Codeword stabilized quantum codes: algorithm and structure

The codeword stabilized ("CWS") quantum codes formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes (arXiv:0708.1021). This formalism reduces the problem of constructing such quantum codes to finding a binary classical code correcting an error pattern induced by a graph state. Finding such a classical code can be very difficult. Here, we consider an algorithm which maps the search for CWS codes to a problem of identifying maximum cliques in a graph. While solving this problem is in general very hard, we prove three structure theorems which reduce the search space, specifying certain admissible and optimal ((n,K,d)) additive codes. In particular, we find there does not exist any ((7,3,3)) CWS code though the linear programming bound does not rule it out. The complexity of the CWS search algorithm is compared with the contrasting method introduced by Aggarwal and Calderbank (arXiv:cs/0610159).Comment: 11 pages, 1 figur

Local unitary versus local Clifford equivalence of stabilizer and graph states

The equivalence of stabilizer states under local transformations is of fundamental interest in understanding properties and uses of entanglement. Two stabilizer states are equivalent under the usual stochastic local operations and classical communication criterion if and only if they are equivalent under local unitary (LU) operations. More surprisingly, under certain conditions, two LU equivalent stabilizer states are also equivalent under local Clifford (LC) operations, as was shown by Van den Nest et al. [Phys. Rev. \textbf{A71}, 062323]. Here, we broaden the class of stabilizer states for which LU equivalence implies LC equivalence ($LU\Leftrightarrow LC$) to include all stabilizer states represented by graphs with neither cycles of length 3 nor 4. To compare our result with Van den Nest et al.'s, we show that any stabilizer state of distance $\delta=2$ is beyond their criterion. We then further prove that $LU\Leftrightarrow LC$ holds for a more general class of stabilizer states of $\delta=2$. We also explicitly construct graphs representing $\delta>2$ stabilizer states which are beyond their criterion: we identify all 58 graphs with up to 11 vertices and construct graphs with $2^m-1$ ($m\geq 4$) vertices using quantum error correcting codes which have non-Clifford transversal gates.Comment: Revised version according to referee's comments. To appear in Physical Review

Fitness function distributions over generalized search neighborhoods in the q-ary hypercube

Evolutionary Computation, 21(4): 561-590, 2013The frequency distribution of a fitness function over regions of its domain is an important quantity for understanding the behavior of algorithms that employ randomized sampling to search the function. In general, exactly characterizing this distribution is at least as hard as the search problem, since the solutions typically live in the tails of the distribution. However, in some cases it is possible to efficiently retrieve a collection of quantities (called moments) that describe the distribution. In this paper, we consider functions of bounded epistasis that are defined over length-n strings from a finite alphabet of cardinality q. Many problems in combinatorial optimization can be specified as search problems over functions of this type. Employing Fourier analysis of functions over finite groups, we derive an efficient method for computing the exact moments of the frequency distribution of fitness functions over Hamming regions of the q-ary hypercube. We then use this approach to derive equations that describe the expected fitness of the offspring of any point undergoing uniform mutation. The results we present provide insight into the statistical structure of the fitness function for a number of combinatorial problems. For the graph coloring problem, we apply our results to efficiently compute the average number of constraint violations that lie within a certain number of steps of any coloring. We derive an expression for the mutation rate that maximizes the expected fitness of an offspring at each fitness level. We also apply the results to the slightly more complex frequency assignment problem, a relevant application in the domain of the telecommunications industry. As with the graph coloring problem, we provide formulas for the average value of the fitness function in Hamming regions around a solution and the expectation-optimal mutation rate.Spanish Ministry of Science and Innovation and FEDER under contract TIN2008-06491-C04-01 (the M∗ project). Andalusian Government under contract P07-TIC-03044 (DIRICOM project). Air Force Office of Scientific Re- search, Air Force Materiel Command, USAF, under grant number FA9550-08-1-0422

CrY2H-seq: a massively multiplexed assay for deep-coverage interactome mapping.

Broad-scale protein-protein interaction mapping is a major challenge given the cost, time, and sensitivity constraints of existing technologies. Here, we present a massively multiplexed yeast two-hybrid method, CrY2H-seq, which uses a Cre recombinase interaction reporter to intracellularly fuse the coding sequences of two interacting proteins and next-generation DNA sequencing to identify these interactions en masse. We applied CrY2H-seq to investigate sparsely annotated Arabidopsis thaliana transcription factors interactions. By performing ten independent screens testing a total of 36 million binary interaction combinations, and uncovering a network of 8,577 interactions among 1,453 transcription factors, we demonstrate CrY2H-seq's improved screening capacity, efficiency, and sensitivity over those of existing technologies. The deep-coverage network resource we call AtTFIN-1 recapitulates one-third of previously reported interactions derived from diverse methods, expands the number of known plant transcription factor interactions by three-fold, and reveals previously unknown family-specific interaction module associations with plant reproductive development, root architecture, and circadian coordination

A reference map of the human binary protein interactome.

