Large Fourier transforms never exactly realized by braiding conformal blocks

Fourier transform is an essential ingredient in Shor's factoring algorithm. In the standard quantum circuit model with the gate set \{\U(2), \textrm{CNOT}\}, the discrete Fourier transforms $F_N=(\omega^{ij})_{N\times N},i,j=0,1,..., N-1, \omega=e^{\frac{2\pi i}{N}}$, can be realized exactly by quantum circuits of size $O(n^2), n=\textrm{log}N$, and so can the discrete sine/cosine transforms. In topological quantum computing, the simplest universal topological quantum computer is based on the Fibonacci (2+1)-topological quantum field theory (TQFT), where the standard quantum circuits are replaced by unitary transformations realized by braiding conformal blocks. We report here that the large Fourier transforms $F_N$ and the discrete sine/cosine transforms can never be realized exactly by braiding conformal blocks for a fixed TQFT. It follows that approximation is unavoidable to implement the Fourier transforms by braiding conformal blocks

Ingestive behaviour and physiology of the medicinal leech

Ingestion lasts 25 min in Hirudo medicinalis and is characterized by pharyngeal peristalsis which fills the crop. This peristalsis has an initial rate of 2.4 Hz which decays smoothly to 1.2 Hz at termination of ingestion. During ingestion, the leech body wall undergoes peristalsis which appears to aid in filling the crop diverticula. Body peristalsis begins at a rate of 10 min^(-1) and decreases linearly to 2 min^(-1) at termination. The body also undergoes dorsoventral flexions when blood flow is occluded. Blood meal size increases slightly with leech size: 8.4 g for 1-g leeches and 9.7 g for 2-g leeches. However, relative meal size decreases markedly with increasing animal size; from 8.15 times body mass for 1-g to 4.80 times for 2-g leeches. When intact leeches were exposed to micromolar concentrations of serotonin, there was an increase in the rate of pharyngeal peristalsis and the size of the blood meals. Leeches excrete the plasma from their ingested blood meals. Excretion is activated during ingestion, which increases feeding efficiency by increasing the proportion of blood cells in the ingestate. Excretion continues for 4–6 days following ingestion, removing all the remaining plasma from the ingestate. Leech ingestion comprises stereotyped muscular movements, secretion of saliva and excretion of plasma. A strikingly similar feeding physiology is seen in the blood-sucking insect Rhodnius, and we suggest that efficient sanguivory may require the convergent evolution of similar ingestive mechanisms

Constructing Functional Braids for Low-Leakage Topological Quantum Computing

We discuss how to significantly reduce leakage errors in topological quantum computation by introducing an irrelevant error in phase, using the construction of a CNOT gate in the Fibonacci anyon model as a concrete example. To be specific, we construct a functional braid in a six-anyon Hilbert space that exchanges two neighboring anyons while conserving the encoded quantum information. The leakage error is $\sim$$10^{-10}$ for a braid of $\sim$100 interchanges of anyons. Applying the braid greatly reduces the leakage error in the construction of generic controlled-rotation gates.Comment: 5 pages, 4 figures, updated, accepeted by Phys. Rev.

Lie Algebras and Suppression of Decoherence in Open Quantum Systems

Since there are many examples in which no decoherence-free subsystems exist (among them all cases where the error generators act irreducibly on the system Hilbert space), it is of interest to search for novel mechanisms which suppress decoherence in these more general cases. Drawing on recent work (quant-ph/0502153) we present three results which indicate decoherence suppression without the need for noiseless subsystems. There is a certain trade-off; our results do not necessarily apply to an arbitrary initial density matrix, or for completely generic noise parameters. On the other hand, our computational methods are novel and the result--suppression of decoherence in the error-algebra approach without noiseless subsystems--is an interesting new direction.Comment: 7 page

Energies of knot diagrams

We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals considered are the sum of two terms, the uniformization term (which tends to make the curvature of the knot uniform) and the resistance term (which, in particular, forbids crossing changes). We define an infinite family of uniformization functionals, depending on an arbitrary smooth function $f$ and study the simplest nontrivial case $f(x)=x^2$, obtaining neat normal forms (corresponding to minima of the functional) by making use of the Gauss representation of immersed curves, of the phase space of the pendulum, and of elliptic functions

SU(m) non-Abelian anyons in the Jain hierarchy of quantum Hall states

We show that different classes of topological order can be distinguished by the dynamical symmetry algebra of edge excitations. Fundamental topological order is realized when this algebra is the largest possible, the algebra of quantum area-preserving diffeomorphisms, called $W_{1+\infty}$. We argue that this order is realized in the Jain hierarchy of fractional quantum Hall states and show that it is more robust than the standard Abelian Chern-Simons order since it has a lower entanglement entropy due to the non-Abelian character of the quasi-particle anyon excitations. These behave as SU($m$) quarks, where $m$ is the number of components in the hierarchy. We propose the topological entanglement entropy as the experimental measure to detect the existence of these quantum Hall quarks. Non-Abelian anyons in the $\nu = 2/5$ fractional quantum Hall states could be the primary candidates to realize qbits for topological quantum computation.Comment: 5 pages, no figures, a few typos corrected, a reference adde

Competency-based assessment for the training of PhD students and early-career scientists.

The training of PhD students and early-career scientists is largely an apprenticeship in which the trainee associates with an expert to become an independent scientist. But when is a PhD student ready to graduate, a postdoctoral scholar ready for an independent position, or an early-career scientist ready for advanced responsibilities? Research training by apprenticeship does not uniformly include a framework to assess if the trainee is equipped with the complex knowledge, skills and attitudes required to be a successful scientist in the 21st century. To address this problem, we propose competency-based assessment throughout the continuum of training to evaluate more objectively the development of PhD students and early-career scientists. © 2018, Verderame et al

Entanglement of Sections, Examples Looking for a Theory

Quantum information is about the entanglement of states. To this starting point we add parameters whereby a single state becomes a non-vanishing section of a bundle. We consider through examples the possible entanglement patterns of sections.Comment: 15 pages, 0 figure