Probabilistic growth of large entangled states with low error accumulation

The creation of complex entangled states, resources that enable quantum computation, can be achieved via simple 'probabilistic' operations which are individually likely to fail. However, typical proposals exploiting this idea carry a severe overhead in terms of the accumulation of errors. Here we describe an method that can rapidly generate large entangled states with an error accumulation that depends only logarithmically on the failure probability. We find that the approach may be practical for success rates in the sub-10% range, while ultimately becoming unfeasible at lower rates. The assumptions that we make, including parallelism and high connectivity, are appropriate for real systems including measurement-induced entanglement. This result therefore shows the feasibility for real devices based on such an approach.Comment: 5 pages, 3 figure

Fully fault tolerant quantum computation with non-deterministic gates

In certain approaches to quantum computing the operations between qubits are non-deterministic and likely to fail. For example, a distributed quantum processor would achieve scalability by networking together many small components; operations between components should assumed to be failure prone. In the logical limit of this architecture each component contains only one qubit. Here we derive thresholds for fault tolerant quantum computation under such extreme paradigms. We find that computation is supported for remarkably high failure rates (exceeding 90%) providing that failures are heralded, meanwhile the rate of unknown errors should not exceed 2 in 10^4 operations.Comment: 5 pages, 3 fig

The reconstruction of digital holograms on a computational grid

Digital holography is greatly extending the range ofholography's applications and moving it from the lab into the field: a single CCD or other solid-state sensor can capture any number of holograms while numerical reconstruction within a computer eliminates the need for chemical development and readily allows further processing and visualisation of the holographic image. The steady increase in sensor pixel count leads to the possibilities of larger sample volumes, while smaller-area pixels enable the practical use of digital off-axis holography. However this increase in pixel count also drives a corresponding expansion of the computational effort needed to numerically reconstruct such holograms to an extent where the reconstruction process for a single depth slice takes significantly longer than the capture process for each single hologram. Grid computing - a recent innovation in large-scale distributed processing - provides a convenient means of harnessing significant computing resources in an ad-hoc fashion that might match the field deployment of a holographic instrument. We describe here the reconstruction of digital holograms on a trans-national computational Grid with over 10 000 nodes available at over 100 sites. A simplistic scheme of deployment was found to provide no computational advantage over a single powerful workstation. Based on these experiences we suggest an improved strategy for workflow and job execution for the replay ofdigital holograms on a Grid

Polarization studies in electromagnetic scattering by small Solar-system particles

