scholarly journals Quantum computation of Groebner basis

2021 ◽  
Author(s):  
Ichio Kikuchi ◽  
Akihito Kikuchi
Keyword(s):  
Algebraic Geometry ◽  
Quantum Computation ◽  
Classical Method ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Full Rank ◽  
Groebner Basis ◽  
Linear Solvers ◽  
Regular Sequences ◽  
Quantum Version

In this essay, we examine the feasibility of quantum computation of Groebner basis which is a fundamental tool of algebraic geometry. The classical method for computing Groebner basis is based on Buchberger's algorithm, and our question is how to adopt quantum algorithm there. A Quantum algorithm for finding the maximum is usable for detecting head terms of polynomials, which are required for the computation of S-polynomials. The reduction of S-polynomials with respect to a Groebner basis could be done by the quantum version of Gauss-Jordan elimination of echelon which represents polynomials. However, the frequent occurrence of zero-reductions of polynomials is an obstacle to the effective application of quantum algorithms. This is because zero-reductions of polynomials occur in non-full-rank echelons, for which quantum linear systems algorithms (through the inversion of matrices) are inadequate, as ever-known quantum linear solvers (such as Harrow-Hassidim-Lloyd) require the clandestine computations of the inverses of eigenvalues. Hence, for the quantum computation of the Groebner basis, the schemes to suppress the zero-reductions are necessary. To this end, the F5 algorithm or its variant (F5C) would be the most promising, as these algorithms have countermeasures against the occurrence of zero-reductions and can construct full-rank echelons whenever the inputs are regular sequences. Between these two algorithms, the F5C is the better match for algorithms involving the inversion of matrices.

scholarly journals An Efficient Quantum Algorithm for the Hidden Subgroup Problem over some Non-Abelian Groups

2017 ◽  
Vol 18 (2) ◽  
pp. 0215 ◽  
Author(s):  
Demerson Nunes Gonçalves ◽  
Tharso D Fernandes ◽  
C M M Cosme
Keyword(s):  
Positive Integer ◽  
Quantum Computation ◽  
Prime Number ◽  
Abelian Groups ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Prime Factorization ◽  
Hidden Subgroup Problem ◽  
Special Cases ◽  
Subgroup Problem

The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms are special cases of the HSP. In this paper we show that there exist a new efficient quantum algorithm for the HSP on groups $\Z_{N}\rtimes\Z_{q^s}$ where $N$ is an integer with a special prime factorization, $q$ prime number and $s$ any positive integer.

scholarly journals Quantum software engineering. Pt. I: Quantum Circuit (Gate) Model based Computing – education Lectures and pedagogical workshop

2020 ◽  
pp. 129-201
Author(s):  
Sergey Ulyanov ◽  
Andrey Reshetnikov ◽  
Olga Tyatyushkina ◽  
Vladimir Korenkov
Keyword(s):  
Software Engineering ◽  
Quantum Computation ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Circuit Model ◽  
Topological Quantum ◽  
Definition Of ◽  
Topological Quantum Computation ◽  
Main Ideas

All the quantum algorithms are based on a certain quantum computing model, varying from the quantum circuit, one-way quantum computation, adiabatic quantum computation and topological quantum computation. These four models are equivalent in computational power; among them, the quantum circuit model is most frequently used. In the circuit model, it has been proved that arbitrary single-qubit rotations plus twoqubit controlled-NOT gates are universal, i.e. they can provide a set of gates to implement any quantum algorithm. This article discusses the goal for this research: it is to given a lightning-fast (as-barebones-as-possible) definition of the quantum circuit model computing and leisurely development of quantum computation before actually getting around to sophisticated algorithms. In this article the main ideas of quantum software engineering is described.

scholarly journals Quantum circuit optimization using quantum Karnaugh map

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
J.-H. Bae ◽  
Paul M. Alsing ◽  
Doyeol Ahn ◽  
Warner A. Miller
Keyword(s):  
Quantum Computation ◽  
Quantum Algorithms ◽  
Optimization Technique ◽  
Circuit Complexity ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Circuit Optimization ◽  
General Technique ◽  
Small Quantum

Abstract Every quantum algorithm is represented by set of quantum circuits. Any optimization scheme for a quantum algorithm and quantum computation is very important especially in the arena of quantum computation with limited number of qubit resources. Major obstacle to this goal is the large number of elemental quantum gates to build even small quantum circuits. Here, we propose and demonstrate a general technique that significantly reduces the number of elemental gates to build quantum circuits. This is impactful for the design of quantum circuits, and we show below this could reduce the number of gates by 60% and 46% for the four- and five-qubit Toffoli gates, two key quantum circuits, respectively, as compared with simplest known decomposition. Reduced circuit complexity often goes hand-in-hand with higher efficiency and bandwidth. The quantum circuit optimization technique proposed in this work would provide a significant step forward in the optimization of quantum circuits and quantum algorithms, and has the potential for wider application in quantum computation.

