52,689 research outputs found
Complexity of Two-Dimensional Patterns
In dynamical systems such as cellular automata and iterated maps, it is often
useful to look at a language or set of symbol sequences produced by the system.
There are well-established classification schemes, such as the Chomsky
hierarchy, with which we can measure the complexity of these sets of sequences,
and thus the complexity of the systems which produce them.
In this paper, we look at the first few levels of a hierarchy of complexity
for two-or-more-dimensional patterns. We show that several definitions of
``regular language'' or ``local rule'' that are equivalent in d=1 lead to
distinct classes in d >= 2. We explore the closure properties and computational
complexity of these classes, including undecidability and L-, NL- and
NP-completeness results.
We apply these classes to cellular automata, in particular to their sets of
fixed and periodic points, finite-time images, and limit sets. We show that it
is undecidable whether a CA in d >= 2 has a periodic point of a given period,
and that certain ``local lattice languages'' are not finite-time images or
limit sets of any CA. We also show that the entropy of a d-dimensional CA's
finite-time image cannot decrease faster than t^{-d} unless it maps every
initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy
Reducing sequencing complexity in dynamical quantum error suppression by Walsh modulation
We study dynamical error suppression from the perspective of reducing
sequencing complexity, in order to facilitate efficient semi-autonomous
quantum-coherent systems. With this aim, we focus on digital sequences where
all interpulse time periods are integer multiples of a minimum clock period and
compatibility with simple digital classical control circuitry is intrinsic,
using so-called em Walsh functions as a general mathematical framework. The
Walsh functions are an orthonormal set of basis functions which may be
associated directly with the control propagator for a digital modulation
scheme, and dynamical decoupling (DD) sequences can be derived from the
locations of digital transitions therein. We characterize the suite of the
resulting Walsh dynamical decoupling (WDD) sequences, and identify the number
of periodic square-wave (Rademacher) functions required to generate a Walsh
function as the key determinant of the error-suppressing features of the
relevant WDD sequence. WDD forms a unifying theoretical framework as it
includes a large variety of well-known and novel DD sequences, providing
significant flexibility and performance benefits relative to basic
quasi-periodic design. We also show how Walsh modulation may be employed for
the protection of certain nontrivial logic gates, providing an implementation
of a dynamically corrected gate. Based on these insights we identify Walsh
modulation as a digital-efficient approach for physical-layer error
suppression.Comment: 15 pages, 3 figure
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