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Abuzer Yakaryılmaz

Publications and source records attributed to Abuzer Yakaryılmaz.

At least 19 recordsLinked to original sources
arXiv2025

QIP $ \subseteq $ AM(2QCFA)

The class of languages having polynomial-time classical or quantum interactive proof systems ($\mathsf{IP}$ or $\mathsf{QIP}$, respectively) is identical to $\mathsf{PSPACE}$. We show that $\mathsf{PSPACE}$ (and so $\mathsf{QIP}$) is subset of $\mathsf{AM(2QCFA)}$, the class of languages having Arthur-Merlin proof systems where the verifiers are two-way finite automata with quantum and classical states (2QCFAs) communicating with the provers classically. Our protocols use only rational-valued quantum transitions and run in double-exponential expected time. Moreover, the member strings are accepted with probability 1 (i.e., perfect-completeness).

quant-ph↗
arXiv2025

Two-way affine automata can verify every language

When used as verifiers in Arthur-Merlin systems, two-way quantum finite automata can verify membership in all languages with bounded error with double-exponential expected running time, which cannot be achieved by their classical counterparts. We obtain the same result for affine automata with single-exponential expected time. We show that every binary (and r-ary) language is verified by some two-way affine finite automata verifiers by presenting two protocols: A weak verification protocol uses a single affine register and the input is read once; and, a strong verification protocol uses two affine registers. These results reflects the remarkable verification capabilities of affine finite automata.

cs.FL↗
arXiv2024

Classical and quantum Merlin-Arthur automata

We introduce Merlin-Arthur (MA) automata where Merlin provides a certificate at the beginning of computation and it is scanned by Arthur before reading the input. We define Merlin-Arthur deterministic, probabilistic, and quantum finite state automata (resp., MA-DFAs, MA-PFAs, and MA-QFAs) and postselecting MA-PFAs and MA-QFAs (resp., MA-PostPFA and MA-PostQFA). We present several results using different certificate lengths. We show that MA-DFAs use constant length certificates, and they are equivalent to multi-entry DFAs. Thus, they recognize all and only regular languages, but they can be exponential and polynomial state efficient over binary and unary languages, respectively. With sublinear length certificates, MA-PFAs can recognize several nonstochastic unary languages with cutpoint 1/2. With linear length certificates, MA-PostPFAs can recognize these nonstochastic unary languages with bounded error. With arbitrarily long certificates, bounded-error MA-PostPFAs can verify every unary decidable language. With sublinear length certificates, bounded-error MA-PostQFAs can verify several nonstochastic unary languages. With linear length certificates, they can verify every unary language and some NP-complete binary languages. With exponential length certificates, they can verify every binary language.

cs.FL↗
arXiv2024

A Representative Framework for Implementing Quantum Finite Automata on Real Devices

We present a framework for the implementation of quantum finite automata algorithms designed for the language $ MOD_p = \{ a^{i\cdot p } \mid i \geq 0 \}$ on gate-based quantum computers. First, we compile the known theoretical results from the literature to reduce the number of CNOT gates. Second, we demonstrate techniques for modifying the algorithms based on the basis gates of available quantum hardware in order to reduce circuit depth. Lastly, we explore how the number of CNOT gates may be reduced further if the topology of the qubits is known.

quant-ph↗
arXiv2023

GAPs for Shallow Implementation of Quantum Finite Automata

Quantum fingerprinting is a technique that maps classical input word to a quantum state. The obtained quantum state is much shorter than the original word, and its processing uses less resources, making it useful in quantum algorithms, communication, and cryptography. One of the examples of quantum fingerprinting is quantum automata algorithms for MOD_p languages, where p is a prime number. However, implementing such an automaton on the current quantum hardware is not efficient. Quantum fingerprinting maps a word of length N to a state of O(log N) qubits, and uses O(N) unitary operations. Computing quantum fingerprint using all available qubits of the current quantum computers is infeasible due to a large number of quantum operations. To make quantum fingerprinting practical, we should optimize the circuit for depth instead of width in contrast to the previous works. We propose explicit methods of quantum fingerprinting based on tools from additive combinatorics, such as generalized arithmetic progressions (GAPs), and prove that these methods provide circuit depth comparable to a probabilistic method. We also compare our method to prior work on explicit quantum fingerprinting methods.

