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arXiv:1309.7739

Communication complexity of promise problems and their applications to finite automata

Jozef Gruska, Daowen Qiu, Shenggen Zheng

Published Sep 30, 2013• 3 citations
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Abstract

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.

DOI
10.48550/arxiv.1309.7739

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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 O(log⁡n){\bf O}(\log {n})O(logn) while the classical deterministic communication complexity is n+1n+1n+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 O(log⁡n){\bf O}(\log {n})O(logn). 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.
In thedistributed Deutsch–Jozsa promise problem, two parties are to determine whether their respective stringsx, y∈ {0,1}nare at theHamming distanceH(x, y) = 0 orH(x, y) = n2\frac{n}{2}2n​ . Buhrmanet al.(STOC' 98) proved that the exactquantum communication complexityof this problem isO(logn) while thedeterministic communication complexityisΩ(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 n2\frac{n}{2}2n​ ⩽k⩽n, whetherH(x, y) = 0 orH(x, y) =k, and show that an exponential gap between exact quantum and deterministic communication complexity still holds ifkis an even such that 12\frac{1}{2}21​ n⩽k< (1 − λ)n, where 0 < λ < 12\frac{1}{2}21​ is given. We also deal with a promise version of the well-knowndisjointnessproblem 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.
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. 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 finite automata with very little quantumness (so-called semi-quantum one- and two-way finite automata); (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.
We investigate the power of quantum communication protocols compared to classical probabilistic protocols.In our first result we describe a total Boolean function that has a quantum Las Vegas protocol communicating at most O(N 1°/11+~) qubits for all e > 0, while any classical probabilistic protocol (with bounded error) needs ~(N/log N) bits.Then we investigate quantum one-way communication complexity.First we show that the VC-dimension lower bound on one-way probabilistic communication of [26] holds for quantum protocols, too.Then we prove that for oneway protocols computing total functions quantum Las Vegas communication is asymptotically as efficient as exact quantum communication, which is exactly as efficient as deterministic communication.We describe applications of the lower bounds for one-way communication complexity to quantum finite automata and quantum formulae.
Communication complexity has become a central completity model.In that model, we count the amount of communication bits needed between two parties in order to solve certain computational problems.We show that for certain communication complezity problems quantum communication protocols are exponentially faster than classical ores.More explicitly, we give an example for a communication complexity relation (0~ promise problem) P such that: 1) The quantum communication complexity of P is O(log m). 2) The classical probabilistic communication complexity of P is Q(m'l'/ logm).(where m is the length of the inputs).This gives an ezponential gap between quantum communication complexity and classical probabilistic communication complexity.Only a quadratic gap was previously known.Our problem P is of geometrical nature, and is a finite precision variation of the following problem: Player I gets as input a unit vector z E R" and two orthogonal subspaces MO, MI c R". Player II gets as input an orthogonal matrixT : R" + R".Their goal is to answer 0 if T(x) E MO and 1 if T(x) E M,, (and any an~luer in any other case).We give an almost tight analysis for the quantum communication complexity and for the classical-probabilistic communication complexity of this problem.
{\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∈Z+m\in {\mathbb{Z}}^+m∈Z+ and any ε<1/2ε<1/2ε<1/2, we show that: {enumerate} there is a promise problem Aeq(m)A^{eq}(m)Aeq(m) which can be solved by a 2QCFA with one-sided error εεε in a polynomial expected running time with a constant number (that depends neither on mmm nor on ε\varepsilonε) of quantum states and O(log⁡1ε)\mathbf{O}(\log{\frac{1}ε)}O(logε1​) 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+22m+22m+2, log⁡m\sqrt{\log{m}}logm​, and (log⁡m)/b3\sqrt[3]{(\log m)/b}3(logm)/b​, respectively; there exists a language Ltwin(m)={wcw∣w∈{a,b}∗}L^{twin}(m)=\{wcw| w\in\{a,b\}^*\}Ltwin(m)={wcw∣w∈{a,b}∗} over the alphabet Σ={a,b,c}Σ=\{a,b,c\}Σ={a,b,c} which can be recognized by a 2QCFA with one-sided error εεε in an exponential expected running time with a constant number of quantum states and O(log⁡1ε)\mathbf{O}(\log{\frac{1}ε)}O(logε1​) classical states, whereas the sizes of the corresponding DFA, 2NFA and polynomial expected running time 2PFA are at least 2m2^m2m, m\sqrt{m}m​, and m/b3\sqrt[3]{m/b}3m/b​, respectively; {enumerate} where bbb is a constant.
One-way quantum finite automata together with classical states (1QFAC) proposed in [Journal of Computer and System Sciences 81(2) (2015) 359--375] is a new one-way quantum finite automata (1QFA) model that integrates quantum finite automata (QFA) and deterministic finite automata (DFA). This model uses classical states to control the evolution and measurement of quantum states. As a quantum-classical hybrid model, 1QFAC recognize all regular languages. It was shown that the state complexity of 1QFAC for some languages is essentially superior to that of DFA and other 1QFA. In this paper, our goal is to clarify state complexity problems for 1QFAC. We obtain the following results: (1) We optimize the bound given by Qiu et al. that characterizes the relationship between quantum basis state number and classical state number of 1QFAC as well as the state number of its corresponding minimal DFA for recognizing any given regular language. (2) We give an upper bound showing that how many classical states are needed if the quantum basis states of 1QFAC are reduced without changing its recognition ability. (3) We give a lower bound of the classical state number of 1QFAC for recognizing any given regular language, and the lower bound is exact if the given language is finite. (4) We show that 1QFAC are exponentially more succinct than DFA and probabilistic finite automata (PFA) for recognizing some regular languages that can not be recognized by measure-once 1QFA (MO-1QFA), measure-many 1QFA (MM-1QFA) or multi-letter 1QFA. (5) We reveal essential relationships between 1QFAC, MO-1QFA and multi-letter 1QFA, and induce a result regarding a quantitative relationship between the state number of multi-letter 1QFA and DFA.