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Despite their remarkable capabilities, LLMs learn word representations that exhibit the undesirable yet poorly understood feature of anisotropy. In this paper, we argue that the second moment in Adam is a cause of anisotropic embeddings, and suggest a modified optimizer called Coupled Adam to mitigate the problem. Our experiments demonstrate that Coupled Adam significantly improves the quality of embeddings, while also leading to better upstream and downstream performance on large enough datasets., Comment: 17 pages, 8 figures; figures corrected

Advances in quantum algorithms suggest a tentative scaling advantage on certain combinatorial optimization problems. Recent work, however, has also reinforced the idea that barren plateaus render variational algorithms ineffective on large Hilbert spaces. Hence, finding annealing protocols by variation ultimately appears to be difficult. Similarly, the adiabatic theorem fails on hard problem instances with first-order quantum phase transitions. Here, we show how to use the spin coherent-state path integral to shape the geometry of quantum adiabatic evolution, leading to annealing protocols at polynomial overhead that provide orders-of-magnitude improvements in the probability to measure optimal solutions, relative to linear protocols. These improvements are not obtained on a controllable toy problem but on randomly generated hard instances (Sherrington-Kirkpatrick and Maximum 2-Satisfiability), making them generic and robust. Our method works for large systems and may thus be used to improve the performance of state-of-the-art quantum devices., Comment: 12 pages, 11 figures

As structural and functional materials, topologically close-packed (TCP) phases of transition metal compounds offer a wide range of attractive properties. Due to their complex crystal structure and the resulting brittleness, the knowledge on their mechanical behaviour is still very limited, especially below the brittle-to-ductile transition temperature. In this study, we systematically analyse the influence of composition and crystal structure on the mechanical properties and deformation mechanisms in the binary Ta-Fe system as well as in the ternary Ta-Fe-Al system as both systems contain a hexagonal C14 Laves and a mu-phase. We use nanoindentation, slip trace analysis and transmission electron microscopy to study the influence of crystal structure, composition and crystal orientation. The composition strongly influences the indentation modulus in the binary Ta-Fe system, showing a decreasing trend with increasing Ta content. The addition of Al, however, does not lead to a significant change of the mechanical properties of the ternary TCP phases. The investigation of the deformation mechanisms revealed that the Laves phase primarily deforms via non-basal slip while the basal plane is the favoured slip plane in the mu-phase. By partly replacing Fe with Al, the plasticity is not affected strongly, but the proportion of non-basal slip slightly increases for both ternary TCP phases compared to the binary ones.

Intermetallics, which encompass a wide range of compounds, often exhibit similar or closely related crystal structures, resulting in various intermetallic systems with structurally derivative phases. This study examines the hypothesis that deformation behavior can be transferred from fundamental building blocks to structurally related phases using the binary samarium-cobalt system. We investigate SmCo$_2$ and SmCo$_5$ as fundamental building blocks and compare them to the structurally related SmCo$_3$ and Sm$_2$Co$_{17}$ phases. Nanoindentation and micropillar compression tests were performed to characterize the primary slip systems, complemented by generalized stacking fault energy calculations via atomic-scale modeling. Our results show that while elastic properties of the structurally complex phases follow a rule of mixtures, their plastic deformation mechanisms are more intricate, influenced by the stacking and bonding nature within the crystal's building blocks. These findings underscore the importance of local bonding environments in predicting the mechanical behavior of structurally related intermetallics, providing crucial insights for the development of high-performance intermetallic materials.

This paper introduces a structure-inspired ansatz for addressing quadratic unconstrained binary optimization problems with the Variational Quantum Eigensolver. We propose a novel warm start technique that is based on imaginary time evolution, and allows for determining a set of initial parameters prioritizing lower energy states in a resource-efficient way. Using classical simulations, we demonstrate that this warm start method significantly improves the success rate and reduces the number of iterations required for the convergence of Variational Quantum Eigensolver. The numerical results also indicate that the warm start approach effectively mitigates statistical errors arising from a finite number of measurements, and to a certain extent alleviates the effect of barren plateaus.