In remote-sensing studies, particles that are comparable to the wavelength exhibit characteristic features in electromagnetic scattering, especially in the degree of linear polarization. These features vary with the physical properties of the particles, such as shape, size, refractive index, and orientation. In the thesis, the direct problem of computing the unknown scattered quantities using the known properties of the particles and the incident radiation is solved at both optical and radar spectral regions in a unique way. The internal electromagnetic fields of wavelength-scale particles are analyzed by using both novel and established methods to show how the internal fields are related to the scattered fields in the far zone. This is achieved by using the tools and methods that were developed specifically to reveal the internal field structure of particles and to study the mechanisms that relate the structure to the scattering characteristics of those particles. It is shown that, for spherical particles, the internal field is a combination of a forward propagating wave with the apparent wavelength determined by the refractive index of the particle, and a standing wave pattern with the apparent wavelength the same as for the incident wave. Due to the surface curvature and dielectric nature of the particle, the incident wave front undergoes a phase shift, and the resulting internal wave is focused mostly at the forward part of the particle similar to an optical lens. This focusing is also seen for irregular particles. It is concluded that, for both spherical and nonspherical particles, the interference at the far field between the partial waves that originate from these concentrated areas in the particle interior, is responsible for the specific polarization features that are common for wavelength-scale particles, such as negative values and local extrema in the degree of linear polarization, asymmetry of the phase function, and enhancement of intensity near the backscattering direction. The papers presented in this thesis solve the direct problem for particles with both simple and irregular shapes to demonstrate that these interference mechanisms are common for all dielectric wavelength-scale particles. Furthermore, it is shown that these mechanisms can be applied to both regolith particles in the optical wavelengths and hydrometeors at microwave frequencies. An advantage from this kind of study is that it does not matter whether the observation is active (e.g., polarimetric radar) or passive (e.g., optical telescope). In both cases, the internal field is computed for two mutually perpendicular incident polarizations, so that the polarization characteristics can then be analyzed according to the relation between these fields and the scattered far field.Kaukokartoitustutkimuksissa aallonpituusluokkaa olevat hiukkaset aiheuttavat niille luonteenomaisia piirteitä sähkömagnettisessa säteilyssä, varsinkin lineaarisen polarisaation asteessa. Nämä piirteet vaihtelevat hiukkasen fyysisten ominaisuuksien, kuten muodon, koon, taitekertoimen ja orientaation myötä. Tässä väitöskirjassa ratkaistaan sähkömagneettisen sironnan suora ongelma uudella tavalla, samalla kun hiukkasten ominaisuudet oletetaan tunnetuiksi. Aallonpituusluokkaa olevien hiukkasten sisäisiä sähkökenttiä analysoidaan sekä uusilla että vakiintuneilla menetelmillä, jotta voidaan osoittaa, mikä on sisäisten kenttien suhde sironneisiin kenttiin kauko-alueessa. Tämä on saavutettu käyttämällä työkaluja ja menetelmiä, jotka on kehitetty paljastamaan sirottajien sisäisen kentän rakenne ja joilla voidaan tutkia mekanismeja, jotka liittävät näiden sirottajien rakenteen niiden sirontaominaisuuksiin. Tutkimuksessa näytetään, että pallomaisten hiukkasten sisäinen kenttä on yhdistelmä eteenpäin etenevää aaltoa, jonka allonpituus määräytyy hiukkasen taitekertoimen mukaan, ja seisovaa aaltoa, jonka aallonpituus on sama kuin tulevan aallon. Koska hiukkanen on eriste ja sen pinta on kaareva, tuleva aaltorintama kokee vaihesiirron ja tuloksena oleva sisäinen aalto fokusoituu pääasiassa hiukkasen etupuolelle optisen linssin tavoin. Tämä fokusointi havaitaan myös epäsäännöllisillä hiukkasilla. Johtopäätöksenä on, sekä pallomaisille että ei-pallomaisille hiukkasille, että kaukokentässä tapahtuva interferenssi osittaisten aaltojen välillä, jotka ovat peräisin näistä fokusoituneista alueista hiukkasen sisällä, on vastuussa tietyistä, aallonpituusluokkaa oleville hiukkasille ominaisista piirteistä lineaarisessa polarisaatiossa, kuten negatiiviset arvot ja paikalliset maksimit, vaihefunktion asymmetria, ja intensiteetin kasvaminen lähellä takaisinsirontasuuntaa. Tässä väitöskirjassa esitellyt paperit ratkaisevat suoran ongelman, sekä yksinkertaisille, että epäsäännöllisille hiukkasille osoittaakseen, että nämä interferenssimekanismit ovat yhteisiä kaikille aallonpituusluokkaa oleville, eristäville sirottajille. Lisäksi näytetään, että näitä mekanismeja voidaan soveltaa sekä regoliittihiukkasille näkyvän valon alueella että hydrometeoriiteille mikroaaltoalueessa. Yksi tällaisen tutkimuksen eduista on, että ei ole merkitystä, onko havaitsija aktiivinen (esim. polarisaatiotutka) vai passiivinen (esim. optinen teleskooppi). Molemmissa tapauksissa sisäinen kenttä lasketaan kahdelle keskenään kohtisuorasti polarisoituneelle tulevalle kentälle, jotta polarisaatiossa havaitut piirteet voidaan analysoida näiden kenttien ja sironneen kentän suhteen avulla

Developing Mathematics Enrichment Workshops for Middle School Students: Philosophy and Sample Workshops

This paper describes our approach to organizing enrichment activities using advanced mathematics topics for diverse audiences of middle school students. We discuss our philosophy and approaches for the structure of these workshops, and then provide sample schedules and resource materials. The workshops cover activities on the following topics: Graphing Calculators; The Chaos Game; Statistical Sampling; CT Scans–the reconstruction problem; The Platonic and Archimedean solids; The Shape of Space; Symmetry; The Binary Number System and the game of NIM; Graph Theory: Proof by Counterexample