Efficiency of Producing Random Unitary Matrices with Quantum Circuits

We study the scaling of the convergence of several statistical properties of a recently introduced random unitary circuit ensemble towards their limits given by the circular unitary ensemble (CUE). Our study includes the full distribution of the absolute square of a matrix element, moments of that distribution up to order eight, as well as correlators containing up to 16 matrix elements in a given column of the unitary matrices. Our numerical scaling analysis shows that all of these quantities can be reproduced efficiently, with a number of random gates which scales at most as $n_q\log (n_q/\epsilon)$ with the number of qubits $n_q$ for a given fixed precision $\epsilon$. This suggests that quantities which require an exponentially large number of gates are of more complex nature.Comment: 18 pages, 10 figure

Exploring pure quantum states with maximally mixed reductions

We investigate multipartite entanglement for composite quantum systems in a pure state. Using the generalized Bloch representation for n-qubit states, we express the condition that all k-qubit reductions of the whole system are maximally mixed, reflecting maximum bipartite entanglement across all k vs. n-k bipartitions. As a special case, we examine the class of balanced pure states, which are constructed from a subset of the Pauli group P_n that is isomorphic to Z_2^n. This makes a connection with the theory of quantum error-correcting codes and provides bounds on the largest allowed k for fixed n. In particular, the ratio k/n can be lower and upper bounded in the asymptotic regime, implying that there must exist multipartite entangled states with at least k=0.189 n when $n\to \infty$. We also analyze symmetric states as another natural class of states with high multipartite entanglement and prove that, surprisingly, they cannot have all maximally mixed k-qubit reductions with k>1. Thus, measured through bipartite entanglement across all bipartitions, symmetric states cannot exhibit large entanglement. However, we show that the permutation symmetry only constrains some components of the generalized Bloch vector, so that very specific patterns in this vector may be allowed even though k>1 is forbidden. This is illustrated numerically for a few symmetric states that maximize geometric entanglement, revealing some interesting structures.Comment: 10 pages, 2 figure

Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections

We consider a stochastic partial differential equation with two logarithmic nonlinearities, with two reflections at 1 and -1 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche, Gouden\`ege and Zambotti, we obtain existence and uniqueness of solution for initial conditions in the interval $(-1,1)$. Finally, we prove that the unique invariant measure is ergodic, and we give a result of exponential mixing

Palvelusetelin ja ostopalvelusopimuksen erot yksityiselle päiväkodille

Opinnäytetyössä on tutkittu palvelusetelin ja ostopalvelusopimuksen erojen vaikutuksia yksityiselle päiväkodille. Opinnäytetyössä on pyritty tutkimaan yksityisten päiväkotien toiminnan yleisimpien asioiden erilaisuutta ostopalvelumallissa sekä palvelusetelimallissa. Viralliset tutkimuskysymykset olivat 1) Mitä tarkoittavat ostopalvelusopimus ja palveluseteli päivähoidossa? 2) Kuinka ostopalvelusopimus ja palveluseteli vaikuttavat yksityisessä päiväkodissa? 3) Kuinka ostopalvelusopimuksen ja palvelusetelin eroavaisuudet vaikuttavat yksityiseen päiväkotiin? Työn tavoitteena oli saada selkeä kuva toimintamallien eroavaisuuksista. Työn empiiriseen osaan haastateltiin ostopalvelupäiväkodin johtajaa sekä palvelusetelipäiväkodin johtajaa. Haastattelut suoritettiin teemahaastatteluina. Haastatteluista kävi selkeästi ilmi suurimmat eroavaisuudet ostopalvelusopimuksen ja palvelusetelin välillä. Toimintaan vaikuttavia tekijöitä rajattiin kuuteen eri osa-alueeseen. Osa-alueita olivat: lasten valinta, rahalliset vaikutukset, vastuu ja valvonta, kilpailu, esiopetus, päivähoidossa olevien lasten tuen tarpeet sekä uhkat. Tuloksista käy ilmi toimintaan eniten vaikuttavat tekijät. Molemmissa malleissa oli hyviä ja huonoja puolia. Toimintaan eniten vaikuttavia eroavaisuuksia ostopalvelusopimusmallin ja palvelusetelimallin välillä olivat rahalliset vaikutukset ja kilpailu. Rahalliset vaikutukset käsiteltiin yleiseltä pohjalta opinnäytetyössä ja aihe olisi hyvä jatkotutkimuksen kohde. Pienen yksityisen päiväkodin toimintaan vaikutti paljon enemmän eroavaisuudet kuin suuren yksityisen päiväkotiketjun. Pienet yksityiset päiväkodit pysyivät parhaiten markkinoilla tarjoamalla perheille erilaista pedagogiikkaa ja erilaisia arvoja

