1,108 research outputs found
On vertices enforcing a Hamiltonian cycle
A nonempty vertex set X ⊆ V(G) of a hamiltonian graph G is called an of G if every X-cycle of G (i.e. a cycle of G containing all vertices of X) is hamiltonian. The h(G) of a graph G is defined to be the smallest cardinality of an H-force set of G. In the paper the study of this parameter is introduced and its value or a lower bound for outerplanar graphs, planar graphs, k-connected graphs and prisms over graphs is determined
Symmetry protected topological order at nonzero temperature
We address the question of whether symmetry-protected topological (SPT) order
can persist at nonzero temperature, with a focus on understanding the thermal
stability of several models studied in the theory of quantum computation. We
present three results in this direction. First, we prove that nontrivial SPT
order protected by a global on-site symmetry cannot persist at nonzero
temperature, demonstrating that several quantum computational structures
protected by such on-site symmetries are not thermally stable. Second, we prove
that the 3D cluster state model used in the formulation of topological
measurement-based quantum computation possesses a nontrivial SPT-ordered
thermal phase when protected by a global generalized (1-form) symmetry. The SPT
order in this model is detected by long-range localizable entanglement in the
thermal state, which compares with related results characterizing SPT order at
zero temperature in spin chains using localizable entanglement as an order
parameter. Our third result is to demonstrate that the high error tolerance of
this 3D cluster state model for quantum computation, even without a protecting
symmetry, can be understood as an application of quantum error correction to
effectively enforce a 1-form symmetry.Comment: 42 pages, 10 figures, comments welcome; v2 published versio
Symmetry-protected self-correcting quantum memories
A self-correcting quantum memory can store and protect quantum information
for a time that increases without bound with the system size and without the
need for active error correction. We demonstrate that symmetry can lead to
self-correction in 3D spin-lattice models. In particular, we investigate codes
given by 2D symmetry-enriched topological (SET) phases that appear naturally on
the boundary of 3D symmetry-protected topological (SPT) phases. We find that
while conventional on-site symmetries are not sufficient to allow for
self-correction in commuting Hamiltonian models of this form, a generalized
type of symmetry known as a 1-form symmetry is enough to guarantee
self-correction. We illustrate this fact with the 3D "cluster-state" model from
the theory of quantum computing. This model is a self-correcting memory, where
information is encoded in a 2D SET-ordered phase on the boundary that is
protected by the thermally stable SPT ordering of the bulk. We also investigate
the gauge color code in this context. Finally, noting that a 1-form symmetry is
a very strong constraint, we argue that topologically ordered systems can
possess emergent 1-form symmetries, i.e., models where the symmetry appears
naturally, without needing to be enforced externally.Comment: 39 pages, 16 figures, comments welcome; v2 includes much more
explicit detail on the main example model, including boundary conditions and
implementations of logical operators through local moves; v3 published
versio
Lombardi Drawings of Graphs
We introduce the notion of Lombardi graph drawings, named after the American
abstract artist Mark Lombardi. In these drawings, edges are represented as
circular arcs rather than as line segments or polylines, and the vertices have
perfect angular resolution: the edges are equally spaced around each vertex. We
describe algorithms for finding Lombardi drawings of regular graphs, graphs of
bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International
Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure
A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
For even , the matchings connectivity matrix encodes which
pairs of perfect matchings on vertices form a single cycle. Cygan et al.
(STOC 2013) showed that the rank of over is
and used this to give an
time algorithm for counting Hamiltonian cycles modulo on graphs of
pathwidth . The same authors complemented their algorithm by an
essentially tight lower bound under the Strong Exponential Time Hypothesis
(SETH). This bound crucially relied on a large permutation submatrix within
, which enabled a "pattern propagation" commonly used in previous
related lower bounds, as initiated by Lokshtanov et al. (SODA 2011).
We present a new technique for a similar pattern propagation when only a
black-box lower bound on the asymptotic rank of is given; no
stronger structural insights such as the existence of large permutation
submatrices in are needed. Given appropriate rank bounds, our
technique yields lower bounds for counting Hamiltonian cycles (also modulo
fixed primes ) parameterized by pathwidth.
To apply this technique, we prove that the rank of over the
rationals is . We also show that the rank of
over is for any prime
and even for some primes.
As a consequence, we obtain that Hamiltonian cycles cannot be counted in time
for any unless SETH fails. This
bound is tight due to a time algorithm by Bodlaender et
al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be
counted modulo primes in time , indicating
that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in
SODA 201
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