arXiv subjects

Jozef Gruska

Publications and source records attributed to Jozef Gruska.

15 recordsLinked to original sources

arXiv2023

Lifting query complexity to time-space complexity for two-way finite automata

Time-space tradeoff has been studied in a variety of models, such as Turing machines, branching programs, and finite automata, etc. While communication complexity as a technique has been applied to study finite automata, it seems it has not been used to study time-space tradeoffs of finite automata. We design a new technique showing that separations of query complexity can be lifted, via communication complexity, to separations of time-space complexity of two-way finite automata. As an application, one of our main results exhibits the first example of a language $L$ such that the time-space complexity of two-way probabilistic finite automata with a bounded error (2PFA) is $\widetildeΩ(n^2)$, while of exact two-way quantum finite automata with classical states (2QCFA) is $\widetilde{O}(n^{5/3})$, that is, we demonstrate for the first time that exact quantum computing has an advantage in time-space complexity comparing to classical computing.

cs.CC↗

arXiv2021

Testing Boolean Functions Properties

The goal in the area of functions property testing is to determine whether a given black-box Boolean function has a particular given property or is $\varepsilon$-far from having that property. We investigate here several types of properties testing for Boolean functions (identity, correlations and balancedness) using the Deutsch-Jozsa algorithm (for the Deutsch-Jozsa (D-J) problem) and also the amplitude amplification technique. At first, we study here a particular testing problem: namely whether a given Boolean function $f$, of $n$ variables, is identical with a given function $g$ or is $\varepsilon$-far from $g$, where $\varepsilon$ is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity $O(\frac{1}{\sqrt{\varepsilon}})$. Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is $Θ(\frac{1}{\varepsilon})$. Secondly, we consider the D-J problem from the perspective of functional correlations and let $C(f,g)$ denote the correlation of $f$ and $g$. We propose an exact quantum algorithm for making distinction between $|C(f,g)|=\varepsilon$ and $|C(f,g)|=1$ using six queries, while the classical deterministic query complexity for this problem is $Θ(2^{n})$ queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus $\varepsilon$-far balanced using $O(\frac{1}{\varepsilon})$ queries. We also prove here that our quantum algorithm for balancedness testing is optimal. At the same time, for this balancedness testing problem we present a classical randomized algorithm with query complexity of $O(1/\varepsilon^{2})$. Also this randomized algorithm is optimal. Besides, we link the problems considered here together and generalize them to the general case.

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arXiv2020

Security Improvements of Several Basic Quantum Private Query Protocols with O(log N) Communication Complexity

New quantum private database (with N elements) query protocols are presented and analyzed. Protocols preserve O(logN) communication complexity of known protocols for the same task, but achieve several significant improvements in security, especially concerning user privacy. For example, the randomized form of our protocol has a cheat-sensitive property - it allows the user to detect a dishonest database with a nonzero probability, while the phase-encoded private query protocols for the same task do not have such a property. Moreover, when the database performs the computational basis measurement, a particular projective measurement which can cause a significant loss of user privacy in the previous private query protocols with O(logN) communication complexity, at most half of the user privacy could leak to such a database in our protocol, while in the QPQ protocol, the entire user privacy could leak out. In addition, it is proved here that for large N, the user could detect a cheating via the computational basis measurement, with a probability close to 1/2 using O(\sqrt{N}) special queries. Finally, it is shown here, for both forms of our protocol, basic and randomized, how a dishonest database has to act in case it could not learn user's queries.

cs.CR↗

arXiv2019

Multiple Pulses Phase-matching Quantum Key Distribution

We propose a multiple pulses phase-matching quantum key distribution protocol (MPPM-QKD) to exceed the linear key rate bound and to achieve higher error tolerance. In our protocol, Alice and Bob generate at first their own train pulses (each train should contain L pulses) as well as random bit sequences, and also encode each pulse of their trains with a randomized phase and a modulation phase. As the next step, both encoded trains are simultaneously sent to Charlie, who performs an interference detection and may be also an eavesdropper. After a successful detection is announced by Charlie, Alice and Bob open the randomized phase of each pulse and keep only communications when the summation of the difference randomized phases at two success detection's time-stamps for Alice and Bob are equal to 0 or pi. Thereafter, Alice and Bob compute the sifted key with the time-stamps. The above procedure is repeated until both Alice and Bob achieve sufficiently long sifted keys. We can also show that the secret key rate of the proposed QKD protocol can beat the rate-loss limit of so far known QKD protocols when the transmission distance is greater than 250 km. Moreover, the proposed protocol has a higher error tolerance, approximately 24%, when the transmission distance is 50 km and L = 128. The secret key rate and the transmission distance of our protocol are superior to that of the round-robin differential-phase-shift quantum key distribution protocol [6], and also of the measurement-device-independent quantum key distribution protocol [4], and the secret key rate performance is better in both cases than that of phase-matching quantum key distribution when bit train length is greater than 32.

