7,646 research outputs found
Connecting the dots (with minimum crossings)
We study a prototype Crossing Minimization problem, defined as follows. Let F be an infinite family of (possibly vertex-labeled) graphs. Then, given a set P of (possibly labeled) n points in the Euclidean plane, a collection L subseteq Lines(P)={l: l is a line segment with both endpoints in P}, and a non-negative integer k, decide if there is a subcollection L'subseteq L such that the graph G=(P,L') is isomorphic to a graph in F and L' has at most k crossings. By G=(P,L'), we refer to the graph on vertex set P, where two vertices are adjacent if and only if there is a line segment that connects them in L'. Intuitively, in Crossing Minimization, we have a set of locations of interest, and we want to build/draw/exhibit connections between them (where L indicates where it is feasible to have these connections) so that we obtain a structure in F. Natural choices for F are the collections of perfect matchings, Hamiltonian paths, and graphs that contain an (s,t)-path (a path whose endpoints are labeled). While the objective of seeking a solution with few crossings is of interest from a theoretical point of view, it is also well motivated by a wide range of practical considerations. For example, links/roads (such as highways) may be cheaper to build and faster to traverse, and signals/moving objects would collide/interrupt each other less often. Further, graphs with fewer crossings are preferred for graphic user interfaces. As a starting point for a systematic study, we consider a special case of Crossing Minimization. Already for this case, we obtain NP-hardness and W[1]-hardness results, and ETH-based lower bounds. Specifically, suppose that the input also contains a collection D of d non-crossing line segments such that each point in P belongs to exactly one line in D, and L does not contain line segments between points on the same line in D. Clearly, Crossing Minimization is the case where d=n - then, P is in general position. The case of d=2 is of interest not only because it is the most restricted non-trivial case, but also since it corresponds to a class of graphs that has been well studied - specifically, it is Crossing Minimization where G=(P,L) is a (bipartite) graph with a so called two-layer drawing. For d=2, we consider three basic choices of F. For perfect matchings, we show (i) NP-hardness with an ETH-based lower bound, (ii) solvability in subexponential parameterized time, and (iii) existence of an O(k^2)-vertex kernel. Second, for Hamiltonian paths, we show (i) solvability in subexponential parameterized time, and (ii) existence of an O(k^2)-vertex kernel. Lastly, for graphs that contain an (s,t)-path, we show (i) NP-hardness and W[1]-hardness, and (ii) membership in XP
Connecting the Dots (with Minimum Crossings)
We study a prototype Crossing Minimization problem, defined as follows. Let F be an infinite family of (possibly vertex-labeled) graphs. Then, given a set P of (possibly labeled) n points in the Euclidean plane, a collection L subseteq Lines(P)={l: l is a line segment with both endpoints in P}, and a non-negative integer k, decide if there is a subcollection L\u27subseteq L such that the graph G=(P,L\u27) is isomorphic to a graph in F and L\u27 has at most k crossings. By G=(P,L\u27), we refer to the graph on vertex set P, where two vertices are adjacent if and only if there is a line segment that connects them in L\u27. Intuitively, in Crossing Minimization, we have a set of locations of interest, and we want to build/draw/exhibit connections between them (where L indicates where it is feasible to have these connections) so that we obtain a structure in F. Natural choices for F are the collections of perfect matchings, Hamiltonian paths, and graphs that contain an (s,t)-path (a path whose endpoints are labeled). While the objective of seeking a solution with few crossings is of interest from a theoretical point of view, it is also well motivated by a wide range of practical considerations. For example, links/roads (such as highways) may be cheaper to build and faster to traverse, and signals/moving objects would collide/interrupt each other less often. Further, graphs with fewer crossings are preferred for graphic user interfaces.