Tensor network algorithms can efficiently simulate complex quantum many-body systems by utilizing knowledge of their structure and entanglement. These methodologies have been adapted recently for solving the Navier-Stokes equations, which describe a spectrum of fluid phenomena, from the aerodynamics of vehicles to weather patterns. Within this quantum-inspired paradigm, velocity is encoded as matrix product states (MPS), effectively harnessing the analogy between interscale correlations of fluid dynamics and entanglement in quantum many-body physics. This particular tensor structure is also called quantics tensor train (QTT). By utilizing NVIDIA's cuQuantum library to perform parallel tensor computations on GPUs, our adaptation speeds up simulations by up to 12.1 times. This allows us to study the algorithm in terms of its applicability, scalability, and performance. By simulating two qualitatively different but commonly encountered 2D flow problems at high Reynolds numbers up to $1\times10^7$ using a fourth-order time stepping scheme, we find that the algorithm has a potential advantage over direct numerical simulations in the turbulent regime as the requirements for grid resolution increase drastically. In addition, we derive the scaling $\chi=\mathcal{O}(\text{poly}(1/\epsilon))$ for the maximum bond dimension $\chi$ of MPS representing turbulent flow fields, with an error $\epsilon$, based on the spectral distribution of turbulent kinetic energy. Our findings motivate further exploration of related quantum algorithms and other tensor network methods.

Parameterized quantum circuits are attractive candidates for potential quantum advantage in the near term and beyond. At the same time, as quantum computing hardware not only continues to improve but also begins to incorporate new features such as mid-circuit measurement and adaptive control, opportunities arise for innovative algorithmic paradigms. In this work we focus on measurement-based quantum computing protocols for approximate optimization, in particular related to quantum alternating operator ans\"atze (QAOA), a popular quantum circuit model approach to combinatorial optimization. For the construction and analysis of our measurement-based protocols we demonstrate that diagrammatic approaches, specifically ZX-calculus and its extensions, are effective for adapting such algorithms to the measurement-based setting. In particular we derive measurement patterns for applying QAOA to the broad and important class of QUBO problems. We further outline how for constrained optimization, hard problem constraints may be directly incorporated into our protocol to guarantee the feasibility of the solution found and avoid the need for dealing with penalties. Finally we discuss the resource requirements and tradeoffs of our approach to that of more traditional quantum circuits., Comment: To appear in the proceedings of the 2024 IPDPS Workshop on Quantum Computing Algorithms, Systems, and Applications (Q-CASA)

Quantum computing (QC) and machine learning (ML), taken individually or combined into quantum-assisted ML (QML), are ascending computing paradigms whose calculations come with huge potential for speedup, increase in precision, and resource reductions. Likely improvements for numerical simulations in engineering imply the possibility of a strong economic impact on the manufacturing industry. In this project report, we propose a framework for a quantum computing-enhanced service ecosystem for simulation in manufacturing, consisting of various layers ranging from hardware to algorithms to service and organizational layers. In addition, we give insight into the current state of the art of applications research based on QC and QML, both from a scientific and an industrial point of view. We further analyse two high-value use cases with the aim of a quantitative evaluation of these new computing paradigms for industrially-relevant settings., Comment: 11 pages, 3 figures. Accepted version

The Quantum Approximate Optimization Algorithm (QAOA) is suggested as a promising application on early quantum computers. Here, a quantum-inspired classical algorithm, the mean-field Approximate Optimization Algorithm (mean-field AOA), is developed by replacing the quantum evolution of the QAOA with classical spin dynamics through the mean-field approximation. Due to the alternating structure of the QAOA, this classical dynamics can be found exactly for any number of QAOA layers. We benchmark its performance against the QAOA on the Sherrington-Kirkpatrick (SK) model and the partition problem, and find that the mean-field AOA outperforms the QAOA in both cases for most instances. Our algorithm can thus serve as a tool to delineate optimization problems that can be solved classically from those that cannot, i.e. we believe that it will help to identify instances where a true quantum advantage can be expected from the QAOA. To quantify quantum fluctuations around the mean-field trajectories, we solve an effective scattering problem in time, which is characterized by a spectrum of time-dependent Lyapunov exponents. These provide an indicator for the hardness of a given optimization problem relative to the mean-field AOA., Comment: 17 pages, 8 figures

We investigate the performance of the variational quantum eigensolver (VQE) for the optimal flight gate assignment problem. This problem is a combinatorial optimization problem that aims at finding an optimal assignment of flights to the gates of an airport, in order to minimize the passenger travel time. To study the problem, we adopt a qubit-efficient binary encoding with a cyclic mapping, which is suitable for a digital quantum computer. Using this encoding in conjunction with the Conditional Value at Risk (CVaR) as an aggregation function, we systematically explore the performance of the approach by classically simulating the CVaR-VQE. Our results indicate that the method allows for finding a good solution with high probability, and the method significantly outperforms the naive VQE approach. We examine the role of entanglement for the performance, and find that ans\"atze with entangling gates allow for better results than pure product states. Studying the problem for various sizes, our numerical data show that the scaling of the number of cost function calls for obtaining a good solution is not exponential for the regimes we investigate in this work.

