scholarly journals Upper bounds on the rate of low density stabilizer codes for the quantum erasure channel

2013 ◽  
Vol 13 (9&10) ◽  
pp. 793-826
Author(s):  
Nicolas Delfosse ◽  
Gilles Zemor
Keyword(s):  
Upper Bound ◽  
Percolation Theory ◽  
Hyperbolic Plane ◽  
Upper Bounds ◽  
Low Density ◽  
Critical Probability ◽  
Stabilizer Codes ◽  
Erasure Channel ◽  
Finite Quotients ◽  
Quantum Erasure

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, $R \leq 1-2p$, for stabilizer codes: we also derive an improved upper bound of the form $R \leq 1-2p-D(p)$ 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.

Upper bounds for the critical probability of oriented percolation in two dimensions

1993 ◽  
Vol 440 (1908) ◽  
pp. 201-220 ◽  
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Two Dimensions ◽  
Oriented Percolation ◽  
Bond Percolation ◽  
Critical Probability ◽  
Site Percolation

We give a method for obtaining upper bounds on the critical probability in oriented bond percolation in two dimensions. This method enables us to prove that the critical probability is at most 0.6863, greatly improving the best published upper bound, 0.84. We also prove that our method can be used to give arbitrarily good upper bounds. We also use a slight variant of our method to obtain an upper bound of 0.72599 for the critical probability in oriented site percolation.

scholarly journals Multilevel polarization for quantum Channels

2021 ◽  
Vol 21 (7&8) ◽  
pp. 577-606
Author(s):  
Ashutosh Goswami ◽  
Mehdi Mhalla ◽  
Valentin Savin
Keyword(s):  
Upper Bound ◽  
Noisy Channels ◽  
Code Construction ◽  
Epr Pairs ◽  
Polar Code ◽  
Alternative Construction ◽  
The Family ◽  
Quantum Version ◽  
Erasure Channel ◽  
Quantum Erasure

Recently, a purely quantum version of polar codes has been proposed in~\cite{DGMS19} based on a quantum channel combining and splitting procedure, where a randomly chosen two-qubit Clifford unitary acts as a channel combining operation. Here, we consider the quantum polar code construction using the same channel combining and splitting procedure as in~\cite{DGMS19}, but with a fixed two-qubit Clifford unitary. For the family of Pauli channels, we show that polarization happens in multi-levels, where synthesized quantum virtual channels tend to become completely noisy, half-noisy, or noiseless. Further, we present a quantum polar code exploiting the multilevel nature of polarization, and provide an efficient decoding for this code. We show that half-noisy channels can be frozen by fixing their inputs in either the amplitude or the phase basis, which allows reducing the number of preshared EPR pairs compared to the construction in~\cite{DGMS19}. We provide an upper bound on the number of preshared EPR pairs, which is an equality in the case of the quantum erasure channel. To improve the speed of polarization, we propose an alternative construction, which again polarizes in multi-levels, and the previous upper bound on the number of preshared EPR pairs also holds. For a quantum erasure channel, we confirm by numerical analysis that the multilevel polarization happens relatively faster for the alternative construction.

Upper bounds on the energy dissipation in turbulent precession

1996 ◽  
Vol 321 ◽  
pp. 335-370 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Experimental Data ◽  
Energy Dissipation ◽  
Viscous Dissipation ◽  
Upper Bound ◽  
Dissipation Rate ◽  
Oblate Spheroid ◽  
Upper Bounds ◽  
Precession Rate ◽  
Long Cylinder

Rigorous upper bounds on the viscous dissipation rate are identified for two commonly studied precessing fluid-filled configurations: an oblate spheroid and a long cylinder. The latter represents an interesting new application of the upper-bounding techniques developed by Howard and Busse. A novel ‘background’ method recently introduced by Doering & Constantin is also used to deduce in both instances an upper bound which is independent of the fluid's viscosity and the forcing precession rate. Experimental data provide some evidence that the observed viscous dissipation rate mirrors this behaviour at sufficiently high precessional forcing. Implications are then discussed for the Earth's precessional response.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