As a starting point for a systematic study, we consider a special case of Crossing Minimization. Already for this case, we obtain NP-hardness and W[1]-hardness results, and ETH-based lower bounds. Specifically, suppose that the input also contains a collection D of d non-crossing line segments such that each point in P belongs to exactly one line in D, and L does not contain line segments between points on the same line in D. Clearly, Crossing Minimization is the case where d=n - then, P is in general position. The case of d=2 is of interest not only because it is the most restricted non-trivial case, but also since it corresponds to a class of graphs that has been well studied - specifically, it is Crossing Minimization where G=(P,L) is a (bipartite) graph with a so called two-layer drawing. For d=2, we consider three basic choices of F. For perfect matchings, we show (i) NP-hardness with an ETH-based lower bound, (ii) solvability in subexponential parameterized time, and (iii) existence of an O(k^2)-vertex kernel. Second, for Hamiltonian paths, we show (i) solvability in subexponential parameterized time, and (ii) existence of an O(k^2)-vertex kernel. Lastly, for graphs that contain an (s,t)-path, we show (i) NP-hardness and W[1]-hardness, and (ii) membership in XP
Impurity effects on semiconductor quantum bits in coupled quantum dots
We theoretically consider the effects of having unintentional charged
impurities in laterally coupled two-dimensional double (GaAs) quantum dot
systems, where each dot contains one or two electrons and a single charged
impurity in the presence of an external magnetic field. Using molecular orbital
and configuration interaction method, we calculate the effect of the impurity
on the 2-electron energy spectrum of each individual dot as well as on the
spectrum of the coupled-double-dot 2-electron system. We find that the
singlet-triplet exchange splitting between the two lowest energy states, both
for the individual dots and the coupled dot system, depends sensitively on the
location of the impurity and its coupling strength (i.e. the effective charge).
A strong electron-impurity coupling breaks down equality of the two
doubly-occupied singlets in the left and the right dot leading to a mixing
between different spin singlets. As a result, the maximally entangled qubit
states are no longer fully obtained in zero magnetic field case. Moreover, a
repulsive impurity results in a triplet-singlet transition as the impurity
effective charge increases or/and the impurity position changes. We comment on
the impurity effect in spin qubit operations in the double dot system based on
our numerical results.Comment: published version on Physical Review B journal, 25 pages, 26 figure
Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops
It is found that the number, , of irreducible multiple zeta values
(MZVs) of weight , is generated by . For
, enumerates positive knots with crossings. Positive
knots to which field theory assigns knot-numbers that are not MZVs first appear
at 10 crossings. We identify all the positive knots, up to 15 crossings, that
are in correspondence with irreducible MZVs, by virtue of the connection
between knots and numbers realized by Feynman diagrams with up to 9 loops.Comment: 15 pages, Latex, figures using EPSF, replaced version has references
and conclusions updated, Eq.(7) revised; as to appear in Phys.Lett.
Correlations of conductance peaks and transmission phases in deformed quantum dots
We investigate the Coulomb blockade resonances and the phase of the
transmission amplitude of a deformed ballistic quantum dot weakly coupled to
leads. We show that preferred single--particle levels exist which stay close to
the Fermi energy for a wide range of values of the gate voltage. These states
give rise to sequences of Coulomb blockade resonances with correlated peak
heights and transmission phases. The correlation of the peak heights becomes
stronger with increasing temperature. The phase of the transmission amplitude
shows lapses by between the resonances. Implications for recent
experiments on ballistic quantum dots are discussed.Comment: 29 pages, 9 eps-figure
Electron transport through double quantum dots
Electron transport experiments on two lateral quantum dots coupled in series
are reviewed. An introduction to the charge stability diagram is given in terms
of the electrochemical potentials of both dots. Resonant tunneling experiments
show that the double dot geometry allows for an accurate determination of the
intrinsic lifetime of discrete energy states in quantum dots. The evolution of
discrete energy levels in magnetic field is studied. The resolution allows to
resolve avoided crossings in the spectrum of a quantum dot. With microwave
spectroscopy it is possible to probe the transition from ionic bonding (for
weak inter-dot tunnel coupling) to covalent bonding (for strong inter-dot
tunnel coupling) in a double dot artificial molecule. This review on the
present experimental status of double quantum dot studies is motivated by their
relevance for realizing solid state quantum bits.Comment: 32 pages, 31 figure
High Resolution Valley Spectroscopy of Si Quantum Dots
We study an accumulation mode Si/SiGe double quantum dot (DQD) containing a
single electron that is dipole coupled to microwave photons in a
superconducting cavity. Measurements of the cavity transmission reveal
dispersive features due to the DQD valley states in Si. The occupation of the
valley states can be increased by raising temperature or applying a finite
source-drain bias across the DQD, resulting in an increased signal. Using
cavity input-output theory and a four-level model of the DQD, it is possible to
efficiently extract valley splittings and the inter- and intra-valley tunnel
couplings
Entropy, fidelity, and double orthogonality for resonance states in two-electron quantum dots
Resonance states of a two-electron quantum dot are studied using a
variational expansion with both real basis-set functions and complex scaling
methods. The two-electron entanglement (linear entropy) is calculated as a
function of the electron repulsion at both sides of the critical value, where
the ground (bound) state becomes a resonance (unbound) state. The linear
entropy and fidelity and double orthogonality functions are compared as methods
for the determination of the real part of the energy of a resonance. The
complex linear entropy of a resonance state is introduced using complex scaling
formalism
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