantum oracle for use with Grover’s algorithm with applications to cryptanalysis of SIKE

2020 ◽  
Vol 15 (1) ◽  
pp. 143-156
Author(s):  
Jean-François Biasse ◽  
Benjamin Pring
Keyword(s):  
Search Algorithm ◽  
Circuit Complexity ◽  
Quantum Circuit ◽  
Grover’S Algorithm ◽  
Search Problems ◽  
Trade Offs ◽  
Single Target ◽  
Grover's Algorithm ◽  
Quantum Oracle ◽  
The Cost

AbstractIn this paper we provide a framework for applying classical search and preprocessing to quantum oracles for use with Grover’s quantum search algorithm in order to lower the quantum circuit-complexity of Grover’s algorithm for single-target search problems. This has the effect (for certain problems) of reducing a portion of the polynomial overhead contributed by the implementation cost of quantum oracles and can be used to provide either strict improvements or advantageous trade-offs in circuit-complexity. Our results indicate that it is possible for quantum oracles for certain single-target preimage search problems to reduce the quantum circuit-size from $O\left(2^{n/2}\cdot mC\right)$ (where C originates from the cost of implementing the quantum oracle) to $O(2^{n/2} \cdot m\sqrt{C})$ without the use of quantum ram, whilst also slightly reducing the number of required qubits.This framework captures a previous optimisation of Grover’s algorithm using preprocessing [21] applied to cryptanalysis, providing new asymptotic analysis. We additionally provide insights and asymptotic improvements on recent cryptanalysis [16] of SIKE [14] via Grover’s algorithm, demonstrating that the speedup applies to this attack and impacting upon quantum security estimates [16] incorporated into the SIKE specification [14].

Global Optimization With Quantum Walk Enhanced Grover Search

2014 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Global Optimization ◽  
Optimization Problems ◽  
Early Stage ◽  
Hybrid Approach ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Tunneling Effect ◽  
Grover’S Algorithm ◽  
Grover's Algorithm ◽  
Grover Search

One of the significant breakthroughs in quantum computation is Grover’s algorithm for unsorted database search. Recently, the applications of Grover’s algorithm to solve global optimization problems have been demonstrated, where unknown optimum solutions are found by iteratively improving the threshold value for the selective phase shift operator in Grover rotation. In this paper, a hybrid approach that combines continuous-time quantum walks with Grover search is proposed. By taking advantage of quantum tunneling effect, local barriers are overcome and better threshold values can be found at the early stage of search process. The new algorithm based on the formalism is demonstrated with benchmark examples of global optimization. The results between the new algorithm and the Grover search method are also compared.

scholarly journals Adaptive Phase Matching in Grover’s Algorithm

2011 ◽  
Vol 01 (02) ◽  
pp. 43-49
Author(s):  
Panchi Li ◽  
Kaoping Song
Keyword(s):  
Phase Matching ◽  
Grover’S Algorithm ◽  
Grover's Algorithm

Impact of various IBM Quantum architectures with different properties on Grover's algorithm

2021 ◽  
Author(s):  
Mohd Harith Akmal Zulfaizal Fadillah ◽  
Bahari Idrus ◽  
Mohammad Khatim Hasan ◽  
Siti Munirah Mohd
Keyword(s):  
Grover’S Algorithm ◽  
Grover's Algorithm

The first iteration of Grover's algorithm using classical light with orbital angular momentum

2018 ◽  
Vol 65 (16) ◽  
pp. 1942-1948 ◽  
Author(s):  
Benjamin Perez-Garcia ◽  
Raul I. Hernandez-Aranda ◽  
Andrew Forbes ◽  
Thomas Konrad
Keyword(s):  
Angular Momentum ◽  
Orbital Angular Momentum ◽  
Grover’S Algorithm ◽  
Grover's Algorithm

Grover’s Algorithm for Unstructured Search

2019 ◽  
pp. 35-55
Author(s):  
Franklin de Lima Marquezino ◽  
Renato Portugal ◽  
Carlile Lavor
Keyword(s):  
Grover’S Algorithm ◽  
Unstructured Search ◽  
Grover's Algorithm

scholarly journals Is Grover’s Algorithm a Quantum Hidden Subgroup Algorithm?

2007 ◽  
Vol 6 (6) ◽  
pp. 461-476 ◽  
Author(s):  
Samuel J. Lomonaco ◽  
Louis H. Kauffman
Keyword(s):  
Grover’S Algorithm ◽  
Grover's Algorithm
Sign in / Sign up

Export Citation Format

Share Document