Statistique de l'interférence quantique et circuits quantiques aléatoires

Cette thèse présente différents résultats portant sur deux thèmes précis dans le domaine de l'information quantique. Le premier de ces thèmes concerne l'interférence présente dans les algorithmes quantiques. Par l'intermédiaire d’une mesure récemment introduite dans la littérature, il est possible de quantifier l'interférence présente dans tout algorithme quantique, dans le but de vérifier si il existe un lien entre interférence et ressource computationnelle. Pour ce faire, deux types de modèles statistiques d'algorithmes quantiques ont été utilisés. Le premier type, issu de la théorie des matrices aléatoires, est l'ensemble circulaire unitaire(CUE) tandis que le second est un ensemble de circuits quantiques, construits par des séquences aléatoires de portes quantiques. Les résultats analytiques et numériques obtenus dans cette thèse montrent qu'en moyenne tout algorithme quantique contient une grande quantité d'interférence. L'étape supplémentaire fût d'étudier l'influence de la décohérence engendrée par un bain thermique sur le comportement statistique de l'interférence. Grâce à l'utilisation de méthodes mathématiques d'intégrations sur le groupe unitaire U (N ), il est possible de généraliser les résultats analytiques et numériques précédents pour inclure les effets de la décohérence. Le deuxième thème étudié concerne la possibilité d'utiliser des algorithmes quantiques pour créer efficacement des ensembles de matrices aléatoires. Pendant les travaux sur l'interférence, une équivalence entre CUE et le modèle de circuits quantiques aléatoires fût observée. On peut attendre que de tels circuits quantiques aléatoires puissent être utilisé pour construire des matrices aléatoires distribuées comme CUE. Les résultats numériques de cette thèse montrent que certaines quantités statistiques propres à CUE sont bien reproduites par le modèle de séquences aléatoires, ceci de manière efficace. L'efficacité signifie que les séquences sont constituées d'un nombre de portes qui augmente comme le logarithme de la taille des matrices produites. Ces résultats sont en parfait accord avec des travaux analytiques récemment publiés.This thesis presents different results about two topics in the field of quantum information. The first of these topics concerns the interference present in quantum algorithms. Thanks to a measure recently introduce in the litterature, it is possible to quantify the interference present in any quantum algorithm, in order to check if there is a link between interference and computationnal ressource. To do this, two kinds of statistical models of quantum algorithms were used. The first one, which comes from the random matrix theory, is the circular unitary ensemble(CUE) while the second one is a set of quantum circuits, built by random sequences of quantum gates.The analytical and numerical results obtained in this thesis show that on average anyquantum algorithm contain a big amount of interference. The next step was to study the influence of the decoherence created by a thermal bath on the statistics of the interference. Thanks to mathematical methods of integration over the unitary group U (N ), it is possible to generalise the previous analytical and numerical results to include the effect of decoherence. The second topic is about the possibility of using quantum algorithms to create efficiently random matrix ensembles. During the works on interference, an equivalence between CUE and the model of random quantum circuits was noticed. One might expect that such random quantum circuits can be used to build random matrix ensembles drawn as CUE. The numerical results of this thesis show that given CUE quantities are well reproduced by the model of random sequences in an efficient way. The efficiency means that the sequences are built with a number of gates which increases as the logarithm of the size of the produced matrices. These results are in agreement with recently published works

Thermo-economic optimization during preliminary design phase of organic Rankine cycle systems for waste heat recovery from exhaust and recirculated gases of heavy duty trucks

peer reviewedWaste heat recovery (WHR) Organic Rankine Cycle (ORC) system is a very promising technology for reducing fuel consumption and consequently the CO2 emissions of future heavy-duty trucks (HDT). Nonetheless, the adoption of this technology in the automotive domain requires specific R&D activities going from the system definition to the on-board integration. This study focuses on the preliminary design phase of ORC systems recovering the heat wasted from two of the sources available on a HDT: the exhaust and recirculated gases. From these heat sources and their combinations, 6 possible architectures are identified. On the other hand, 4 volumetric expansion machine technologies are considered (scroll, screw, piston and vane Expanders). At the end, 24 topologies are therefore modelled considering the main components (Pump, Heat exchangers, Expansion machines). A three-step optimization method is proposed to identify the most promising system. First, the most suitable conditions are identified for the design of the ORC systems using a simple model of volumetric expansion machine. In a second step, the design phase, using more detailed models for the expansion machines, a thermos-economic optimization is performed. Finally, in a third step, the output power of the latter system models is maximized in off-design conditions, optimizing the evaporating pressure and the overheating degree