Quantum Computing Approach for Alignment-Free Sequence Search and Classification

2012 ◽  
pp. 279-300
Author(s):  
Rao M. Kotamarti ◽  
Mitchell A. Thornton ◽  
Margaret H. Dunham
Keyword(s):  
Quantum Computing ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Search Space ◽  
Classification Problem ◽  
Sequence Classification ◽  
Sequence Search ◽  
Quantum Version ◽  
Computing Approach ◽  
Candidate Set

Many classes of algorithms that suffer from large complexities when implemented on conventional computers may be reformulated resulting in greatly reduced complexity when implemented on quantum computers. The dramatic reductions in complexity for certain types of quantum algorithms coupled with the computationally challenging problems in some bioinformatics problems motivates researchers to devise efficient quantum algorithms for sequence (DNA, RNA, protein) analysis. This chapter shows that the important sequence classification problem in bioinformatics is suitable for formulation as a quantum algorithm. This chapter leverages earlier research for sequence classification based on Extensible Markov Model (EMM) and proposes a quantum computing alternative. The authors utilize sequence family profiles built using EMM methodology which is based on using pre-counted word data for each sequence. Then a new method termed quantum seeding is proposed for generating a key based on high frequency words. The key is applied in a quantum search based on Grover algorithm to determine a candidate set of models resulting in a significantly reduced search space. Given Z as a function of M models of size N, the quantum version of the seeding algorithm has a time complexity in the order of as opposed to O(Z) for the standard classic version for large values of Z.

Quantum Computing Approach for Alignment-Free Sequence Search and Classification

2013 ◽  
pp. 1705-1726
Author(s):  
Rao M. Kotamarti ◽  
Mitchell A. Thornton ◽  
Margaret H. Dunham
Keyword(s):  
Quantum Computing ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Search Space ◽  
Classification Problem ◽  
Sequence Classification ◽  
Sequence Search ◽  
Quantum Version ◽  
Computing Approach ◽  
Candidate Set

Many classes of algorithms that suffer from large complexities when implemented on conventional computers may be reformulated resulting in greatly reduced complexity when implemented on quantum computers. The dramatic reductions in complexity for certain types of quantum algorithms coupled with the computationally challenging problems in some bioinformatics problems motivates researchers to devise efficient quantum algorithms for sequence (DNA, RNA, protein) analysis. This chapter shows that the important sequence classification problem in bioinformatics is suitable for formulation as a quantum algorithm. This chapter leverages earlier research for sequence classification based on Extensible Markov Model (EMM) and proposes a quantum computing alternative. The authors utilize sequence family profiles built using EMM methodology which is based on using pre-counted word data for each sequence. Then a new method termed quantum seeding is proposed for generating a key based on high frequency words. The key is applied in a quantum search based on Grover algorithm to determine a candidate set of models resulting in a significantly reduced search space. Given Z as a function of M models of size N, the quantum version of the seeding algorithm has a time complexity in the order of as opposed to O(Z) for the standard classic version for large values of Z.

scholarly journals Practical Quantum Computing

2021 ◽  
Vol 2 (1) ◽  
pp. 1-35
Author(s):  
Adrien Suau ◽  
Gabriel Staffelbach ◽  
Henri Calandra
Keyword(s):  
Wave Equation ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Equation Solving ◽  
Small Quantum ◽  
Quantum Wave ◽  
Variational Algorithms ◽  
The One