Global insights into cellular organization and genome function require comprehensive understanding of the interactome networks that mediate genotype-phenotype relationships(1,2). Here we present a human 'all-by-all' reference interactome map of human binary protein interactions, or 'HuRI'. With approximately 53,000 protein-protein interactions, HuRI has approximately four times as many such interactions as there are high-quality curated interactions from small-scale studies. The integration of HuRI with genome(3), transcriptome(4) and proteome(5) data enables cellular function to be studied within most physiological or pathological cellular contexts. We demonstrate the utility of HuRI in identifying the specific subcellular roles of protein-protein interactions. Inferred tissue-specific networks reveal general principles for the formation of cellular context-specific functions and elucidate potential molecular mechanisms that might underlie tissue-specific phenotypes of Mendelian diseases. HuRI is a systematic proteome-wide reference that links genomic variation to phenotypic outcomes

A model for understanding the causes and consequences of walking impairments

Walking is an important skill with positive impacts on health, function, and well-being. Many disorders impair walking and its positive impacts through a variety of complex and interrelated mechanisms. Any attempt to understand walking impairments, or the effects of interventions intended to treat these impairments, must respect this complexity. Therefore, our main objectives in conducting this study were to (1) propose a comprehensive model for quantifying the causes and consequences of walking impairments and (2) demonstrate the potential utility of the model for supporting clinical care and addressing basic scientific questions related to walking. To achieve these goals, we introduced a model, described by a directed acyclic graph, consisting of 10 nodes and 23 primary causal paths. We gave detailed descriptions of each node and path based on domain knowledge. We then demonstrated the model’s utility using a large sample of gait data (N = 9504) acquired as part of routine care at a regional referral center. We analyzed five relevant examples that involved many of the model’s nodes and paths. We computed causal effect magnitudes as Shapley values and displayed the overall importance of variables (mean absolute Shapley value), the variation of Shapley values with respect to underlying variables, and Shapley values for individual observations (case studies). We showed that the model was plausible, captured some well-known cause-effect relationships, provided new insights into others, and generated novel hypotheses requiring further testing through simulation or experiment. To aid in transparency, reproducibility, and future enhancements we have included an extensively commented Rmarkdown file and a deidentified data set

Bounds for binary codes of length less than 25

Abstract-Improved bounds for A(n,d), the maximum number of codewords in a (linear or nonlinear) binary code of word length n and minimum distance d, and for A (n&u), the maximum number of binary vectors of length n, distance d, and constant weight w in the range n 5 24 and d 5 10 are presented. Some of the new values are A (9,4) = 20 (which was previously believed to follow from the results of Wax), A (13,6) = 32 (which proves that the Nadler code is optimal), A (17,8) = 36 or 37, and A (21,8) = 512. The upper bounds on A (n,d) are found with the help of linear programming, making use of the values of A(n,d,w). T I

Microstructure—Thermal Property Relationships of Poly (Ethylene Glycol-b-Caprolactone) Copolymers and Their Micelles

The crystallinity of polymers strongly affects their properties. For block copolymers, whereby two crystallisable blocks are covalently tethered to one another, the molecular weight of the individual blocks and their relative weight fraction are important structural parameters that control their crystallisation. In the case of block copolymer micelles, these parameters can influence the crystallinity of the core, which has implications for drug encapsulation and release. Therefore, in this study, we aimed to determine how the microstructure of poly(ethylene glycol-b-caprolactone) (PEG-b-PCL) copolymers contributes to the crystallinity of their hydrophobic PCL micelle cores. Using a library of PEG-b-PCL copolymers with PEG number-average molecular weight (Mn) values of 2, 5, and 10 kDa and weight fractions of PCL (fPCL) ranging from 0.11 to 0.67, the thermal behaviour and morphology were studied in blends, bulk, and micelles using differential scanning calorimetry (DSC), wide-angle X-ray diffraction (WXRD), and Synchrotron wide-angle X-ray scattering (WAXS). Compared to PEG and PCL homopolymers, the block copolymers displayed reduced crystallinity in the bulk phase and the individual blocks had a large influence on the crystallisation of one another. The fPCL was determined to be the dominant contributor to the extent and order of crystallisation of the two blocks. When fPCL < 0.35, the initial crystallisation of PEG led to an amorphous PCL phase. At fPCL values between 0.35 and 0.65, PEG crystallisation was followed by PCL crystallisation, whereas this behaviour was reversed when fPCL > 0.65. For lyophilised PEG-b-PCL micelles, the crystallinity of the core increased with increasing fPCL, although the core was predominately amorphous for micelles with fPCL < 0.35. These findings contribute to understanding the relationships between copolymer microstructure and micelle core crystallinity that are important for the design and performance of micellar drug delivery systems, and the broader application of polymer micelles