4,170 research outputs found
Lower Bounds for Graph-Walking Automata
Graph-walking automata (GWA) traverse graphs by moving between the nodes following the edges, using a finite-state control to decide where to go next. It is known that every GWA can be transformed to a GWA that halts on every input, to a GWA returning to the initial node in order to accept, as well as to a reversible GWA. This paper establishes lower bounds on the state blow-up of these transformations: it is shown that making an n-state GWA traversing k-ary graphs return to the initial node requires at least 2(n-1)(k-3) states in the worst case; the same lower bound holds for the transformation to halting automata. Automata satisfying both properties at once must have at least 4(n-1)(k-3) states. A reversible automaton must have at least 4(n-1)(k-3)-1 states. These bounds are asymptotically tight to the upper bounds proved using the methods from the literature
Sum of exit times in series of metastable states in probabilistic cellular automata
Reversible Probabilistic Cellular Automata are a special class
of automata whose stationary behavior is described by Gibbs--like
measures. For those models the dynamics can be trapped for a very
long time in states which are very different from the ones typical
of stationarity.
This phenomenon can be recasted in the framework of metastability
theory which is typical of Statistical Mechanics.
In this paper we consider a model presenting two not degenerate in
energy
metastable states which form a series, in the sense that,
when the dynamics is started at one of them, before reaching
stationarity, the system must necessarily visit the second one.
We discuss a rule for combining the exit times
from each of the metastable states
Optimal lower bounds for quantum automata and random access codes
Consider the finite regular language L_n = {w0 : w \in {0,1}^*, |w| \le n}.
It was shown by Ambainis, Nayak, Ta-Shma and Vazirani that while this language
is accepted by a deterministic finite automaton of size O(n), any one-way
quantum finite automaton (QFA) for it has size 2^{Omega(n/log n)}. This was
based on the fact that the evolution of a QFA is required to be reversible.
When arbitrary intermediate measurements are allowed, this intuition breaks
down. Nonetheless, we show a 2^{Omega(n)} lower bound for such QFA for L_n,
thus also improving the previous bound. The improved bound is obtained by
simple entropy arguments based on Holevo's theorem. This method also allows us
to obtain an asymptotically optimal (1-H(p))n bound for the dense quantum codes
(random access codes) introduced by Ambainis et al. We then turn to Holevo's
theorem, and show that in typical situations, it may be replaced by a tighter
and more transparent in-probability bound.Comment: 8 pages, 1 figure, Latex2e. Extensive modifications have been made to
increase clarity. To appear in FOCS'9
Dense Quantum Coding and a Lower Bound for 1-way Quantum Automata
We consider the possibility of encoding m classical bits into much fewer n
quantum bits so that an arbitrary bit from the original m bits can be recovered
with a good probability, and we show that non-trivial quantum encodings exist
that have no classical counterparts. On the other hand, we show that quantum
encodings cannot be much more succint as compared to classical encodings, and
we provide a lower bound on such quantum encodings. Finally, using this lower
bound, we prove an exponential lower bound on the size of 1-way quantum finite
automata for a family of languages accepted by linear sized deterministic
finite automata.Comment: 12 pages, 3 figures. Defines random access codes, gives upper and
lower bounds for the number of bits required for such (possibly quantum)
codes. Derives the size lower bound for quantum finite automata of the
earlier version of the paper using these result
Quantum computation with devices whose contents are never read
In classical computation, a "write-only memory" (WOM) is little more than an
oxymoron, and the addition of WOM to a (deterministic or probabilistic)
classical computer brings no advantage. We prove that quantum computers that
are augmented with WOM can solve problems that neither a classical computer
with WOM nor a quantum computer without WOM can solve, when all other resource
bounds are equal. We focus on realtime quantum finite automata, and examine the
increase in their power effected by the addition of WOMs with different access
modes and capacities. Some problems that are unsolvable by two-way
probabilistic Turing machines using sublogarithmic amounts of read/write memory
are shown to be solvable by these enhanced automata.Comment: 32 pages, a preliminary version of this work was presented in the 9th
International Conference on Unconventional Computation (UC2010
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