ances and covariances obtained from REML are normally distributed with expectation vector and variance-covariance matrix equal to the fol-low  ing, r  espectiv    ely,   When σˆ > 0.04, let νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − (1 + c (7.6) be an estimate for the (7.3) reference-scaled metric in accordance with FDA Guidance (2001) and using a REML UN model. Then (Patter-son, 2003; Patterson and Jones, 2002b), this estimate is asymptotically normally distributed and unbiased with E[νˆ ] = δ +σ − (1 + c and Var[νˆ ] = 4σ + l + 4l + (1 + c ) (l )+ 2l −2(1+c − 2(1+c +4(1+c −2(1+c . Similarly, for the constant-scaled metric, when σˆ ≤ 0.04, νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − σˆ − 0.04(c ) (7.7) E[νˆ ] = δ +σ − 0.04(c ) Var[νˆ ] = 4σ + l + 4l + 2l − 2l − 4l + 4l − 2l . The required asymptotic upper bound √ of the 90% confidence interval can √ then be calculated as νˆ + 1.645× V̂ ar[νˆ ] or νˆ + 1.645× V̂ ar[νˆ ], where the variances are obtained by ‘plugging in’ the estimated values of the variances and covariances obtained from SAS proc mixed into the formulae for Var[νˆ ] or Var[νˆ ]. The necessary SAS code to do this is given in Appendix B. The output reveals that σˆ = 0.0714 and the upper bound is−0.060 for log(AUC). For log(Cmax), σˆ = 0.1060 and the upper bound is −0.055. As both of these upper bounds are below zero, IBE can be claimed.

2003 ◽  
pp. 367-367
Keyword(s):  
Confidence Interval ◽  
Covariance Matrix ◽  
Upper Bound ◽  
Upper Bounds ◽  
Fda Guidance ◽  
Asymptotic Upper Bound ◽  
Variance Covariance Matrix ◽  
Normally Distributed

Models of arithmetic and upper bounds for arithmetic sets

1994 ◽  
Vol 59 (3) ◽  
pp. 977-983 ◽  
Author(s):  
Alistair H. Lachlan ◽  
Robert I. Soare
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Models Of Arithmetic

AbstractWe settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

ON THE QUASI-STATIONARY DISTRIBUTION OF SIS MODELS

2016 ◽  
Vol 30 (4) ◽  
pp. 622-639 ◽  
Author(s):  
Gaofeng Da ◽  
Maochao Xu ◽  
Shouhuai Xu
Keyword(s):  
Stationary Distribution ◽  
Upper Bound ◽  
Hazard Rate ◽  
State Of The Art ◽  
Upper Bounds ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Novel Method ◽  
Better Than ◽  
Quasi Stationary Distribution

In this paper, we propose a novel method for constructing upper bounds of the quasi-stationary distribution of SIS processes. Using this method, we obtain an upper bound that is better than the state-of-the-art upper bound. Moreover, we prove that the fixed point map Φ [7] actually preserves the equilibrium reversed hazard rate order under a certain condition. This allows us to further improve the upper bound. Some numerical results are presented to illustrate the results.

scholarly journals Upper bounds for geodesic periods over hyperbolic manifolds

2018 ◽  
Vol 29 (01) ◽  
pp. 1850009
Author(s):  
Feng Su
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Hyperbolic Manifolds ◽  
Maass Forms ◽  
Maass Form ◽  
Ambient Manifold

We prove an upper bound for geodesic periods of Maass forms over hyperbolic manifolds. By definition, such periods are integrals of Maass forms restricted to a special geodesic cycle of the ambient manifold, against a Maass form on the cycle. Under certain restrictions, the bound will be uniform.

Approximate upper bounds for the critical probability of oriented percolation in two dimensions based on rapidly mixing Markov chains

1997 ◽  
Vol 34 (04) ◽  
pp. 859-867
Author(s):  
Béla Bollabás ◽  
Alan Stacey
Keyword(s):  
Monte Carlo ◽  
Markov Chains ◽  
Monte Carlo Simulations ◽  
Statistical Tests ◽  
Upper Bounds ◽  
Two Dimensions ◽  
Oriented Percolation ◽  
Critical Probability ◽  
Confidence Levels

We develop a technique for establishing statistical tests with precise confidence levels for upper bounds on the critical probability in oriented percolation. We use it to give pc &lt; 0.647 with a 99.999967% confidence. As Monte Carlo simulations suggest that pc ≈ 0.6445, this bound is fairly tight.

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