Superlinear advantage for exact quantum algorithms

A quantum algorithm is exact if, on any input data, it outputs the correct answer with certainty (probability 1). A key question is: how big is the advantage of exact quantum algorithms over their classical counterparts: deterministic algorithms. For total Boolean functions in the query model, the biggest known gap was just a factor of 2: PARITY of N inputs bits requires $N$ queries classically but can be computed with N/2 queries by an exact quantum algorithm. We present the first example of a Boolean function f(x_1, ..., x_N) for which exact quantum algorithms have superlinear advantage over the deterministic algorithms. Any deterministic algorithm that computes our function must use N queries but an exact quantum algorithm can compute it with O(N^{0.8675...}) queries.Comment: 20 pages, v6: small number of small correction

From Quantum Query Complexity to State Complexity

State complexity of quantum finite automata is one of the interesting topics in studying the power of quantum finite automata. It is therefore of importance to develop general methods how to show state succinctness results for quantum finite automata. One such method is presented and demonstrated in this paper. In particular, we show that state succinctness results can be derived out of query complexity results.Comment: Some typos in references were fixed. To appear in Gruska Festschrift (2014). Comments are welcome. arXiv admin note: substantial text overlap with arXiv:1402.7254, arXiv:1309.773

Exact Quantum Query Complexity of EXACT and THRESHOLD

A quantum algorithm is exact if it always produces the correct answer, on any input. Coming up with exact quantum algorithms that substantially outperform the best classical algorithm has been a quite challenging task. In this paper, we present two new exact quantum algorithms for natural problems: - for the problem EXACT_k^n in which we have to determine whether the sequence of input bits x_1, ..., x_n contains exactly k values x_i=1; - for the problem THRESHOLD_k^n in which we have to determine if at least k of n input bits are equal to 1

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.Comment: We have improved the presentation of the paper. Accepted to International Journal of Foundation of Computer Scienc

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

Exact quantum query complexity of EXACTk,ln\rm{EXACT}_{k,l}^n

In the exact quantum query model a successful algorithm must always output the correct function value. We investigate the function that is true if exactly $k$ or $l$ of the $n$ input bits given by an oracle are 1. We find an optimal algorithm (for some cases), and a nontrivial general lower and upper bound on the minimum number of queries to the black box.Comment: 19 pages, fixed some typos and constraint