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arXiv2018

Synthesis of Quantum Images Using Phase Rotation

A topic about synthesis of quantum images is proposed, and a specific phase rotation transform constructed is adopted to theoretically realise the synthesis of two quantum images. The synthesis strategy of quantum images comprises three steps, which include: (1) In the stage of phase extraction, we obtain the phases of the state of the quantum image by transforming the state of the quantum image to prepare the conditions for multiple phases extraction. (2) In the stage of rotation operator construction, the phases obtained in the first stage are used to construct the rotation operator where a mechanism is introduced into it to reduce the phase overflow. (3) In the stage of application of the rotation operator, we apply the operator constructed in the second stage on the state of quantum image to get a goal state. Additionally, numerical analysis gives the joint uncertainty relation of the pixel of the synthesized quantum image. The analysis result about the compression ratio indicates that the phase rotation transform and the overflow control mechanism are effective.

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arXiv2016

Time-space tradeoffs for two-way finite automata

We explore bounds of {\em time-space tradeoffs} in language recognition on {\em two-way finite automata} for some special languages. We prove: (1) a time-space tradeoff upper bound for recognition of the languages $L_{EQ}(n)$ on {\em two-way probabilistic finite automata} (2PFA): $TS={\bf O}(n\log n)$, whereas a time-space tradeoff lower bound on {\em two-way deterministic finite automata} is ${\bf Ω}(n^2)$, (2) a time-space tradeoff upper bound for recognition of the languages $L_{INT}(n)$ on {\em two-way finite automata with quantum and classical states} (2QCFA): $TS={\bf O}(n^{3/2}\log n)$, whereas a lower bound on 2PFA is $TS={\bf Ω}(n^2)$, (3) a time-space tradeoff upper bound for recognition of the languages $L_{NE}(n)$ on exact 2QCFA: $TS={\bf O}(n^{1.87} \log n)$, whereas a lower bound on 2PFA is $TS={\bf Ω}(n^2)$. It has been proved (Klauck, STOC'00) that the exact one-way quantum finite automata have no advantage comparing to classical finite automata in recognizing languages. However, the result (3) shows that the exact 2QCFA do have an advantage in comparison with their classical counterparts, which has been the first example showing that the exact quantum computing have advantage in time-space tradeoff comparing to classical computing. Usually, two communicating parties, Alice and Bob, are supposed to have an access to arbitrary computational power in {\em communication complexity} model that is used. Instead of that we will consider communication complexity in such a setting that two parties are using only finite automata and we prove in this setting that quantum automata are better than classical automata and also probabilistic automata are better than deterministic automata for some well known tasks.

quant-ph↗

arXiv2015

Promise problems solved by quantum and classical finite automata

The concept of promise problems was introduced and started to be systematically explored by Even, Selman, Yacobi, Goldreich, and other scholars. It has been argued that promise problems should be seen as partial decision problems and as such that they are more fundamental than decision problems and formal languages that used to be considered as the basic ones for complexity theory. The main purpose of this paper is to explore the promise problems accepted by classical, quantum and also semi-quantum finite automata. More specifically, we first introduce two acceptance modes of promise problems, recognizability and solvability, and explore their basic properties. Afterwards, we show several results concerning descriptional complexity on promise problems. In particular, we prove: (1) there is a promise problem that can be recognized exactly by measure-once one-way quantum finite automata (MO-1QFA), but no deterministic finite automata (DFA) can recognize it; (2) there is a promise problem that can be solved with error probability $ε\leq 1/3$ by one-way finite automaton with quantum and classical states (1QCFA), but no one-way probability finite automaton (PFA) can solve it with error probability $ε\leq 1/3$; and especially, (3) there are promise problems $A(p)$ with prime $p$ that can be solved {\em with any error probability} by MO-1QFA with only two quantum basis states, but they can not be solved exactly by any MO-1QFA with two quantum basis states; in contrast, the minimal PFA solving $A(p)$ with any error probability (usually smaller than $1/2$) has $p$ states. Finally, we mention a number of problems related to promise for further study.

