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Abuzer Yakaryilmaz

Publications and source records attributed to Abuzer Yakaryilmaz.

33 records · Page 2Linked to original sources
arXiv2012

NP has log-space verifiers with fixed-size public quantum registers

In classical Arthur-Merlin games, the class of languages whose membership proofs can be verified by Arthur using logarithmic space (AM(log-space)) coincides with the class P \cite{Co89}. In this note, we show that if Arthur has a fixed-size quantum register (the size of the register does not depend on the length of the input) instead of another source of random bits, membership in any language in NP can be verified with any desired error bound.

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arXiv2011

Proving the power of postselection

It is a widely believed, though unproven, conjecture that the capability of postselection increases the language recognition power of both probabilistic and quantum polynomial-time computers. It is also unknown whether polynomial-time quantum machines with postselection are more powerful than their probabilistic counterparts with the same resource restrictions. We approach these problems by imposing additional constraints on the resources to be used by the computer, and are able to prove for the first time that postselection does augment the computational power of both classical and quantum computers, and that quantum does outperform probabilistic in this context, under simultaneous time and space bounds in a certain range. We also look at postselected versions of space-bounded classes, as well as those corresponding to error-free and one-sided error recognition, and provide classical characterizations. It is shown that $\mathsf{NL}$ would equal $\mathsf{RL}$ if the randomized machines had the postselection capability.

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arXiv2011

Quantum counter automata

The question of whether quantum real-time one-counter automata (rtQ1CAs) can outperform their probabilistic counterparts has been open for more than a decade. We provide an affirmative answer to this question, by demonstrating a non-context-free language that can be recognized with perfect soundness by a rtQ1CA. This is the first demonstration of the superiority of a quantum model to the corresponding classical one in the real-time case with an error bound less than 1. We also introduce a generalization of the rtQ1CA, the quantum one-way one-counter automaton (1Q1CA), and show that they too are superior to the corresponding family of probabilistic machines. For this purpose, we provide general definitions of these models that reflect the modern approach to the definition of quantum finite automata, and point out some problems with previous results. We identify several remaining open problems.

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arXiv2011

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.

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arXiv2011

Superiority of exact quantum automata for promise problems

In this note, we present an infinite family of promise problems which can be solved exactly by just tuning transition amplitudes of a two-state quantum finite automata operating in realtime mode, whereas the size of the corresponding classical automata grow without bound.

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arXiv2011

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.

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arXiv2011

Unbounded-error quantum computation with small space bounds

We prove the following facts about the language recognition power of quantum Turing machines (QTMs) in the unbounded error setting: QTMs are strictly more powerful than probabilistic Turing machines for any common space bound $ s $ satisfying $ s(n)=o(\log \log n) $. For "one-way" Turing machines, where the input tape head is not allowed to move left, the above result holds for $s(n)=o(\log n) $. We also give a characterization for the class of languages recognized with unbounded error by real-time quantum finite automata (QFAs) with restricted measurements. It turns out that these automata are equal in power to their probabilistic counterparts, and this fact does not change when the QFA model is augmented to allow general measurements and mixed states. Unlike the case with classical finite automata, when the QFA tape head is allowed to remain stationary in some steps, more languages become recognizable. We define and use a QTM model that generalizes the other variants introduced earlier in the study of quantum space complexity.

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arXiv2011

Probabilistic and quantum finite automata with postselection

We prove that endowing a real-time probabilistic or quantum computer with the ability of postselection increases its computational power. For this purpose, we provide a new model of finite automata with postselection, and compare it with the model of Lāce et al. We examine the related language classes, and also establish separations between the classical and quantum versions, and between the zero-error vs. bounded-error modes of recognition in this model.

