320,832 research outputs found
Hierarchical interpolative factorization for elliptic operators: differential equations
This paper introduces the hierarchical interpolative factorization for
elliptic partial differential equations (HIF-DE) in two (2D) and three
dimensions (3D). This factorization takes the form of an approximate
generalized LU/LDL decomposition that facilitates the efficient inversion of
the discretized operator. HIF-DE is based on the multifrontal method but uses
skeletonization on the separator fronts to sparsify the dense frontal matrices
and thus reduce the cost. We conjecture that this strategy yields linear
complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity
in 3D can be achieved by skeletonizing the compressed fronts themselves, which
amounts geometrically to a recursive dimensional reduction scheme. Numerical
experiments support our claims and further demonstrate the performance of our
algorithm as a fast direct solver and preconditioner. MATLAB codes are freely
available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math.
arXiv admin note: substantial text overlap with arXiv:1307.266
Atomic line radiative transfer with MCFOST I. Code description and benchmarking
Aims. We present MCFOST-art, a new non-local thermodynamic equilibrium
radiative transfer solver for multilevel atomic systems. The code is embedded
in the 3D radiative transfer code MCFOST and is compatible with most of the
MCFOST modules. The code is versatile and designed to model the close
environment of stars in 3D. Methods. The code solves for the statistical
equilibrium and radiative transfer equations using the Multilevel Accelerated
Lambda Iteration (MALI) method. We tested MCFOST-art on spherically symmetric
models of stellar photospheres as well as on a standard model of the solar
atmosphere. We computed atomic level populations and outgoing fluxes and
compared these values with the results of the TURBOspectrum and RH codes.
Calculations including expansion and rotation of the atmosphere were also
performed. We tested both the pure local thermodynamic equilibrium and the
out-of-equilibrium problems. Results. In all cases, the results from all codes
agree within a few percent at all wavelengths and reach the sub-percent level
between RH and MCFOST-art. We still note a few marginal discrepancies between
MCFOST-art and TURBOspectrum as a result of different treatments of background
opacities at some critical wavelength ranges
Tailoring surface codes for highly biased noise
The surface code, with a simple modification, exhibits ultra-high error
correction thresholds when the noise is biased towards dephasing. Here, we
identify features of the surface code responsible for these ultra-high
thresholds. We provide strong evidence that the threshold error rate of the
surface code tracks the hashing bound exactly for all biases, and show how to
exploit these features to achieve significant improvement in logical failure
rate. First, we consider the infinite bias limit, meaning pure dephasing. We
prove that the error threshold of the modified surface code for pure dephasing
noise is , i.e., that all qubits are fully dephased, and this threshold
can be achieved by a polynomial time decoding algorithm. We demonstrate that
the sub-threshold behavior of the code depends critically on the precise shape
and boundary conditions of the code. That is, for rectangular surface codes
with standard rough/smooth open boundaries, it is controlled by the parameter
, where and are dimensions of the surface code lattice. We
demonstrate a significant improvement in logical failure rate with pure
dephasing for co-prime codes that have , and closely-related rotated
codes, which have a modified boundary. The effect is dramatic: the same logical
failure rate achievable with a square surface code and physical qubits can
be obtained with a co-prime or rotated surface code using only
physical qubits. Finally, we use approximate maximum likelihood decoding to
demonstrate that this improvement persists for a general Pauli noise biased
towards dephasing. In particular, comparing with a square surface code, we
observe a significant improvement in logical failure rate against biased noise
using a rotated surface code with approximately half the number of physical
qubits.Comment: 18+4 pages, 24 figures; v2 includes additional coauthor (ASD) and new
results on the performance of surface codes in the finite-bias regime,
obtained with beveled surface codes and an improved tensor network decoder;
v3 published versio
Quantum Random Access Codes with Shared Randomness
We consider a communication method, where the sender encodes n classical bits
into 1 qubit and sends it to the receiver who performs a certain measurement
depending on which of the initial bits must be recovered. This procedure is
called (n,1,p) quantum random access code (QRAC) where p > 1/2 is its success
probability. It is known that (2,1,0.85) and (3,1,0.79) QRACs (with no
classical counterparts) exist and that (4,1,p) QRAC with p > 1/2 is not
possible.
We extend this model with shared randomness (SR) that is accessible to both
parties. Then (n,1,p) QRAC with SR and p > 1/2 exists for any n > 0. We give an
upper bound on its success probability (the known (2,1,0.85) and (3,1,0.79)
QRACs match this upper bound). We discuss some particular constructions for
several small values of n.
We also study the classical counterpart of this model where n bits are
encoded into 1 bit instead of 1 qubit and SR is used. We give an optimal
construction for such codes and find their success probability exactly--it is
less than in the quantum case.
Interactive 3D quantum random access codes are available on-line at
http://home.lanet.lv/~sd20008/racs .Comment: 51 pages, 33 figures. New sections added: 1.2, 3.5, 3.8.2, 5.4 (paper
appears to be shorter because of smaller margins). Submitted as M.Math thesis
at University of Waterloo by M
Performance and structure of single-mode bosonic codes
The early Gottesman, Kitaev, and Preskill (GKP) proposal for encoding a qubit
in an oscillator has recently been followed by cat- and binomial-code
proposals. Numerically optimized codes have also been proposed, and we
introduce new codes of this type here. These codes have yet to be compared
using the same error model; we provide such a comparison by determining the
entanglement fidelity of all codes with respect to the bosonic pure-loss
channel (i.e., photon loss) after the optimal recovery operation. We then
compare achievable communication rates of the combined encoding-error-recovery
channel by calculating the channel's hashing bound for each code. Cat and
binomial codes perform similarly, with binomial codes outperforming cat codes
at small loss rates. Despite not being designed to protect against the
pure-loss channel, GKP codes significantly outperform all other codes for most
values of the loss rate. We show that the performance of GKP and some binomial
codes increases monotonically with increasing average photon number of the
codes. In order to corroborate our numerical evidence of the cat/binomial/GKP
order of performance occurring at small loss rates, we analytically evaluate
the quantum error-correction conditions of those codes. For GKP codes, we find
an essential singularity in the entanglement fidelity in the limit of vanishing
loss rate. In addition to comparing the codes, we draw parallels between
binomial codes and discrete-variable systems. First, we characterize one- and
two-mode binomial as well as multi-qubit permutation-invariant codes in terms
of spin-coherent states. Such a characterization allows us to introduce check
operators and error-correction procedures for binomial codes. Second, we
introduce a generalization of spin-coherent states, extending our
characterization to qudit binomial codes and yielding a new multi-qudit code.Comment: 34 pages, 11 figures, 4 tables. v3: published version. See related
talk at https://absuploads.aps.org/presentation.cfm?pid=1351
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