1,042 research outputs found
Computation in Finitary Stochastic and Quantum Processes
We introduce stochastic and quantum finite-state transducers as
computation-theoretic models of classical stochastic and quantum finitary
processes. Formal process languages, representing the distribution over a
process's behaviors, are recognized and generated by suitable specializations.
We characterize and compare deterministic and nondeterministic versions,
summarizing their relative computational power in a hierarchy of finitary
process languages. Quantum finite-state transducers and generators are a first
step toward a computation-theoretic analysis of individual, repeatedly measured
quantum dynamical systems. They are explored via several physical systems,
including an iterated beam splitter, an atom in a magnetic field, and atoms in
an ion trap--a special case of which implements the Deutsch quantum algorithm.
We show that these systems' behaviors, and so their information processing
capacity, depends sensitively on the measurement protocol.Comment: 25 pages, 16 figures, 1 table; http://cse.ucdavis.edu/~cmg; numerous
corrections and update
Quantum finite automata: survey, status and research directions
Quantum computing is concerned with computer technology based on the
principles of quantum mechanics, with operations performed at the quantum
level. Quantum computational models make it possible to analyze the resources
required for computations. Quantum automata can be classified thusly: quantum
finite automata, quantum sequential machine, quantum pushdown automata, quantum
Turing machine and orthomodular lattice-valued automata. These models are
useful for determining the expressive power and boundaries of various
computational features. In light of the current state of quantum computation
theory research, a systematic review of the literature seems timely. This
article seeks to provide a comprehensive and systematic analysis of quantum
finite automata models, quantum finite automata models with density operators
and quantum finite automata models with classical states, interactive proof
systems, quantum communication complexity and query complexity as described in
the literature. The statistics of quantum finite automata related papers are
shown and open problems are identified for more advanced research. The current
status of quantum automata theory is distributed into various categories. This
research work also highlights the previous research, current status and future
directions of quantum automata models.Comment:
1-way quantum finite automata: strengths, weaknesses and generalizations
We study 1-way quantum finite automata (QFAs). First, we compare them with
their classical counterparts. We show that, if an automaton is required to give
the correct answer with a large probability (over 0.98), then the power of
1-way QFAs is equal to the power of 1-way reversible automata. However, quantum
automata giving the correct answer with smaller probabilities are more powerful
than reversible automata.
Second, we show that 1-way QFAs can be very space-efficient. Namely, we
construct a 1-way QFA which is exponentially smaller than any equivalent
classical (even randomized) finite automaton. This construction may be useful
for design of other space-efficient quantum algorithms.
Third, we consider several generalizations of 1-way QFAs. Here, our goal is
to find a model which is more powerful than 1-way QFAs keeping the quantum part
as simple as possible.Comment: 23 pages LaTeX, 1 figure, to appear at FOCS'9
The class of languages recognizable by 1-way quantum finite automata is not closed under union
In this paper we develop little further the theory of quantum finite automata
(QFA). There are already few properties of QFA known, that deterministic and
probabilistic finite automata do not have e.g. they cannot recognize all
regular languages. In this paper we show, that class of languages recognizable
by QFA is not closed under union, even not under any Boolean operation, where
both arguments are significant.Comment: 13 pages, 9 figure
Quantum Automata and Quantum Grammars
To study quantum computation, it might be helpful to generalize structures
from language and automata theory to the quantum case. To that end, we propose
quantum versions of finite-state and push-down automata, and regular and
context-free grammars. We find analogs of several classical theorems, including
pumping lemmas, closure properties, rational and algebraic generating
functions, and Greibach normal form. We also show that there are quantum
context-free languages that are not context-free.Comment: 21 page
Succinctness of two-way probabilistic and quantum finite automata
We prove that two-way probabilistic and quantum finite automata (2PFA's and
2QFA's) can be considerably more concise than both their one-way versions
(1PFA's and 1QFA's), and two-way nondeterministic finite automata (2NFA's). For
this purpose, we demonstrate several infinite families of regular languages
which can be recognized with some fixed probability greater than by
just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with
a constant number of states, whereas the sizes of the corresponding 1PFA's,
1QFA's and 2NFA's grow without bound. We also show that 2QFA's with mixed
states can support highly efficient probability amplification. The weakest
known model of computation where quantum computers recognize more languages
with bounded error than their classical counterparts is introduced.Comment: A new version, 21 pages, late
Quantum finite automata: A modern introduction
We present five examples where quantum finite automata (QFAs) outperform
their classical counterparts. This may be useful as a relatively simple
technique to introduce quantum computation concepts to computer scientists. We
also describe a modern QFA model involving superoperators that is able to
simulate all known QFA and classical finite automaton variants.Comment: 15 page
On the class of languages recognizable by 1-way quantum finite automata
It is an open problem to characterize the class of languages recognized by
quantum finite automata (QFA). We examine some necessary and some sufficient
conditions for a (regular) language to be recognizable by a QFA. For a subclass
of regular languages we get a condition which is necessary and sufficient.
Also, we prove that the class of languages recognizable by a QFA is not
closed under union or any other binary Boolean operation where both arguments
are significant.Comment: 18 pages, 16 figures, extends quant-ph/000100
Quantum finite multitape automata
Quantum finite automata were introduced by C.Moore, J.P. Crutchfield, and by
A.Kondacs and J.Watrous. This notion is not a generalization of the
deterministic finite automata. Moreover, it was proved that not all regular
languages can be recognized by quantum finite automata. A.Ambainis and
R.Freivalds proved that for some languages quantum finite automata may be
exponentially more concise rather than both deterministic and probabilistic
finite automata. In this paper we introduce the notion of quantum finite
multitape automata and prove that there is a language recognized by a quantum
finite automaton but not by a deterministic or probabilistic finite automata.
This is the first result on a problem which can be solved by a quantum computer
but not by a deterministic or probabilistic computer. Additionally we discover
unexpected probabilistic automata recognizing complicated languages.Comment: 14 pages, LaTe
Some observations on two-way finite automata with quantum and classical states
{\it Two-way finite automata with quantum and classical states} (2qcfa's)
were introduced by Ambainis and Watrous. Though this computing model is more
restricted than the usual {\it two-way quantum finite automata} (2qfa's) first
proposed by Kondacs and Watrous, it is still more powerful than the classical
counterpart. In this note, we focus on dealing with the operation properties of
2qcfa's. We prove that the Boolean operations (intersection, union, and
complement) and the reversal operation of the class of languages recognized by
2qcfa's with error probabilities are closed; as well, we verify that the
catenation operation of such class of languages is closed under certain
restricted condition. The numbers of states of these 2qcfa's for the above
operations are presented. Some examples are included, and \{xx^{R}|x\in
\{a,b\}^{*},#_{x}(a)=#_{x}(b)\} is shown to be recognized by 2qcfa with
one-sided error probability, where is the reversal of , and
#_{x}(a) denotes the 's number in string .Comment: Comments and suggestions are welcom
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