960 research outputs found
On U-Statistics and Compressed Sensing II: Non-Asymptotic Worst-Case Analysis
In another related work, U-statistics were used for non-asymptotic
"average-case" analysis of random compressed sensing matrices. In this
companion paper the same analytical tool is adopted differently - here we
perform non-asymptotic "worst-case" analysis.
Simple union bounds are a natural choice for "worst-case" analyses, however
their tightness is an issue (and questioned in previous works). Here we focus
on a theoretical U-statistical result, which potentially allows us to prove
that these union bounds are tight. To our knowledge, this kind of (powerful)
result is completely new in the context of CS. This general result applies to a
wide variety of parameters, and is related to (Stein-Chen) Poisson
approximation. In this paper, we consider i) restricted isometries, and ii)
mutual coherence. For the bounded case, we show that k-th order restricted
isometry constants have tight union bounds, when the measurements m =
\mathcal{O}(k (1 + \log(n/k))). Here we require the restricted isometries to
grow linearly in k, however we conjecture that this result can be improved to
allow them to be fixed. Also, we show that mutual coherence (with the standard
estimate \sqrt{(4\log n)/m}) have very tight union bounds.
For coherence, the normalization complicates general discussion, and we
consider only Gaussian and Bernoulli cases here.Comment: 12 pages. Submitted to IEEE Transactions on Signal Processin
Polaronic signatures and spectral properties of graphene antidot lattices
We explore the consequences of electron-phonon (e-ph) coupling in graphene
antidot lattices (graphene nanomeshes), i.e., triangular superlattices of
circular holes (antidots) in a graphene sheet. They display a direct band gap
whose magnitude can be controlled via the antidot size and density. The
relevant coupling mechanism in these semiconducting counterparts of graphene is
the modulation of the nearest-neighbor electronic hopping integrals due to
lattice distortions (Peierls-type e-ph coupling). We compute the full momentum
dependence of the e-ph vertex functions for a number of representative antidot
lattices. Based on the latter, we discuss the origins of the previously found
large conduction-band quasiparticle spectral weight due to e-ph coupling. In
addition, we study the nonzero-momentum quasiparticle properties with the aid
of the self-consistent Born approximation, yielding results that can be
compared with future angle-resolved photoemission spectroscopy measurements.
Our principal finding is a significant e-ph mass enhancement, an indication of
polaronic behavior. This can be ascribed to the peculiar momentum dependence of
the e-ph interaction in these narrow-band systems, which favors small phonon
momentum scattering. We also discuss implications of our study for recently
fabricated large-period graphene antidot lattices.Comment: published versio
Bare-excitation ground state of a spinless-fermion -- boson model and W-state engineering in an array of superconducting qubits and resonators
This work unravels an interesting property of a one-dimensional lattice model
that describes a single itinerant spinless fermion (excitation) coupled to
zero-dimensional (dispersionless) bosons through two different
nonlocal-coupling mechanisms. Namely, below a critical value of the effective
excitation-boson coupling strength the exact ground state of this model is the
zero-quasimomentum Bloch state of a bare (i.e., completely undressed)
excitation. It is demonstrated here how this last property of the lattice model
under consideration can be exploited for a fast, deterministic preparation of
multipartite states in a readily realizable system of inductively-coupled
superconducting qubits and microwave resonators.Comment: final, published versio
Extracting spectral properties of small Holstein polarons from a transmon-based analog quantum simulator
The Holstein model, which describes purely local coupling of an itinerant
excitation (electron, hole, exciton) with zero-dimensional (dispersionless)
phonons, represents the paradigm for short-range excitation-phonon
interactions. It is demonstrated here how spectral properties of small Holstein
polarons -- heavily phonon-dressed quasiparticles, formed in the
strong-coupling regime of the Holstein model -- can be extracted from an analog
quantum simulator of this model. This simulator, which is meant to operate in
the dispersive regime of circuit quantum electrodynamics, has the form of an
array of capacitively coupled superconducting transmon qubits and microwave
resonators, the latter being subject to a weak external driving. The magnitude
of -type coupling between adjacent qubits in this system can be tuned
through an external flux threading the SQUID loops between those qubits; this
translates into an {\em in-situ} flux-tunable hopping amplitude of a fictitious
itinerant spinless-fermion excitation, allowing one to access all the relevant
physical regimes of the Holstein model. By employing the kernel-polynomial
method, based on expanding dynamical response functions in Chebyshev
polynomials of the first kind and their recurrence relation, the relevant
single-particle momentum-frequency resolved spectral function of this system is
computed here for a broad range of parameter values. To complement the
evaluation of the spectral function, it is also explained how -- by making use
of the many-body version of the Ramsey interference protocol -- this
dynamical-response function can be measured in the envisioned analog simulator.Comment: 17 pages, 7 figure
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