In the last few years, several quantum algorithms that try to address the problem of partial differential equation solving have been devised: on the one hand, “direct” quantum algorithms that aim at encoding the solution of the PDE by executing one large quantum circuit; on the other hand, variational algorithms that approximate the solution of the PDE by executing several small quantum circuits and making profit of classical optimisers. In this work, we propose an experimental study of the costs (in terms of gate number and execution time on a idealised hardware created from realistic gate data) associated with one of the “direct” quantum algorithm: the wave equation solver devised in [32]. We show that our implementation of the quantum wave equation solver agrees with the theoretical big-O complexity of the algorithm. We also explain in great detail the implementation steps and discuss some possibilities of improvements. Finally, our implementation proves experimentally that some PDE can be solved on a quantum computer, even if the direct quantum algorithm chosen will require error-corrected quantum chips, which are not believed to be available in the short-term.

scholarly journals Deep neural networks for quantum circuit mapping

2021 ◽  
Author(s):  
Giovanni Acampora ◽  
Roberto Schiattarella
Keyword(s):  
Neural Networks ◽  
Deep Neural Networks ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Machine Learning Techniques ◽  
Quantum Computers ◽  
Physical Constraints ◽  
Proper Mapping ◽  
Correct Execution

AbstractQuantum computers have become reality thanks to the effort of some majors in developing innovative technologies that enable the usage of quantum effects in computation, so as to pave the way towards the design of efficient quantum algorithms to use in different applications domains, from finance and chemistry to artificial and computational intelligence. However, there are still some technological limitations that do not allow a correct design of quantum algorithms, compromising the achievement of the so-called quantum advantage. Specifically, a major limitation in the design of a quantum algorithm is related to its proper mapping to a specific quantum processor so that the underlying physical constraints are satisfied. This hard problem, known as circuit mapping, is a critical task to face in quantum world, and it needs to be efficiently addressed to allow quantum computers to work correctly and productively. In order to bridge above gap, this paper introduces a very first circuit mapping approach based on deep neural networks, which opens a completely new scenario in which the correct execution of quantum algorithms is supported by classical machine learning techniques. As shown in experimental section, the proposed approach speeds up current state-of-the-art mapping algorithms when used on 5-qubits IBM Q processors, maintaining suitable mapping accuracy.

Efficient linear optics quantum computation

2001 ◽  
Vol 1 (Special) ◽  
pp. 13-19
Author(s):  
G.J. Milburn ◽  
T. Ralph ◽  
A. White ◽  
E. Knill ◽  
R. Laflamme
Keyword(s):  
Quantum Computation ◽  
Single Photon ◽  
Quantum Algorithms ◽  
Optical Nonlinearities ◽  
Linear Optics ◽  
Optical Elements ◽  
Particle Detectors ◽  
Feed Forward ◽  
Single Electrons ◽  
Photon Source

Two qubit gates for photons are generally thought to require exotic materials with huge optical nonlinearities. We show here that, if we accept two qubit gates that only work conditionally, single photon sources, passive linear optics and particle detectors are sufficient for implementing reliable quantum algorithms. The conditional nature of the gates requires feed-forward from the detectors to the optical elements. Without feed forward, non-deterministic quantum computation is possible. We discuss one proposed single photon source based on the surface acoustic wave guiding of single electrons.

TOWARDS A QUANTUM ALGORITHM FOR THE PERMANENT

2007 ◽  
Vol 05 (01n02) ◽  
pp. 223-228 ◽  
Author(s):  
ANNALISA MARZUOLI ◽  
MARIO RASETTI
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Computation ◽  
Quantum Computer ◽  
Quantum Algorithm ◽  
Topological Quantum Field Theory ◽  
Quantum Field ◽  
Topological Quantum

We resort to considerations based on topological quantum field theory to outline the development of a possible quantum algorithm for the evaluation of the permanent of a 0 - 1 matrix. Such an algorithm might represent a breakthrough for quantum computation, since computing the permanent is considered a "universal problem", namely, one among the hardest problems that a quantum computer can efficiently handle.

scholarly journals Exact quantum algorithms have advantage for almost all Boolean functions

2015 ◽  
pp. 435-452
Author(s):  
Andris Ambainis ◽  
Jozef Gruska ◽  
Shenggen Zheng
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Almost All ◽  
Quantum Query

It has been proved that almost all n-bit Boolean functions have exact classical query complexity n. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all n-bit Boolean functions can be computed by an exact quantum algorithm with less than n queries. More exactly, we prove that ANDn is the only n-bit Boolean function, up to isomorphism, that requires n queries.

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