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arXiv2011

Classical and quantum computation with small space bounds (PhD thesis)

In this thesis, we introduce a new quantum Turing machine (QTM) model that supports general quantum operators, together with its pushdown, counter, and finite automaton variants, and examine the computational power of classical and quantum machines using small space bounds in many different cases. The main contributions are summarized below. Firstly, we consider QTMs in the unbounded error setting: (i) in some cases of sublogarithmic space bounds, the class of languages recognized by QTMs is shown to be strictly larger than that of classical ones; (ii) in constant space bounds, the same result can still be obtained for restricted QTMs; (iii) the complete characterization of the class of languages recognized by realtime constant space nondeterministic QTMs is given. Secondly, we consider constant space-bounded QTMs in the bounded error setting: (i) we introduce a new type of quantum and probabilistic finite automata (QFAs and PFAs, respectively,) with a special two-way input head which is not allowed to be stationary or move to the left but has the capability to reset itself to its starting position; (ii) the computational power of this type of quantum machine is shown to be superior to that of the probabilistic machine; (iii) based on these models, two-way PFAs and two-way classical-head QFAs are shown to be more succinct than two-way nondeterministic finite automata and their one-way variants; (iv) we also introduce PFAs and QFAs with postselection with their bounded error language classes, and give many characterizations of them. Thirdly, the computational power of realtime QFAs augmented with a write-only memory is investigated by showing many simulation results for different kinds of counter automata. Finally, some lower bounds of realtime classical Turing machines in order to recognize a nonregular language are shown to be tight.

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arXiv2010

Quantum computation with devices whose contents are never read

In classical computation, a "write-only memory" (WOM) is little more than an oxymoron, and the addition of WOM to a (deterministic or probabilistic) classical computer brings no advantage. We prove that quantum computers that are augmented with WOM can solve problems that neither a classical computer with WOM nor a quantum computer without WOM can solve, when all other resource bounds are equal. We focus on realtime quantum finite automata, and examine the increase in their power effected by the addition of WOMs with different access modes and capacities. Some problems that are unsolvable by two-way probabilistic Turing machines using sublogarithmic amounts of read/write memory are shown to be solvable by these enhanced automata.

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arXiv2010

Language recognition by generalized quantum finite automata with unbounded error (abstract & poster)

In this note, we generalize the results of arXiv:0901.2703v1 We show that all one-way quantum finite automaton (QFA) models that are at least as general as Kondacs-Watrous QFA's are equivalent in power to classical probabilistic finite automata in this setting. Unlike their probabilistic counterparts, allowing the tape head to stay put for some steps during its traversal of the input does enlarge the class of languages recognized by such QFA's with unbounded error. (Note that, the proof of Theorem 1 in the abstract was presented in the previous version (arXiv:0901.2703v1).)

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arXiv2010

Languages recognized by nondeterministic quantum finite automata

The nondeterministic quantum finite automaton (NQFA) is the only known case where a one-way quantum finite automaton (QFA) model has been shown to be strictly superior in terms of language recognition power to its probabilistic counterpart. We give a characterization of the class of languages recognized by NQFA's, demonstrating that it is equal to the class of exclusive stochastic languages. We also characterize the class of languages that are recognized necessarily by two-sided error by QFA's. It is shown that these classes remain the same when the QFA's used in their definitions are replaced by several different model variants that have appeared in the literature. We prove several closure properties of the related classes. The ramifications of these results about classical and quantum sublogarithmic space complexity classes are examined.

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arXiv2009

Succinctness of two-way probabilistic and quantum finite automata

We prove that two-way probabilistic and quantum finite automata (2PFA's and 2QFA's) can be considerably more concise than both their one-way versions (1PFA's and 1QFA's), and two-way nondeterministic finite automata (2NFA's). For this purpose, we demonstrate several infinite families of regular languages which can be recognized with some fixed probability greater than $ {1/2} $ by just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with a constant number of states, whereas the sizes of the corresponding 1PFA's, 1QFA's and 2NFA's grow without bound. We also show that 2QFA's with mixed states can support highly efficient probability amplification. The weakest known model of computation where quantum computers recognize more languages with bounded error than their classical counterparts is introduced.

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