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 ${\bf \Omega}(n)$. 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 $\frac{1}{2}n\leq k<(1-\lambda) n$, where $0< \lambda<\frac{1}{2}$ 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.Comment: we correct some errors of and improve the presentation the previous version. arXiv admin note: substantial text overlap with arXiv:1309.773

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 $\epsilon<1/2$, we show that: {enumerate} there is a promise problem $A^{eq}(m)$ which can be solved by a 2QCFA with one-sided error $\epsilon$ in a polynomial expected running time with a constant number (that depends neither on $m$ nor on $\varepsilon$) of quantum states and $\mathbf{O}(\log{\frac{1}{\epsilon})}$ 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 $\Sigma=\{a,b,c\}$ which can be recognized by a 2QCFA with one-sided error $\epsilon$ in an exponential expected running time with a constant number of quantum states and $\mathbf{O}(\log{\frac{1}{\epsilon})}$ 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.Comment: 26pages, comments and suggestions are welcom

Learning Quantum Finite Automata with Queries

{\it Learning finite automata} (termed as {\it model learning}) has become an important field in machine learning and has been useful realistic applications. Quantum finite automata (QFA) are simple models of quantum computers with finite memory. Due to their simplicity, QFA have well physical realizability, but one-way QFA still have essential advantages over classical finite automata with regard to state complexity (two-way QFA are more powerful than classical finite automata in computation ability as well). As a different problem in {\it quantum learning theory} and {\it quantum machine learning}, in this paper, our purpose is to initiate the study of {\it learning QFA with queries} (naturally it may be termed as {\it quantum model learning}), and the main results are regarding learning two basic one-way QFA: (1) We propose a learning algorithm for measure-once one-way QFA (MO-1QFA) with query complexity of polynomial time; (2) We propose a learning algorithm for measure-many one-way QFA (MM-1QFA) with query complexity of polynomial-time, as well.Comment: 18pages; comments are welcom