2,643 research outputs found
Unary probabilistic and quantum automata on promise problems
We continue the systematic investigation of probabilistic and quantum finite
automata (PFAs and QFAs) on promise problems by focusing on unary languages. We
show that bounded-error QFAs are more powerful than PFAs. But, in contrary to
the binary problems, the computational powers of Las-Vegas QFAs and
bounded-error PFAs are equivalent to deterministic finite automata (DFAs).
Lastly, we present a new family of unary promise problems with two parameters
such that when fixing one parameter QFAs can be exponentially more succinct
than PFAs and when fixing the other parameter PFAs can be exponentially more
succinct than DFAs.Comment: Minor correction
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.Comment: 21 page
Implications of quantum automata for contextuality
We construct zero-error quantum finite automata (QFAs) for promise problems
which cannot be solved by bounded-error probabilistic finite automata (PFAs).
Here is a summary of our results:
- There is a promise problem solvable by an exact two-way QFA in exponential
expected time, but not by any bounded-error sublogarithmic space probabilistic
Turing machine (PTM).
- There is a promise problem solvable by an exact two-way QFA in quadratic
expected time, but not by any bounded-error -space PTMs in
polynomial expected time. The same problem can be solvable by a one-way Las
Vegas (or exact two-way) QFA with quantum head in linear (expected) time.
- There is a promise problem solvable by a Las Vegas realtime QFA, but not by
any bounded-error realtime PFA. The same problem can be solvable by an exact
two-way QFA in linear expected time but not by any exact two-way PFA.
- There is a family of promise problems such that each promise problem can be
solvable by a two-state exact realtime QFAs, but, there is no such bound on the
number of states of realtime bounded-error PFAs solving the members this
family.
Our results imply that there exist zero-error quantum computational devices
with a \emph{single qubit} of memory that cannot be simulated by any finite
memory classical computational model. This provides a computational perspective
on results regarding ontological theories of quantum mechanics \cite{Hardy04},
\cite{Montina08}. As a consequence we find that classical automata based
simulation models \cite{Kleinmann11}, \cite{Blasiak13} are not sufficiently
powerful to simulate quantum contextuality. We conclude by highlighting the
interplay between results from automata models and their application to
developing a general framework for quantum contextuality.Comment: 22 page
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
Solving -SUM using few linear queries
The -SUM problem is given input real numbers to determine whether any
of them sum to zero. The problem is of tremendous importance in the
emerging field of complexity theory within , and it is in particular open
whether it admits an algorithm of complexity with . Inspired by an algorithm due to Meiser (1993), we show
that there exist linear decision trees and algebraic computation trees of depth
solving -SUM. Furthermore, we show that there exists a
randomized algorithm that runs in
time, and performs linear queries on the input. Thus, we show
that it is possible to have an algorithm with a runtime almost identical (up to
the ) to the best known algorithm but for the first time also with the
number of queries on the input a polynomial that is independent of . The
bound on the number of linear queries is also a tighter bound
than any known algorithm solving -SUM, even allowing unlimited total time
outside of the queries. By simultaneously achieving few queries to the input
without significantly sacrificing runtime vis-\`{a}-vis known algorithms, we
deepen the understanding of this canonical problem which is a cornerstone of
complexity-within-.
We also consider a range of tradeoffs between the number of terms involved in
the queries and the depth of the decision tree. In particular, we prove that
there exist -linear decision trees of depth
Quantum Branching Programs and Space-Bounded Nonuniform Quantum Complexity
In this paper, the space complexity of nonuniform quantum computations is
investigated. The model chosen for this are quantum branching programs, which
provide a graphic description of sequential quantum algorithms. In the first
part of the paper, simulations between quantum branching programs and
nonuniform quantum Turing machines are presented which allow to transfer lower
and upper bound results between the two models. In the second part of the
paper, different variants of quantum OBDDs are compared with their
deterministic and randomized counterparts. In the third part, quantum branching
programs are considered where the performed unitary operation may depend on the
result of a previous measurement. For this model a simulation of randomized
OBDDs and exponential lower bounds are presented.Comment: 45 pages, 3 Postscript figures. Proofs rearranged, typos correcte
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 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 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
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