Reed-Muller codes for random erasures and errors

This paper studies the parameters for which Reed-Muller (RM) codes over $GF(2)$ can correct random erasures and random errors with high probability, and in particular when can they achieve capacity for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate $GF(2)$ polynomials on random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about square root of the number of errors at capacity. The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about $E(m,r)$, the matrix whose rows are truth tables of all monomials of degree $\leq r$ in $m$ variables. What is the most (resp. least) number of random columns in $E(m,r)$ that define a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees $r$, which we use to show that RM codes achieve capacity for erasures in these regimes. Our decoding from random errors follows from the following novel reduction. For every linear code $C$ of sufficiently high rate we construct a new code $C'$, also of very high rate, such that for every subset $S$ of coordinates, if $C$ can recover from erasures in $S$, then $C'$ can recover from errors in $S$. Specializing this to RM codes and using our results for erasures imply our result on unique decoding of RM codes at high rate. Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent \cite{KLP} bounds from constant degree to linear degree polynomials

Optimal and Efficient Decoding of Concatenated Quantum Block Codes

We consider the problem of optimally decoding a quantum error correction code -- that is to find the optimal recovery procedure given the outcomes of partial "check" measurements on the system. In general, this problem is NP-hard. However, we demonstrate that for concatenated block codes, the optimal decoding can be efficiently computed using a message passing algorithm. We compare the performance of the message passing algorithm to that of the widespread blockwise hard decoding technique. Our Monte Carlo results using the 5 qubit and Steane's code on a depolarizing channel demonstrate significant advantages of the message passing algorithms in two respects. 1) Optimal decoding increases by as much as 94% the error threshold below which the error correction procedure can be used to reliably send information over a noisy channel. 2) For noise levels below these thresholds, the probability of error after optimal decoding is suppressed at a significantly higher rate, leading to a substantial reduction of the error correction overhead.Comment: Published versio

Characterization of a Chromosomally Encoded 2,4-Dichlorophenoxyacetic Acid/a-Ketoglutarate Dioxygenase from \u3ci\u3eBurkholderia\u3c/i\u3e sp. Strain RASC

The findings of previous studies indicate that the genes required for metabolism of the pesticide 2,4-dichlorophenoxyacetic acid (2,4-D) are typically encoded on broad-host-range plasmids. However, characterization of plasmid-cured strains of Burkholderia sp. strain RASC, as well as mutants obtained by transposon mutagenesis, suggested that the 2,4-D catabolic genes were located on the chromosome of this strain. Mutants of Burkholderia strain RASC unable to degrade 2,4-D (2,4-D- strains) were obtained by insertional inactivation with Tn5. One such mutant (d1) was shown to have Tn5 inserted in tfdARASC, which encodes 2,4-D/alpha-ketoglutarate dioxygenase. This is the first reported example of a chromosomally encoded tfdA. The tfdARASC gene was cloned from a library of wild-type Burkholderia strain RASC DNA and shown to express 2,4-D/alpha-ketoglutarate dioxygenase activity in Escherichia coli. The DNA sequence of the gene was determined and shown to be similar, although not identical, to those of isofunctional genes from other bacteria. Moreover, the gene product (TfdARASC) was purified and shown to be similar in molecular weight, amino-terminal sequence, and reaction mechanism to the canonical TfdA of Alcaligenes eutrophus JMP134. The data presented here indicate that tfdA genes can be found on the chromosome of some bacterial species and suggest that these catabolic genes are rather mobile and may be transferred by means other than conjugation

Good Quantum Convolutional Error Correction Codes And Their Decoding Algorithm Exist

Quantum convolutional code was introduced recently as an alternative way to protect vital quantum information. To complete the analysis of quantum convolutional code, I report a way to decode certain quantum convolutional codes based on the classical Viterbi decoding algorithm. This decoding algorithm is optimal for a memoryless channel. I also report three simple criteria to test if decoding errors in a quantum convolutional code will terminate after a finite number of decoding steps whenever the Hilbert space dimension of each quantum register is a prime power. Finally, I show that certain quantum convolutional codes are in fact stabilizer codes. And hence, these quantum stabilizer convolutional codes have fault-tolerant implementations.Comment: Minor changes, to appear in PR

Quantifying the Performance of Quantum Codes

We study the properties of error correcting codes for noise models in the presence of asymmetries and/or correlations by means of the entanglement fidelity and the code entropy. First, we consider a dephasing Markovian memory channel and characterize the performance of both a repetition code and an error avoiding code in terms of the entanglement fidelity. We also consider the concatenation of such codes and show that it is especially advantageous in the regime of partial correlations. Finally, we characterize the effectiveness of the codes and their concatenation by means of the code entropy and find, in particular, that the effort required for recovering such codes decreases when the error probability decreases and the memory parameter increases. Second, we consider both symmetric and asymmetric depolarizing noisy quantum memory channels and perform quantum error correction via the five qubit stabilizer code. We characterize this code by means of the entanglement fidelity and the code entropy as function of the asymmetric error probabilities and the degree of memory. Specifically, we uncover that while the asymmetry in the depolarizing errors does not affect the entanglement fidelity of the five qubit code, it becomes a relevant feature when the code entropy is used as a performance quantifier.Comment: 21 pages, 10 figure

Multi-Exciton Spectroscopy of a Single Self Assembled Quantum Dot

We apply low temperature confocal optical microscopy to spatially resolve, and spectroscopically study a single self assembled quantum dot. By comparing the emission spectra obtained at various excitation levels to a theoretical many body model, we show that: Single exciton radiative recombination is very weak. Sharp spectral lines are due to optical transitions between confined multiexcitonic states among which excitons thermalize within their lifetime. Once these few states are fully occupied, broad bands appear due to transitions between states which contain continuum electrons.Comment: 12 pages, 4 figures, submitted for publication on Jan,28 199

Mixed quantum state detection with inconclusive results

We consider the problem of designing an optimal quantum detector with a fixed rate of inconclusive results that maximizes the probability of correct detection, when distinguishing between a collection of mixed quantum states. We develop a sufficient condition for the scaled inverse measurement to maximize the probability of correct detection for the case in which the rate of inconclusive results exceeds a certain threshold. Using this condition we derive the optimal measurement for linearly independent pure-state sets, and for mixed-state sets with a broad class of symmetries. Specifically, we consider geometrically uniform (GU) state sets and compound geometrically uniform (CGU) state sets with generators that satisfy a certain constraint. We then show that the optimal measurements corresponding to GU and CGU state sets with arbitrary generators are also GU and CGU respectively, with generators that can be computed very efficiently in polynomial time within any desired accuracy by solving a semidefinite programming problem.Comment: Submitted to Phys. Rev.

Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

Probing the role of proton cross-shell excitations in Ni 70 using nucleon knockout reactions

The neutron-rich Ni isotopes have attracted attention in recent years because of the occurrence of shape or configuration coexistence. We report on the difference in population of excited final states in Ni70 following γ-ray tagged one-proton, one-neutron, and two-proton knockout from Cu71, Ni71, and Zn72 rare-isotope beams, respectively. Using variations observed in the relative transition intensities, signaling the changed population of specific final states in the different reactions, the role of neutron and proton configurations in excited states of Ni70 is probed schematically, with the goal of identifying those that carry, as leading configuration, proton excitations across the Z=28 shell closure. Such states are suggested in the literature to form a collective structure associated with prolate deformation. Adding to the body of knowledge for Ni70, 29 new transitions are reported, of which 15 are placed in its level scheme