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 robustness of bucket brigade quantum RAM

We study the robustness of the bucket brigade quantum random access memory model introduced by Giovannetti, Lloyd, and Maccone [Phys. Rev. Lett. 100, 160501 (2008)]. Due to a result of Regev and Schiff [ICALP '08 pp. 773], we show that for a class of error models the error rate per gate in the bucket brigade quantum memory has to be of order $o(2^{-n/2})$ (where $N=2^n$ is the size of the memory) whenever the memory is used as an oracle for the quantum searching problem. We conjecture that this is the case for any realistic error model that will be encountered in practice, and that for algorithms with super-polynomially many oracle queries the error rate must be super-polynomially small, which further motivates the need for quantum error correction. By contrast, for algorithms such as matrix inversion [Phys. Rev. Lett. 103, 150502 (2009)] or quantum machine learning [Phys. Rev. Lett. 113, 130503 (2014)] that only require a polynomial number of queries, the error rate only needs to be polynomially small and quantum error correction may not be required. We introduce a circuit model for the quantum bucket brigade architecture and argue that quantum error correction for the circuit causes the quantum bucket brigade architecture to lose its primary advantage of a small number of "active" gates, since all components have to be actively error corrected.Comment: Replaced with the published version. 13 pages, 9 figure

Grover's search with faults on some marked elements

Grover's algorithm is a quantum query algorithm solving the unstructured search problem of size $N$ using $O(\sqrt{N})$ queries. It provides a significant speed-up over any classical algorithm \cite{Gro96}. The running time of the algorithm, however, is very sensitive to errors in queries. It is known that if query may fail (report all marked elements as unmarked) the algorithm needs $\Omega(N)$ queries to find a marked element \cite{RS08}. \cite{AB+13} have proved the same result for the model where each marked element has its own probability to be reported as unmarked. We study the behavior of Grover's algorithm in the model where the search space contains both faulty and non-faulty marked elements. We show that in this setting it is indeed possible to find one of non-faulty marked items in $O(\sqrt{N})$ queries. We also analyze the limiting behavior of the algorithm for a large number of steps and show the existence and the structure of limiting state $\rho_{lim}$.Comment: 17 pages, 6 figure

Fault-ignorant Quantum Search

We investigate the problem of quantum searching on a noisy quantum computer. Taking a 'fault-ignorant' approach, we analyze quantum algorithms that solve the task for various different noise strengths, which are possibly unknown beforehand. We prove lower bounds on the runtime of such algorithms and thereby find that the quadratic speedup is necessarily lost (in our noise models). However, for low but constant noise levels the algorithms we provide (based on Grover's algorithm) still outperform the best noiseless classical search algorithm.Comment: v1: 15+8 pages, 4 figures; v2: 19+8 pages, 4 figures, published version (Introduction section significantly expanded, presentation clarified, results and order unchanged