Macroeconomic Forecasting and Business Cycle Analysis with Nonlinear Models

The Great Recession of 2008/09 led to a sharp and persistent decline in real output growth in many countries, resulting in a global economic crisis. This economic crisis had its worst slump in economic activity in late 2008 and early 2009. At this time economists failed to predict the economic slump, though some economic indicators have shown first stirrings for the sharp downturn and thus, a severe recession. This failure raises two important questions: First, did economists use improper forecasting models? For instance, did these forecasting models not include information about the current state of the business cycle? Or did these forecasting models not account for the nonlinear dynamics inherent in the business cycle due to expansions and recessions? Second, did economists select the most informative economic indicators for their forecasting models at that time? Not all economic indicators are available at the same frequency and many models can not handle the frequency mismatch in the data. Furthermore, not all recessions are of the same origins, e.g. they might stem from financial, demand or supply shocks. Hence, the importance of economic indicators for predicting recessions might change over time and economist need an appropriate variable selection method. I examine in three independent articles different approaches to address these aspects and the problems associated with them in a real-time data analysis

Effect of the iron valence in the two types of layers in LiFeO2_2Fe2_2Se2_2

We perform electronic structure calculations for the recently synthesized iron-based superconductor LiFeO$_2$Fe$_2$Se$_2$. In contrast to other iron-based superconductors, this material comprises two different iron atoms in 3$d^5$ and 3$d^6$ configurations. In band theory, both contribute to the low-energy electronic structure. Spin-polarized density functional theory calculations predict an antiferromagnetic metallic ground state with different moments on the two Fe sites. However, several other almost degenerate magnetic configurations exist. Due to their different valences, the two iron atoms behave very differently when local quantum correlations are included through the dynamical mean-field theory. The contributions from the half-filled 3$d^5$ atoms in the LiFeO$_2$ layer are suppressed and the 3$d^6$ states from the FeSe layer restore the standard iron-based superconductor fermiology.Comment: 9 pages, 11 figure

Time-optimal multi-qubit gates: Complexity, efficient heuristic and gate-time bounds

Multi-qubit interactions are omnipresent in quantum computing hardware, and they can generate multi-qubit entangling gates. Such gates promise advantages over traditional two-qubit gates. In this work, we focus on the quantum gate synthesis with multi-qubit Ising-type interactions and single-qubit gates. These interactions can generate global ZZ-gates (GZZ gates). We show that the synthesis of time-optimal multi-qubit gates is NP-hard. However, under certain assumptions we provide explicit constructions of time-optimal multi-qubit gates allowing for efficient synthesis. These constructed multi-qubit gates have a constant gate time and can be implemented with linear single-qubit gate layers. Moreover, a heuristic algorithm with polynomial runtime for synthesizing fast multi-qubit gates is provided. Finally, we prove lower and upper bounds on the optimal GZZ gate-time. Furthermore, we conjecture that any GZZ gate can be executed in a time O(n) for n qubits. We support this claim with theoretical and numerical results.Comment: 11+2 pages, 2 figure

The axiomatic and the operational approaches to resource theories of magic do not coincide

Stabiliser operations occupy a prominent role in the theory of fault-tolerant quantum computing. They are defined operationally: by the use of Clifford gates, Pauli measurements and classical control. Within the stabiliser formalism, these operations can be efficiently simulated on a classical computer, a result which is known as the Gottesman-Knill theorem. However, an additional supply of magic states is enough to promote them to a universal, fault-tolerant model for quantum computing. To quantify the needed resources in terms of magic states, a resource theory of magic has been developed during the last years. Stabiliser operations (SO) are considered free within this theory, however they are not the most general class of free operations. From an axiomatic point of view, these are the completely stabiliser-preserving (CSP) channels, defined as those that preserve the convex hull of stabiliser states. It has been an open problem to decide whether these two definitions lead to the same class of operations. In this work, we answer this question in the negative, by constructing an explicit counter-example. This indicates that recently proposed stabiliser-based simulation techniques of CSP maps are strictly more powerful than Gottesman-Knill-like methods. The result is analogous to a well-known fact in entanglement theory, namely that there is a gap between the class of local operations and classical communication (LOCC) and the class of separable channels. Along the way, we develop a number of auxiliary techniques which allow us to better characterise the set of CSP channels.Comment: 10 pages + 12 pages appendix, 2 figures. Fixed grammar mistake in title. Added one and two-qubit case. Corrected wrong equatio

General guarantees for randomized benchmarking with random quantum circuits

In its many variants, randomized benchmarking (RB) is a broadly used technique for assessing the quality of gate implementations on quantum computers. A detailed theoretical understanding and general guarantees exist for the functioning and interpretation of RB protocols if the gates under scrutiny are drawn uniformly at random from a compact group. In contrast, many practically attractive and scalable RB protocols implement random quantum circuits with local gates randomly drawn from some gate-set. Despite their abundance in practice, for those non-uniform RB protocols, general guarantees under experimentally plausible assumptions are missing. In this work, we derive such guarantees for a large class of RB protocols for random circuits that we refer to as filtered RB. Prominent examples include linear cross-entropy benchmarking, character benchmarking, Pauli-noise tomography and variants of simultaneous RB. Building upon recent results for random circuits, we show that many relevant filtered RB schemes can be realized with random quantum circuits in linear depth, and we provide explicit small constants for common instances. We further derive general sample complexity bounds for filtered RB. We show filtered RB to be sample-efficient for several relevant groups, including protocols addressing higher-order cross-talk. Our theory for non-uniform filtered RB is, in principle, flexible enough to design new protocols for non-universal and analog quantum simulators.Comment: 77 pages, 3 figures. Accepted for a talk at QIP 202

Totalsynthese von (S)-Halitulin, (S)-Haliclorensin und verwandten Alkaloiden und Aminosäuren

Die vorliegende Arbeit beschäftigt sich mit der Totalsynthese der Meeresalkaloide Haliclorensin und Halitulin. Die verschiedenen Synthesestrategien, die zum Aufbau der Halitulin-Struktur verfolgt werden, basieren teilweise auf Überlegungen zur Biosynthese. Neben den Synthesezielen konnten deshalb zahlreiche neuartige Alkaloide und Aminosäuren synthetisiert werden, die als Biosynthesevorläufer von Halitulin denkbar sind. Sie stehen als Testsubstanzen für die Struktur-Aktivitäts-Optimierung des hochwirksamen Cytostatikums Halitulin zur Verfügung. Entscheidend für die erfolgreiche Synthese vieler in dieser Arbeit beschriebener Verbindungen sind zwei neu entwickelte Varianten altbekannter Reaktionen. So lassen sich 7,8-Dihydroxychinoline aus 3-Aminocatecholen durch eine milde Variante der Skraup- Reaktion herstellen. Der effektive Zugang zu den bisher nur aufwendig synthetisierbaren 3- Aminocatecholen wird durch eine modifizierte Baeyer-Villiger-Oxidation ermöglicht. Das letzte Kapitel des speziellen Teils beschäftigt sich mit der katalytischen Wirkung von neuartigen 4-Aminopyridin-Derivaten, die in einer Reihe mit den weithin bekannten und vielseitig eingesetzten Katalysatoren 4-(Dimethylamino)pyridin (DMAP) und 4-Pyrrolidinopyridin (PPY) stehen. Es konnte gezeigt werden, dass − verglichen mit dem bisher wirksamsten Katalysator PPY − weitere Effektivitätssteigerungen möglich sind