cs.FL↗

arXiv2015

Power of the interactive proof systems with verifiers modeled by semi-quantum two-way finite automata

In this paper we explore the power of AM for the case that verifiers are {\em two-way finite automata with quantum and classical states} (2QCFA)--introduced by Ambainis and Watrous in 2002--and the communications are classical. It is of interest to consider AM with such "semi-quantum" verifiers because they use only limited quantum resources. Our main result is that such Quantum Arthur-Merlin proof systems (QAM(2QCFA)) with polynomial expected running time are more powerful than in the case verifiers are two-way probabilistic finite automata (AM(2PFA)) with polynomial expected running time. Moreover, we prove that there is a language which can be recognized by an exponential expected running time QAM(2QCFA), but can not be recognized by any AM(2PFA), and that the NP-complete language $L_{knapsack}$ can also be recognized by a QAM(2QCFA) working only on quantum pure states using unitary operators.

cs.CC↗

arXiv2015

Generalizations of the distributed Deutsch-Jozsa promise problem

In the {\em distributed Deutsch-Jozsa promise problem}, two parties are to determine whether their respective strings $x,y\in\{0,1\}^n$ are at the {\em Hamming distance} $H(x,y)=0$ or $H(x,y)=\frac{n}{2}$. Buhrman et al. (STOC' 98) proved that the exact {\em quantum communication complexity} of this problem is ${\bf O}(\log {n})$ while the {\em deterministic communication complexity} is ${\bf Ω}(n)$. This was the first impressive (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch-Jozsa promise problem to determine, for any fixed $\frac{n}{2}\leq k\leq n$, whether $H(x,y)=0$ or $H(x,y)= k$, and show that an exponential gap between exact quantum and deterministic communication complexity still holds if $k$ is an even such that $\frac{1}{2}n\leq k<(1-λ) n$, where $0< λ<\frac{1}{2}$ is given. We also deal with a promise version of the well-known {\em disjointness} problem and show also that for this promise problem there exists an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the promise version of such a disjointness problem. Finally, some applications to quantum, probabilistic and deterministic finite automata of the results obtained are demonstrated.

quant-ph↗

arXiv2014

Potential of quantum finite automata with exact acceptance

The potential of the exact quantum information processing is an interesting, important and intriguing issue. For examples, it has been believed that quantum tools can provide significant, that is larger than polynomial, advantages in the case of exact quantum computation only, or mainly, for problems with very special structures. We will show that this is not the case. In this paper the potential of quantum finite automata producing outcomes not only with a (high) probability, but with certainty (so called exactly) is explored in the context of their uses for solving promise problems and with respect to the size of automata. It is shown that for solving particular classes $\{A^n\}_{n=1}^{\infty}$ of promise problems, even those without some very special structure, that succinctness of the exact quantum finite automata under consideration, with respect to the number of (basis) states, can be very small (and constant) though it grows proportional to $n$ in the case deterministic finite automata (DFAs) of the same power are used. This is here demonstrated also for the case that the component languages of the promise problems solvable by DFAs are non-regular. The method used can be applied in finding more exact quantum finite automata or quantum algorithms for other promise problems.

cs.FL↗

arXiv2014

Exact quantum algorithms have advantage for almost all Boolean functions

It has been proved that almost all $n$-bit Boolean functions have exact classical query complexity $n$. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries. More exactly, we prove that ${AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that requires $n$ queries.

cs.CC↗

arXiv2014

On the state complexity of semi-quantum finite automata

Some of the most interesting and important results concerning quantum finite automata are those showing that they can recognize certain languages with (much) less resources than corresponding classical finite automata \cite{Amb98,Amb09,AmYa11,Ber05,Fre09,Mer00,Mer01,Mer02,Yak10,ZhgQiu112,Zhg12}. This paper shows three results of such a type that are stronger in some sense than other ones because (a) they deal with models of quantum automata with very little quantumness (so-called semi-quantum one- and two-way automata with one qubit memory only); (b) differences, even comparing with probabilistic classical automata, are bigger than expected; (c) a trade-off between the number of classical and quantum basis states needed is demonstrated in one case and (d) languages (or the promise problem) used to show main results are very simple and often explored ones in automata theory or in communication complexity, with seemingly little structure that could be utilized.