quant-ph↗
arXiv2022

Cost-efficient QFA Algorithm for Quantum Computers

The study of quantum finite automata (QFAs) is one of the possible approaches in exploring quantum computers with finite memory. Despite being one of the most restricted models, Moore-Crutchfield quantum finite automaton (MCQFA) is proven to be exponentially more succinct than classical finite automata models in recognizing certain languages such as $\mathtt{MOD}_p = \{ a^{j} \mid j \equiv 0 \mod p\}$, where $p$ is a prime number. In this paper, we present a modified MCQFA algorithm for the language $\mathtt{MOD}_p$, the operators of which are selected based on the basis gates on the available real quantum computers. As a consequence, we obtain shorter quantum programs using fewer basis gates compared to the implementation of the original algorithm given in the literature.

quant-ph↗
arXiv2022

Quantum advantage using high-dimensional twisted photons as quantum finite automata

Quantum finite automata (QFA) are basic computational devices that make binary decisions using quantum operations. They are known to be exponentially memory efficient compared to their classical counterparts. Here, we demonstrate an experimental implementation of multi-qubit QFAs using the orbital angular momentum (OAM) of single photons. We implement different high-dimensional QFAs encoded on a single photon, where multiple qubits operate in parallel without the need for complicated multi-partite operations. Using two to eight OAM quantum states to implement up to four parallel qubits, we show that a high-dimensional QFA is able to detect the prime numbers 5 and 11 while outperforming classical finite automata in terms of the required memory. Our work benefits from the ease of encoding, manipulating, and deciphering multi-qubit states encoded in the OAM degree of freedom of single photons, demonstrating the advantages structured photons provide for complex quantum information tasks.

quant-ph↗
arXiv2021

Improved constructions for succinct affine automata

Affine finite automata (AfA) can be more succinct than probabilistic and quantum finite automata when recognizing some regular languages with bounded-error. In this paper, we improve previously known constructions given for the succinctness of AfAs in three ways. First, we replace some of fixed error bounds with arbitrarily small error bounds. Second, we present new constructions by using less states than the previous constructions. Third, we show that any language recognized by a nondeterministic finite automaton (NFA) is also recognized by bounded-error AfAs having one more state, and so, AfAs inherit all succinct results by NFAs. As a special case, we also show that any language recognized by a NFA is recognized by AfAs with zero error if the number of accepting path(s) for each member is exactly the same number.

cs.FL↗
arXiv2021

Implementing Quantum Finite Automata Algorithms on Noisy Devices

Quantum finite automata (QFAs) literature offers an alternative mathematical model for studying quantum systems with finite memory. As a superiority of quantum computing, QFAs have been shown exponentially more succinct on certain problems such as recognizing the language $ MOD_p = \{a^j \mid j \equiv 0 \mod p\} $ with bounded error, where $p$ is a prime number. In this paper we present improved circuit based implementations for QFA algorithms recognizing the $ MOD_p $ problem using the Qiskit framework. We focus on the case $p=11$ and provide a 3 qubit implementation for the $MOD_{11}$ problem reducing the total number of required gates using alternative approaches. We run the circuits on real IBM quantum devices but due to the limitation of the real quantum devices in the NISQ era, the results are heavily affected by the noise. This limitation reveals once again the need for algorithms using less amount of resources. Consequently, we consider an alternative 3 qubit implementation which works better in practice and obtain promising results even for the problem $ MOD_{31} $.

quant-ph↗
arXiv2021

Quantum Logarithmic Space and Post-selection

Post-selection, the power of discarding all runs of a computation in which an undesirable event occurs, is an influential concept introduced to the field of quantum complexity theory by Aaronson (Proceedings of the Royal Society A, 2005). In the present paper, we initiate the study of post-selection for space-bounded quantum complexity classes. Our main result shows the identity $\sf PostBQL=PL$, i.e., the class of problems that can be solved by a bounded-error (polynomial-time) logarithmic-space quantum algorithm with post-selection ($\sf PostBQL$) is equal to the class of problems that can be solved by unbounded-error logarithmic-space classical algorithms ($\sf PL$). This result gives a space-bounded version of the well-known result $\sf PostBQP=PP$ proved by Aaronson for polynomial-time quantum computation. As a by-product, we also show that $\sf PL$ coincides with the class of problems that can be solved by bounded-error logarithmic-space quantum algorithms that have no time bound.

