Effects of time reversal symmetry in dynamical decoupling

Dynamical decoupling (DD) is a technique for preserving the coherence of quantum mechanical states in the presence of a noisy environment. It uses sequences of inversion pulses to suppress the environmental perturbations by periodically refocusing them. It has been shown that different sequences of inversion pulses show vastly different performance, in particular also concerning the correction of experimental pulse imperfections. Here, we investigate specifically the role of time-reversal symmetry in the building-blocks of the pulse sequence. We show that using time symmetric building blocks often improves the performance of the sequence compared to sequences formed by time asymmetric building blocks. Using quantum state tomography of the echoes generated by the sequences, we analyze the mechanisms that lead to loss of fidelity and show how they can be compensated by suitable concatenation of symmetry-related blocks of decoupling pulses

Computing the blocks of a quasi-median graph

Quasi-median graphs are a tool commonly used by evolutionary biologists to visualise the evolution of molecular sequences. As with any graph, a quasi-median graph can contain cut vertices, that is, vertices whose removal disconnect the graph. These vertices induce a decomposition of the graph into blocks, that is, maximal subgraphs which do not contain any cut vertices. Here we show that the special structure of quasi-median graphs can be used to compute their blocks without having to compute the whole graph. In particular we present an algorithm that, for a collection of $n$ aligned sequences of length $m$, can compute the blocks of the associated quasi-median graph together with the information required to correctly connect these blocks together in run time $\mathcal O(n^2m^2)$, independent of the size of the sequence alphabet. Our primary motivation for presenting this algorithm is the fact that the quasi-median graph associated to a sequence alignment must contain all most parsimonious trees for the alignment, and therefore precomputing the blocks of the graph has the potential to help speed up any method for computing such trees.Comment: 17 pages, 2 figure

Sparse Long Blocks and the Micro-Structure of the Longest Common Subsequences

Consider two random strings having the same length and generated by an iid sequence taking its values uniformly in a fixed finite alphabet. Artificially place a long constant block into one of the strings, where a constant block is a contiguous substring consisting only of one type of symbol. The long block replaces a segment of equal size and its length is smaller than the length of the strings, but larger than its square-root. We show that for sufficiently long strings the optimal alignment corresponding to a Longest Common Subsequence (LCS) treats the inserted block very differently depending on the size of the alphabet. For two-letter alphabets, the long constant block gets mainly aligned with the same symbol from the other string, while for three or more letters the opposite is true and the block gets mainly aligned with gaps. We further provide simulation results on the proportion of gaps in blocks of various lengths. In our simulations, the blocks are "regular blocks" in an iid sequence, and are not artificially inserted. Nonetheless, we observe for these natural blocks a phenomenon similar to the one shown in case of artificially-inserted blocks: with two letters, the long blocks get aligned with a smaller proportion of gaps; for three or more letters, the opposite is true. It thus appears that the microscopic nature of two-letter optimal alignments and three-letter optimal alignments are entirely different from each other.Comment: To appear: Journal of Statistical Physic

Evolutionary dynamics on strongly correlated fitness landscapes

We study the evolutionary dynamics of a maladapted population of self-replicating sequences on strongly correlated fitness landscapes. Each sequence is assumed to be composed of blocks of equal length and its fitness is given by a linear combination of four independent block fitnesses. A mutation affects the fitness contribution of a single block leaving the other blocks unchanged and hence inducing correlations between the parent and mutant fitness. On such strongly correlated fitness landscapes, we calculate the dynamical properties like the number of jumps in the most populated sequence and the temporal distribution of the last jump which is shown to exhibit a inverse square dependence as in evolution on uncorrelated fitness landscapes. We also obtain exact results for the distribution of records and extremes for correlated random variables

Novel Fmoc-Polyamino Acids for Solid-Phase Synthesis of Defined Polyamidoamines

A versatile solid-phase approach to sequence-defined polyamidoamines was developed. Four different Fmoc-polyamino acid building blocks were synthesized by selective protection of symmetrical oligoethylenimine precursors followed by introduction of a carboxylic acid handle using cyclic anhydrides and subsequent Fmoc-protection. The novel Fmoc-polyamino acids were used to construct polyamidoamines demonstrating complete compatibility to standard Fmoc reaction conditions. The straightforward synthesis of the building blocks and the high efficiency of the solid-phase coupling reactions allow the versatile synthesis of defined polycations

A Central Limit Theorem for non-overlapping return times

Define the non-overlapping return time of a random process to be the number of blocks that we wait before a particular block reappears. We prove a Central Limit Theorem based on these return times. This result has applications to entropy estimation, and to the problem of determining if digits have come from an independent equidistribted sequence. In the case of an equidistributed sequence, we use an argument based on negative association to prove convergence under weaker conditions