Computation with narrow CTCs

We examine some variants of computation with closed timelike curves (CTCs), where various restrictions are imposed on the memory of the computer, and the information carrying capacity and range of the CTC. We give full characterizations of the classes of languages recognized by polynomial time probabilistic and quantum computers that can send a single classical bit to their own past. Such narrow CTCs are demonstrated to add the power of limited nondeterminism to deterministic computers, and lead to exponential speedup in constant-space probabilistic and quantum computation. We show that, given a time machine with constant negative delay, one can implement CTC-based computations without the need to know about the runtime beforehand.Comment: 16 pages. A few typo was correcte

Superiority of one-way and realtime quantum machines and new directions

In automata theory, the quantum computation has been widely examined for finite state machines, known as quantum finite automata (QFAs), and less attention has been given to the QFAs augmented with counters or stacks. Moreover, to our knowledge, there is no result related to QFAs having more than one input head. In this paper, we focus on such generalizations of QFAs whose input head(s) operate(s) in one-way or realtime mode and present many superiority of them to their classical counterparts. Furthermore, we propose some open problems and conjectures in order to investigate the power of quantumness better. We also give some new results on classical computation.Comment: A revised edition with some correction

Quantum hedging in two-round prover-verifier interactions

We consider the problem of a particular kind of quantum correlation that arises in some two-party games. In these games, one player is presented with a question they must answer, yielding an outcome of either 'win' or 'lose'. Molina and Watrous (arXiv:1104.1140) studied such a game that exhibited a perfect form of hedging, where the risk of losing a first game can completely offset the corresponding risk for a second game. This is a non-classical quantum phenomenon, and establishes the impossibility of performing strong error-reduction for quantum interactive proof systems by parallel repetition, unlike for classical interactive proof systems. We take a step in this article towards a better understanding of the hedging phenomenon by giving a complete characterization of when perfect hedging is possible for a natural generalization of the game in arXiv:1104.1140. Exploring in a different direction the subject of quantum hedging, and motivated by implementation concerns regarding loss-tolerance, we also consider a variation of the protocol where the player who receives the question can choose to restart the game rather than return an answer. We show that in this setting there is no possible hedging for any game played with state spaces corresponding to finite-dimensional complex Euclidean spaces.Comment: 34 pages, 1 figure. Added work on connections with other result

Promocijas darbs

Elektroniskā versija nesatur pielikumusKvantu modelis ar pēcatlasi tiek definēts Scott Aaronson darbā. Beigu stāvokļu kopa, kas parasti sastāv no akceptējošiem un noraidošiem stāvokļiem, tiek papildināta ar parametru, kas norāda, vai dotais beigu stāvoklis ietilpst atlases kopā. Mērījumi tiek veikti tikai atlases kopas beigu stāvokļos. Tiek ieviests papildus stāvoklis + q , un ja visu pēcatlases stāvokļu amplitūdas ir 0, tad + q amplitūda saņem vērtību 1. Pēcatlase ļauj pētīt ne tikai kvantu, bet arī tradicionālo algoritumu īpašības. Pētījuma mērķis ir salīdzināt varbūtisko un kvantu galīgo pēcatlases automātu klases un aprakstīt valodu klases, ko atpazīst kvantu galīgs automāts ar pēcatlasi. Pētījuma procesā iegūti šādi rezultāti: • Definēts kvantu galīgā automāta ar pēcatlasi jēdziens; • Aprakstīta valoda PALINDROMES, ko atpazīst galīgs kvantu automāts ar pēcatlasi ar mērījumu katrā solī un galīgs kvantu automāts ar pēcatlasi ar mērījumu beigās; • Aprakstīta valoda, kuru nevar atpazīt galīgs kvantu automāts ar pēcatlasi ar mērījumu katrā solī un galīgs kvantu automāts ar pēcatlasi ar mērījumu beigās: L = {w | w∈{0,1}* and there exist x, y,u, z such that w = x1y = u1z and x = z } Viens no promocijas darba uzdevumiem ir aplūkot kvantu vaicājošos algoritmus Bula funkciju rēķināšanai. Darba sākumā tiek pierādīti kvantu algoritmu apakšējie novērtējumi dažādām funkcijām, kas apraksta grafu problēmas. Ir izveidoti efektīvi kvantu vaicājošie algoritmi. Šajā sadaļā iegūti rezultāti sekojošām funkcijām: • 3-sum problēma, • Hamiltona ceļš, • Hamiltona aplis, • Ceļojošais pārdevējs. Vēl promocijas darbā tiek apskatīta reāla laika Tjūringa mašīnas kvantu analoģija. Tiek parādīts, ka eksistē valoda, kuru pazīst reāla laika kvantu Tjūringa mašīna un nepazīst reāla laika determinēta Tjūringa mašīna.Postselection quantum model is defined by Scott Aaronson. A new parameter is added to a halting set of states, that consists of accepting and rejecting states, which defines if the state is in postselection set. Only states in postselection set are measured. New state + q is added and if all postselection states amplitudes are equal to 0, then + q amplitude is set to 1. Postelection appears to be very useful to study not only quantum, but also traditional algorithms . Paper goal is to compare probabilistic and quantum finite automata with postselection and define language class, that can be recognized by quantum finite automata with postselection. The following results are obtained: • The notion of quantum finite automata with postselection is given; • Language PALINDROMES is defined, that can be recognized by MO- and MM- quantum finite automata with postselection; • Language is defined, that cannot be recognized by MO- and MM- quantum finite automata with postselection: L = {w | w∈{0,1}* and there exist x, y,u, z such that w = x1y = u1z and x = z } One of the research object of this work is find quantum query algorithms to compute Boolean functions. At first we prove higher lower bounds of quantum query algorithms for some of graph problems. Effective quantum query algorithms are created with complexity lower than deterministic one. Results for the following functions are obtained: • 3-sum problem, • Hamiltonian path, • Hamiltonian circuit, • Travelling salesman. Another aim of this paper is to introduce a quantum counterpart for real – time Turing machine. The recognition of a special kind of language, that can’t be recognized by a deterministic real – time Turing machine, is shown