cs.CC↗
arXiv2021

Affine automata verifiers

We initiate the study of the verification power of AfAs as part of Arthur-Merlin (AM) proof systems. We show that every unary language is verified by a real-valued AfA verifier. Then, we focus on the verifiers restricted to have only integer-valued or rational-valued transitions. We observe that rational-valued verifiers can be simulated by integer-valued verifiers, and, their protocols can be simulated in nondeterministic polynomial time. We show that this bound tight by presenting an AfA verifier for NP-complete problem SUBSETSUM. We also show that AfAs can verify certain non-affine and non-stochastic unary languages.

cs.FL↗
arXiv2020

The minimal probabilistic and quantum finite automata recognizing uncountably many languages with fixed cutpoints

It is known that 2-state binary and 3-state unary probabilistic finite automata and 2-state unary quantum finite automata recognize uncountably many languages with cutpoints. These results have been obtained by associating each recognized language with a cutpoint and then by using the fact that there are uncountably many cutpoints. In this note, we prove the same results for fixed cutpoints: each recognized language is associated with an automaton (i.e., algorithm), and the proofs use the fact that there are uncountably many automata. For each case, we present a new construction.

cs.FL↗
arXiv2019

New results on classical and quantum counter automata

We show that one-way quantum one-counter automaton with zero-error is more powerful than its probabilistic counterpart on promise problems. Then, we obtain a similar separation result between Las Vegas one-way probabilistic one-counter automaton and one-way deterministic one-counter automaton. We also obtain new results on classical counter automata regarding language recognition. It was conjectured that one-way probabilistic one blind-counter automata cannot recognize Kleene closure of equality language [A. Yakaryilmaz: Superiority of one-way and realtime quantum machines. RAIRO - Theor. Inf. and Applic. 46(4): 615-641 (2012)]. We show that this conjecture is false, and also show several separation results for blind/non-blind counter automata.

cs.FL↗
arXiv2019

Alternating, private alternating, and quantum alternating realtime automata

We present new results on realtime alternating, private alternating, and quantum alternating automaton models. Firstly, we show that the emptiness problem for alternating one-counter automata on unary alphabets is undecidable. Then, we present two equivalent definitions of realtime private alternating finite automata (PAFAs). We show that the emptiness problem is undecidable for PAFAs. Furthermore, PAFAs can recognize some nonregular unary languages, including the unary squares language, which seems to be difficult even for some classical counter automata with two-way input. Regarding quantum finite automata (QFAs), we show that the emptiness problem is undecidable both for universal QFAs on general alphabets, and for alternating QFAs with two alternations on unary alphabets. On the other hand, the same problem is decidable for nondeterministic QFAs on general alphabets. We also show that the unary squares language is recognized by alternating QFAs with two alternations.

cs.FL↗
arXiv2019

New Results on Vector and Homing Vector Automata

We present several new results and connections between various extensions of finite automata through the study of vector automata and homing vector automata. We show that homing vector automata outperform extended finite automata when both are defined over $ 2 \times 2 $ integer matrices. We study the string separation problem for vector automata and demonstrate that generalized finite automata with rational entries can separate any pair of strings using only two states. Investigating stateless homing vector automata, we prove that a language is recognized by stateless blind deterministic real-time version of finite automata with multiplication iff it is commutative and its Parikh image is the set of nonnegative integer solutions to a system of linear homogeneous Diophantine equations.

cs.FL↗
arXiv2019

Computational Limitations of Affine Automata

We present two new results on the computational limitations of affine automata. First, we show that the computation of bounded-error rational-values affine automata is simulated in logarithmic space. Second, we give an impossibility result for algebraic-valued affine automata. As a result, we identify some unary languages (in logarithmic space) that are not recognized by algebraic-valued affine automata with cutpoints.

cs.FL↗
arXiv2018

Postselecting probabilistic finite state recognizers and verifiers

In this paper, we investigate the computational and verification power of bounded-error postselecting realtime probabilistic finite state automata (PostPFAs). We show that PostPFAs using rational-valued transitions can do different variants of equality checks and they can verify some nonregular unary languages. Then, we allow them to use real-valued transitions (magic-coins) and show that they can recognize uncountably many binary languages by help of a counter and verify uncountably many unary languages by help of a prover. We also present some corollaries on probabilistic counter automata.

cs.CC↗