Kvantu automātu un meklēšanas algoritmu iespējas un ierobežojumi

Kvantu skaitļošana ir nozare, kas pēta uz kvantu mehānikas likumiem balstīto skaitļošanas modeļu īpašības. Disertācija ir veltīta kvantu skaitļošanas algoritmiskiem aspektiem. Piedāvāti rezultāti trijos virzienos: Kvantu galīgi automāti Analizēta stāvokļu efektivitāte kvantu vienvirziena galīgam automātam. Uzlabota labāka zināmā eksponenciālā atšķirība [AF98] starp kvantu un klasiskajiem galīgajiem automātiem. Grovera algoritma analīze Pētīta Grovera algoritma noturība pret kļūdām. Vispārināts [RS08] loģisko kļūdu modelis un piedāvāti vairāki jauni rezultāti. Kvantu klejošana Pētīta meklēšana 2D režģī izmantojot kvantu klejošanu. Paātrināts [AKR05] kvantu klejošanas meklēšanas algoritms. Atslēgas vārdi: Kvantu galīgi automāti, eksponenciālā atšķirība, Grovera algoritms, noturība pret kļūdām, kvantu klejošana LITERATŪRA [AF98] A. Ambainis, R. Freivalds. 1-way quantum finite automata: strengths, weaknesses and generalizations. Proceedings of the 39th IEEE Conference on Foundations of Computer Science, 332-341, 1998. arXiv:quant-ph/9802062v3 [AKR05] A. Ambainis, J. Kempe, A. Rivosh. Coins make quantum walks faster. Proceedings of SODA’05, 1099-1108, 2005. [RS08] O. Regev, L. Schiff. Impossibility of a Quantum Speed-up with a Faulty Oracle. Proceedings of ICALP’2008, Lecture Notes in Computer Science, 5125:773-781, 2008.Quantum computation is the eld that investigates properties of models of computation based on the laws of the quantum mechanics. The thesis is ded- icated to algorithmic aspects of quantum computation and provides results in three directions: Quantum nite automata We study space-eciency of one-way quantum nite automata. We improve best known exponential separation [AF98] between quantum and classical one-way nite automata. Analysis of Grover's algorithm We study fault-tolerance of Grover's algorithm. We generalize the model of logical faults by [RS08] and present several new results. Quantum walks We study search by quantum walks on two-dimensional grid. We im- prove (speed-up) quantum walk search algorithm by [AKR05]. Keywords: Quantum nite automata, exponential separation, Grover's al- gorithm, fault-tolerance, quantum walks BIBLIOGRAPHY [AF98] A. Ambainis, R. Freivalds. 1-way quantum nite automata: strengths, weaknesses and gen- eralizations. Proceedings of the 39th IEEE Conference on Foundations of Computer Science, 332-341, 1998. arXiv:quant-ph/9802062v3 [AKR05] A. Ambainis, J. Kempe, A. Rivosh. Coins make quantum walks faster. Proceedings of SODA'05, 1099-1108, 2005. [RS08] O. Regev, L. Schi. Impossibility of a Quantum Speed-up with a Faulty Oracle. Proceedings of ICALP'2008, Lecture Notes in Computer Science, 5125:773-781, 2008

On the probability of finding marked connected components using quantum walks

Finding a marked vertex in a graph can be a complicated task when using quantum walks. Recent results show that for two or more adjacent marked vertices search by quantum walk with Grover's coin may have no speed-up over classical exhaustive search. In this paper, we analyze the probability of finding a marked vertex for a set of connected components of marked vertices. We prove two upper bounds on the probability of finding a marked vertex and sketch further research directions.Comment: 13 pages. To appear at Lobachevskii Journal of Mathematic