Kvantu skaitļošanas konstrukcijas

Kvantu modelis ar pēcatlasi tiek definēts Scott Aaronson darbā. Beigu stāvokĜu kopa, kas parasti sastāv no akceptējošiem un noraidošiem stāvokĜiem, tiek papildināta ar parametru, kas norāda, vai dotais beigu stāvoklis ietilpst atlases kopā. Mērījumi tiek veikti tikai atlases kopas beigu stāvokĜos. Tiek ieviests papildus stāvoklis + q , un ja visu pēcatlases stāvokĜu amplitūdas ir 0, tad + q amplitūda saĦem vērtību 1. Pēcatlase Ĝauj pētīt ne tikai kvantu, bet arī tradicionālo algoritumu īpašības. Pētījuma mērėis ir salīdzināt varbūtisko un kvantu galīgo pēcatlases automātu klases un aprakstīt valodu klases, ko atpazīst kvantu galīgs automāts ar pēcatlasi. Pētījuma procesā iegūti šādi rezultāti: • Definēts kvantu galīgā automāta ar pēcatlasi jēdziens; • Aprakstīta valoda PALINDROMES, ko atpazīst galīgs kvantu automāts ar pēcatlasi ar mērījumu katrā solī un galīgs kvantu automāts ar pēcatlasi ar mērījumu beigās; • Aprakstīta valoda, kuru nevar atpazīt galīgs kvantu automāts ar pēcatlasi ar mērījumu katrā solī un galīgs kvantu automāts ar pēcatlasi ar mērījumu beigās: L = {w | w∈{0,1}* and there exist x, y,u, z such that w = x1y = u1z and x = z } Viens no promocijas darba uzdevumiem ir aplūkot kvantu vaicājošos algoritmus Bula funkciju rēėināšanai. Darba sākumā tiek pierādīti kvantu algoritmu apakšējie novērtējumi dažādām funkcijām, kas apraksta grafu problēmas. Ir izveidoti efektīvi kvantu vaicājošie algoritmi. Šajā sadaĜā iegūti rezultāti sekojošām funkcijām: • 3-sum problēma, • Hamiltona ceĜš, • Hamiltona aplis, • CeĜojošais pārdevējs. Vēl promocijas darbā tiek apskatīta reāla laika Tjūringa mašīnas kvantu analoăija. Tiek parādīts, ka eksistē valoda, kuru pazīst reāla laika kvantu Tjūringa mašīna un nepazīst reāla laika determinēta Tjūringa mašīna.Postselection quantum model is defined by Scott Aaronson. A new parameter is added to a halting set of states, that consists of accepting and rejecting states, which defines if the state is in postselection set. Only states in postselection set are measured. New state + q is added and if all postselection states amplitudes are equal to 0, then + q amplitude is set to 1. Postelection appears to be very useful to study not only quantum, but also traditional algorithms . Paper goal is to compare probabilistic and quantum finite automata with postselection and define language class, that can be recognized by quantum finite automata with postselection. The following results are obtained: • The notion of quantum finite automata with postselection is given; • Language PALINDROMES is defined, that can be recognized by MO- and MM- quantum finite automata with postselection; • Language is defined, that cannot be recognized by MO- and MM- quantum finite automata with postselection: L = {w | w∈{0,1}* and there exist x, y,u, z such that w = x1y = u1z and x = z } One of the research object of this work is find quantum query algorithms to compute Boolean functions. At first we prove higher lower bounds of quantum query algorithms for some of graph problems. Effective quantum query algorithms are created with complexity lower than deterministic one. Results for the following functions are obtained: • 3-sum problem, • Hamiltonian path, • Hamiltonian circuit, • Travelling salesman. Another aim of this paper is to introduce a quantum counterpart for real – time Turing machine. The recognition of a special kind of language, that can’t be recognized by a deterministic real – time Turing machine, is shown