37,764 research outputs found
Constructions and Noise Threshold of Hyperbolic Surface Codes
We show how to obtain concrete constructions of homological quantum codes
based on tilings of 2D surfaces with constant negative curvature (hyperbolic
surfaces). This construction results in two-dimensional quantum codes whose
tradeoff of encoding rate versus protection is more favorable than for the
surface code. These surface codes would require variable length connections
between qubits, as determined by the hyperbolic geometry. We provide numerical
estimates of the value of the noise threshold and logical error probability of
these codes against independent X or Z noise, assuming noise-free error
correction
Tradeoffs for reliable quantum information storage in surface codes and color codes
The family of hyperbolic surface codes is one of the rare families of quantum
LDPC codes with non-zero rate and unbounded minimum distance. First, we
introduce a family of hyperbolic color codes. This produces a new family of
quantum LDPC codes with non-zero rate and with minimum distance logarithmic in
the blocklength. Second, we study the tradeoff between the length n, the number
of encoded qubits k and the distance d of surface codes and color codes. We
prove that kd^2 is upper bounded by C(log k)^2n, where C is a constant that
depends only on the row weight of the parity-check matrix. Our results prove
that the best asymptotic minimum distance of LDPC surface codes and color codes
with non-zero rate is logarithmic in the length.Comment: 10 page
Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel
Using combinatorial arguments, we determine an upper bound on achievable
rates of stabilizer codes used over the quantum erasure channel. This allows us
to recover the no-cloning bound on the capacity of the quantum erasure channel,
R is below 1-2p, for stabilizer codes: we also derive an improved upper bound
of the form : R is below 1-2p-D(p) with a function D(p) that stays positive for
0 < p < 1/2 and for any family of stabilizer codes whose generators have
weights bounded from above by a constant - low density stabilizer codes.
We obtain an application to percolation theory for a family of self-dual
tilings of the hyperbolic plane. We associate a family of low density
stabilizer codes with appropriate finite quotients of these tilings. We then
relate the probability of percolation to the probability of a decoding error
for these codes on the quantum erasure channel. The application of our upper
bound on achievable rates of low density stabilizer codes gives rise to an
upper bound on the critical probability for these tilings.Comment: 32 page
Morse coding for a Fuchsian group of a finite covolume
We consider the Morse coding of the geodesic flow on the hyperbolic plane
with respect to a Dirichlet fundamental domain of a Fuchsian group
. The main theorem states that the codes of all the generic geodesics
constitute a -step topological Markov chain, if and only if the fundamental
domain is an ideal polygon (i.e. has all of its vertices on the absolute).Comment: 11 pages, 4 figure
Development and application of unified algorithms for problems in computational science
A framework is presented for developing computationally unified numerical algorithms for solving nonlinear equations that arise in modeling various problems in mathematical physics. The concept of computational unification is an attempt to encompass efficient solution procedures for computing various nonlinear phenomena that may occur in a given problem. For example, in Computational Fluid Dynamics (CFD), a unified algorithm will be one that allows for solutions to subsonic (elliptic), transonic (mixed elliptic-hyperbolic), and supersonic (hyperbolic) flows for both steady and unsteady problems. The objectives are: development of superior unified algorithms emphasizing accuracy and efficiency aspects; development of codes based on selected algorithms leading to validation; application of mature codes to realistic problems; and extension/application of CFD-based algorithms to problems in other areas of mathematical physics. The ultimate objective is to achieve integration of multidisciplinary technologies to enhance synergism in the design process through computational simulation. Specific unified algorithms for a hierarchy of gas dynamics equations and their applications to two other areas: electromagnetic scattering, and laser-materials interaction accounting for melting
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