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.

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.

scholarly journals Communication Complexity And Linearly Ordered Sets

2015 ◽  
Vol 29 (1) ◽  
pp. 93-117
Author(s):  
Mieczysław Kula ◽  
Małgorzata Serwecińska
Keyword(s):  
Upper Bound ◽  
Open Problem ◽  
Numerical Experiments ◽  
Communication Complexity ◽  
Ordered Sets ◽  
Finite Sets ◽  
The Family ◽  
Linear Lattices

AbstractThe paper is devoted to the communication complexity of lattice operations in linearly ordered finite sets. All well known techniques ([4, Chapter 1]) to determine the communication complexity of the infimum function in linear lattices disappoint, because a gap between the lower and upper bound is equal to O(log2n), where n is the cardinality of the lattice. Therefore our aim will be to investigate the communication complexity of the function more carefully. We consider a family of so called interval protocols and we construct the interval protocols for the infimum. We prove that the constructed protocols are optimal in the family of interval protocols. It is still open problem to compute the communication complexity of constructed protocols but the numerical experiments show that their complexity is less than the complexity of known protocols for the infimum function.

scholarly journals Codes for the quantum erasure channel

1997 ◽  
Vol 56 (1) ◽  
pp. 33-38 ◽  
Author(s):  
M. Grassl ◽  
Th. Beth ◽  
T. Pellizzari
Keyword(s):  
Erasure Channel ◽  
Quantum Erasure

scholarly journals Durfee Polynomials

10.37236/1370 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
E. Rodney Canfield ◽  
Sylvie Corteel ◽  
Carla D. Savage
Keyword(s):  
Asymptotic Formula ◽  
Experimental Evidence ◽  
Upper Bound ◽  
Asymptotic Form ◽  
Asymptotically Normal ◽  
The Family ◽  
Average Size ◽  
Durfee Square ◽  
Analytical Expressions ◽  
Addition Formulas

Let ${\bf F}(n)$ be a family of partitions of $n$ and let ${\bf F}(n,d)$ denote the set of partitions in ${\bf F}(n)$ with Durfee square of size $d$. We define the Durfee polynomial of ${\bf F}(n)$ to be the polynomial $P_{{\bf F},n}= \sum |{\bf F}(n,d)|y^d$, where $ 0 \leq d \leq \lfloor \sqrt{n} \rfloor.$ The work in this paper is motivated by empirical evidence which suggests that for several families ${\bf F}$, all roots of the Durfee polynomial are real. Such a result would imply that the corresponding sequence of coefficients $\{|{\bf F}(n,d)|\}$ is log-concave and unimodal and that, over all partitions in ${\bf F}(n)$ for fixed $n$, the average size of the Durfee square, $a_{{\bf F}}(n)$, and the most likely size of the Durfee square, $m_{{\bf F}}(n)$, differ by less than 1. In this paper, we prove results in support of the conjecture that for the family of ordinary partitions, ${\bf P}(n)$, the Durfee polynomial has all roots real. Specifically, we find an asymptotic formula for $|{\bf P}(n,d)|$, deriving in the process a simple upper bound on the number of partitions of $n$ with at most $k$ parts which generalizes the upper bound of Erdös for $|{\bf P}(n)|$. We show that as $n$ tends to infinity, the sequence $\{|{\bf P}(n,d)|\},\ 1 \leq d \leq \sqrt{n},$ is asymptotically normal, unimodal, and log concave; in addition, formulas are found for $a_{{\bf P}}(n)$ and $m_{{\bf P}}(n)$ which differ asymptotically by at most 1. Experimental evidence also suggests that for several families ${\bf F}(n)$ which satisfy a recurrence of a certain form, $m_{{\bf F}}(n)$ grows as $c \sqrt{n}$, for an appropriate constant $c=c_{{\bf F}}$. We prove this under an assumption about the asymptotic form of $|{\bf F}(n,d)|$ and show how to produce, from recurrences for the $|{\bf F}(n,d)|$, analytical expressions for the constants $c_{{\bf F}}$ which agree numerically with the observed values.

Polar Code Construction for BP Decoder

2019 ◽  
pp. 232-240
Author(s):  
Xuan Yi ◽  
Aijun Liu ◽  
Qingshuang Zhang ◽  
Xiaohu Liang ◽  
Zhiyong Chen
Keyword(s):  
Code Construction ◽  
Polar Code
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