cs.FL↗

arXiv2013

Communication complexity of promise problems and their applications to finite automata

Equality and disjointness are two of the most studied problems in communication complexity. They have been studied for both classical and also quantum communication and for various models and modes of communication. Buhrman et al. [Buh98] proved that the exact quantum communication complexity for a promise version of the equality problem is ${\bf O}(\log {n})$ while the classical deterministic communication complexity is $n+1$ for two-way communication, which was the first impressively large (exponential) gap between quantum and classical (deterministic and probabilistic) communication complexity. If an error is tolerated, both quantum and probabilistic communication complexities for equality are ${\bf O}(\log {n})$. However, even if an error is tolerated, the gaps between quantum (probabilistic) and deterministic complexity are not larger than quadratic for the disjointness problem. It is therefore interesting to ask whether there are some promise versions of the disjointness problem for which bigger gaps can be shown. We give a positive answer to such a question. Namely, we prove that there exists an exponential gap between quantum (even probabilistic) communication complexity and classical deterministic communication complexity of some specific versions of the disjointness problem. Klauck [Kla00] proved, for any language, that the state complexity of exact quantum/classical finite automata, which is a general model of one-way quantum finite automata, is not less than the state complexity of an equivalent one-way deterministic finite automata (1DFA). In this paper we show, using a communication complexity result, that situation may be different for some promise problems. Namely, we show for certain promise problem that the gap between the state complexity of exact one-way quantum finite automata and 1DFA can be exponential.

cs.CC↗

arXiv2012

State succinctness of two-way finite automata with quantum and classical states

{\it Two-way quantum automata with quantum and classical states} (2QCFA) were introduced by Ambainis and Watrous in 2002. In this paper we study state succinctness of 2QCFA. For any $m\in {\mathbb{Z}}^+$ and any $ε<1/2$, we show that: {enumerate} there is a promise problem $A^{eq}(m)$ which can be solved by a 2QCFA with one-sided error $ε$ in a polynomial expected running time with a constant number (that depends neither on $m$ nor on $\varepsilon$) of quantum states and $\mathbf{O}(\log{\frac{1}ε)}$ classical states, whereas the sizes of the corresponding {\it deterministic finite automata} (DFA), {\it two-way nondeterministic finite automata} (2NFA) and polynomial expected running time {\it two-way probabilistic finite automata} (2PFA) are at least $2m+2$, $\sqrt{\log{m}}$, and $\sqrt[3]{(\log m)/b}$, respectively; there exists a language $L^{twin}(m)=\{wcw| w\in\{a,b\}^*\}$ over the alphabet $Σ=\{a,b,c\}$ which can be recognized by a 2QCFA with one-sided error $ε$ in an exponential expected running time with a constant number of quantum states and $\mathbf{O}(\log{\frac{1}ε)}$ classical states, whereas the sizes of the corresponding DFA, 2NFA and polynomial expected running time 2PFA are at least $2^m$, $\sqrt{m}$, and $\sqrt[3]{m/b}$, respectively; {enumerate} where $b$ is a constant.

quant-ph↗

arXiv2011

One-way finite automata with quantum and classical states

In this paper, we introduce and explore a new model of {\it quantum finite automata} (QFA). Namely, {\it one-way finite automata with quantum and classical states} (1QCFA), a one way version of {\it two-way finite automata with quantum and classical states} (2QCFA) introduced by Ambainis and Watrous in 2002 \cite{AJ}. First, we prove that {\it one-way probabilistic finite automata} (1PFA) \cite{AP} and {\it one-way quantum finite automata with control language} (1QFACL) \cite{ACB} as well as several other models of QFA, can be simulated by 1QCFA. Afterwards, we explore several closure properties for the family of languages accepted by 1QCFA. Finally, the state complexity of 1QCFA is explored and the main succinctness result is derived. Namely, for any prime $m$ and any $ε_1>0$, there exists a language $L_{m}$ that cannot be recognized by any {\it measure-many one-way quantum finite automata} (MM-1QFA) \cite{Kon97} with bounded error $7/9+ε_1$, and any 1PFA recognizing it has at last $m$ states, but $L_{m}$ can be recognized by a 1QCFA for any error bound $ε>0$ with $\bf{O}(\log{m})$ quantum states and 12 classical states.

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