The Power and the Limits of Quantum Automata and Search Algorithms

ANOTĀCIJA Kvantu skaitļošana ir nozare, kas pēta uz kvantu mehānikas likumiem balstīto skaitļošanas modeļu īpašības. Disertācija ir veltīta kvantu skaitļošanas algoritmiskiem aspektiem. Piedāvāti rezultāti trijos virzienos: Kvantu galīgi automāti Analizēta stāvokļu efektivitāte kvantu vienvirziena galīgam automātam. Uzlabota labāka zināmā eksponenciālā atšķirība [AF98] starp kvantu un klasiskajiem galīgajiem automātiem. Grovera algoritma analīze Pētīta Grovera algoritma noturība pret kļūdām. Vispārināts [RS08] loģisko kļūdu modelis un piedāvāti vairāki jauni rezultāti. Kvantu klejošana Pētīta meklēšana 2D režģī izmantojot kvantu klejošanu. Paātrināts [AKR05] kvantu klejošanas meklēšanas algoritms. Atslēgas vārdi: Kvantu galīgi automāti, eksponenciālā atšķirība, Grovera algoritms, noturība pret kļūdām, kvantu klejošana LITERATŪRA [AF98] A. Ambainis, R. Freivalds. 1-way quantum finite automata: strengths, weaknesses and generalizations. Proceedings of the 39th IEEE Conference on Foundations of Computer Science, 332-341, 1998. arXiv:quant-ph/9802062v3 [AKR05] A. Ambainis, J. Kempe, A. Rivosh. Coins make quantum walks faster. Proceedings of SODA’05, 1099-1108, 2005. [RS08] O. Regev, L. Schiff. Impossibility of a Quantum Speed-up with a Faulty Oracle. Proceedings of ICALP’2008, Lecture Notes in Computer Science, 5125:773-781, 2008.ABSTRACT Quantum computation is the eld that investigates properties of models of computation based on the laws of the quantum mechanics. The thesis is ded- icated to algorithmic aspects of quantum computation and provides results in three directions: Quantum nite automata We study space-eciency of one-way quantum nite automata. We improve best known exponential separation [AF98] between quantum and classical one-way nite automata. Analysis of Grover's algorithm We study fault-tolerance of Grover's algorithm. We generalize the model of logical faults by [RS08] and present several new results. Quantum walks We study search by quantum walks on two-dimensional grid. We im- prove (speed-up) quantum walk search algorithm by [AKR05]. Keywords: Quantum nite automata, exponential separation, Grover's al- gorithm, fault-tolerance, quantum walks BIBLIOGRAPHY [AF98] A. Ambainis, R. Freivalds. 1-way quantum nite automata: strengths, weaknesses and gen- eralizations. Proceedings of the 39th IEEE Conference on Foundations of Computer Science, 332-341, 1998. arXiv:quant-ph/9802062v3 [AKR05] A. Ambainis, J. Kempe, A. Rivosh. Coins make quantum walks faster. Proceedings of SODA'05, 1099-1108, 2005. [RS08] O. Regev, L. Schi. Impossibility of a Quantum Speed-up with a Faulty Oracle. Proceedings of ICALP'2008, Lecture Notes in Computer Science, 5125:773-781, 2008

The Power and the Limits of Quantum Automata and Search Algorithms

ANOTĀCIJA Kvantu skaitļošana ir nozare, kas pēta uz kvantu mehānikas likumiem balstīto skaitļošanas modeļu īpašības. Disertācija ir veltīta kvantu skaitļošanas algoritmiskiem aspektiem. Piedāvāti rezultāti trijos virzienos: Kvantu galīgi automāti Analizēta stāvokļu efektivitāte kvantu vienvirziena galīgam automātam. Uzlabota labāka zināmā eksponenciālā atšķirība [AF98] starp kvantu un klasiskajiem galīgajiem automātiem. Grovera algoritma analīze Pētīta Grovera algoritma noturība pret kļūdām. Vispārināts [RS08] loģisko kļūdu modelis un piedāvāti vairāki jauni rezultāti. Kvantu klejošana Pētīta meklēšana 2D režģī izmantojot kvantu klejošanu. Paātrināts [AKR05] kvantu klejošanas meklēšanas algoritms. Atslēgas vārdi: Kvantu galīgi automāti, eksponenciālā atšķirība, Grovera algoritms, noturība pret kļūdām, kvantu klejošana LITERATŪRA [AF98] A. Ambainis, R. Freivalds. 1-way quantum finite automata: strengths, weaknesses and generalizations. Proceedings of the 39th IEEE Conference on Foundations of Computer Science, 332-341, 1998. arXiv:quant-ph/9802062v3 [AKR05] A. Ambainis, J. Kempe, A. Rivosh. Coins make quantum walks faster. Proceedings of SODA’05, 1099-1108, 2005. [RS08] O. Regev, L. Schiff. Impossibility of a Quantum Speed-up with a Faulty Oracle. Proceedings of ICALP’2008, Lecture Notes in Computer Science, 5125:773-781, 2008.ABSTRACT Quantum computation is the eld that investigates properties of models of computation based on the laws of the quantum mechanics. The thesis is ded- icated to algorithmic aspects of quantum computation and provides results in three directions: Quantum nite automata We study space-eciency of one-way quantum nite automata. We improve best known exponential separation [AF98] between quantum and classical one-way nite automata. Analysis of Grover's algorithm We study fault-tolerance of Grover's algorithm. We generalize the model of logical faults by [RS08] and present several new results. Quantum walks We study search by quantum walks on two-dimensional grid. We im- prove (speed-up) quantum walk search algorithm by [AKR05]. Keywords: Quantum nite automata, exponential separation, Grover's al- gorithm, fault-tolerance, quantum walks BIBLIOGRAPHY [AF98] A. Ambainis, R. Freivalds. 1-way quantum nite automata: strengths, weaknesses and gen- eralizations. Proceedings of the 39th IEEE Conference on Foundations of Computer Science, 332-341, 1998. arXiv:quant-ph/9802062v3 [AKR05] A. Ambainis, J. Kempe, A. Rivosh. Coins make quantum walks faster. Proceedings of SODA'05, 1099-1108, 2005. [RS08] O. Regev, L. Schi. Impossibility of a Quantum Speed-up with a Faulty Oracle. Proceedings of ICALP'2008, Lecture Notes in Computer Science, 5125:773-781, 2008