Diagrammatic representations of quantum algorithms and circuits offer novel approaches to their design and analysis. In this work, we describe extensions of the ZX-calculus especially suitable for parameterized quantum circuits, in particular for computing observable expectation values as functions of or for fixed parameters, which are important algorithmic quantities in a variety of applications ranging from combinatorial optimization to quantum chemistry. We provide several new ZX-diagram rewrite rules and generalizations for this setting. In particular, we give formal rules for dealing with linear combinations of ZX-diagrams, where the relative complex-valued scale factors of each diagram must be kept track of, in contrast to most previously studied single-diagram realizations where these coefficients can be effectively ignored. This allows us to directly import a number useful relations from the operator analysis to ZX-calculus setting, including causal cone and quantum gate commutation rules. We demonstrate that the diagrammatic approach offers useful insights into algorithm structure and performance by considering several ansatze from the literature including realizations of hardware-efficient ansatze and QAOA. We find that by employing a diagrammatic representation, calculations across different ansatze can become more intuitive and potentially easier to approach systematically than by alternative means. Finally, we outline how diagrammatic approaches may aid in the design and study of new and more effective quantum circuit ansatze., Comment: In Proceedings QPL 2022, arXiv:2311.08375

In a previous version of this document we misinterpreted the runtime of a part of the described algorithm. Indeed, the runtime is not better than the Grover-Algorithm. We therefor withdraw this work. We present a novel algorithmic approach to find a proper vertex colouring of a graph with $d$ colours, if it exists. We associate a $d$-dimensional quantum system with each vertex and the initial state is a mixture of all possible colourings, from which we obtain a random proper colouring of the graph by measurements. The non-deterministic nature of the quantum measurement is tackled by a reset operation, which can revert the effect of unwanted projections. As in the classical case, we find that the runtime scales exponentially with the number of vertices. However, we provide numerical evidence that the average runtime of the problem-specific part of the algorithm scales polynomially in the number of edges., Comment: We withdraw this work, because, in contrast to what previous versions of this document suggested, the proposed algorithm is not faster than state-of-the-art approaches

In this paper we experimentally test the performance of the recently proposed domain-wall encoding of discrete variables from [Chancellor Quantum Sci. Technol. 4 045004] on Ising model flux qubit quantum annealers. We compare this encoding with the traditional one-hot methods and find that they outperform the one-hot encoding for three different problems at different sizes both of the problem and of the variables. From these results we conclude that the domain-wall encoding yields superior performance against a variety of metrics furthermore, we do not find a single metric by which one hot performs better. We even find that a 2000Q quantum annealer with a drastically less connected hardware graph but using the domain-wall encoding can outperform the next generation Advantage processor if that processor uses one-hot encoding., Comment: 16 pages, 8 figures, new analysis of binary encoding added in v2, associated data can be found at: http://doi.org/10.15128/r11j92g748d

In order to solve real world combinatorial optimization problems with a D-Wave quantum annealer it is necessary to embed the problem at hand into the D-Wave hardware graph, namely Chimera or Pegasus. Most hard real world problems exhibit a strong connectivity. For the worst case scenario of a complete graph, there exists an efficient solution for the embedding into the ideal Chimera graph. However, since real machines almost always have broken qubits it is necessary to find an embedding into the broken hardware graph. We present a new approach to the problem of embedding complete graphs into broken Chimera graphs. This problem can be formulated as an optimization problem, more precisely as a matching problem with additional linear constraints. Although being NP-hard in general it is fixed parameter tractable in the number of inaccessible vertices in the Chimera graph. We tested our exact approach on various instances of broken hardware graphs, both related to real hardware as well as randomly generated. For fixed runtime, we were able to embed larger complete graphs compared to previous, heuristic approaches. As an extension, we developed a fast heuristic algorithm which enables us to solve even larger instances. We compared the performance of our heuristic and exact approaches., Comment: 26 pages, 9 figures, 2 tables

We present a comparison study of state-of-the-art classical optimisation methods to a D-Wave 2000Q quantum annealer for the planning of Earth observation missions. The problem is to acquire high value images while obeying the attitude manoeuvring constraint of the satellite. In order to investigate close to real-world problems, we created benchmark problems by simulating realistic scenarios. Our results show that a tuned quantum annealing approach can run faster than a classical exact solver for some of the problem instances. Moreover, we find that the solution quality of the quantum annealer is comparable to the heuristic method used operationally for small problem instances, but degrades rapidly due to the limited precision of the quantum annealer.

Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue.

Optimal flight gate assignment is a highly relevant optimization problem from airport management. Among others, an important goal is the minimization of the total transit time of the passengers. The corresponding objective function is quadratic in the binary decision variables encoding the flight-to-gate assignment. Hence, it is a quadratic assignment problem being hard to solve in general. In this work we investigate the solvability of this problem with a D-Wave quantum annealer. These machines are optimizers for quadratic unconstrained optimization problems (QUBO). Therefore the flight gate assignment problem seems to be well suited for these machines. We use real world data from a mid-sized German airport as well as simulation based data to extract typical instances small enough to be amenable to the D-Wave machine. In order to mitigate precision problems, we employ bin packing on the passenger numbers to reduce the precision requirements of the extracted instances. We find that, for the instances we investigated, the bin packing has little effect on the solution quality. Hence, we were able to solve small problem instances extracted from real data with the D-Wave 2000Q quantum annealer., Comment: Updated figure 2

This paper introduces the software TiGL: TiGL is an open source high-fidelity geometry modeler that is used in the conceptual and preliminary aircraft and helicopter design phase. It creates full three-dimensional models of aircraft from their parametric CPACS description. Due to its parametric nature, it is typically used for aircraft design analysis and optimization. First, we present the use-case and architecture of TiGL. Then, we discuss it's geometry module, which is used to generate the B-spline based surfaces of the aircraft. The backbone of TiGL is its surface generator for curve network interpolation, based on Gordon surfaces. One major part of this paper explains the mathematical foundation of Gordon surfaces on B-splines and how we achieve the required curve network compatibility. Finally, TiGL's aircraft component module is introduced, which is used to create the external and internal parts of aircraft, such as wings, flaps, fuselages, engines or structural elements.

We present the mapping of a class of simplified air traffic management (ATM) problems (strategic conflict resolution) to quadratic unconstrained boolean optimization (QUBO) problems. The mapping is performed through an original representation of the conflict-resolution problem in terms of a conflict graph, where nodes of the graph represent flights and edges represent a potential conflict between flights. The representation allows a natural decomposition of a real world instance related to wind-optimal trajectories over the Atlantic ocean into smaller subproblems, that can be discretized and are amenable to be programmed in quantum annealers. In the study, we tested the new programming techniques and we benchmark the hardness of the instances using both classical solvers and the D-Wave 2X and D-Wave 2000Q quantum chip. The preliminary results show that for reasonable modeling choices the most challenging subproblems which are programmable in the current devices are solved to optimality with 99% of probability within a second of annealing time., Comment: Paper accepted for publication on: IEEE Transactions on Intelligent Transportation Systems

We present a comparative, theoretical study of the doping dependence of the critical temperature $T_C$ of the ferromagnetic insulator-metal transition in Gd-doped and O-deficient EuO, respectively. The strong $T_C$ enhancement in Eu$_{1-x}$Gd$_x$O is due to Kondo-like spin fluctuations on the Gd sites, which are absent in EuO$_{1-x}$. Moreover, we find that the $T_C$ saturation in Eu$_{1-x}$Gd$_x$O for large $x$ is due to a reduced activation of dopant electrons into the conduction band, in agreement with experiments, rather than antiferromagnetic long-range contributions of the RKKY interaction. The results shed light on possibilities for further increasing $T_C$., Comment: Published version; 3 references added

Wannier function expansions are well suited for the description of photonic- crystal-based defect structures, but constructing maximally localized Wannier functions by optimizing the phase degree of freedom of the Bloch modes is crucial for the efficiency of the approach. We systematically analyze different locality criteria for maximally localized Wannier functions in two- dimensional square and triangular lattice photonic crystals, employing (local) conjugate-gradient as well as (global) genetic-algorithm-based, stochastic methods. Besides the commonly used second moment (SM) locality measure, we introduce a new locality measure, namely the integrated modulus (IM) of the Wannier function. We show numerically that, in contrast to the SM criterion, the IM criterion leads to an optimization problem with a single extremum, thus allowing for fast and efficient construction of maximally localized Wannier functions using local optimization techniques. We also present an analytical formula for the initial choice of Bloch phases, which under certain conditions represents the global maximum of the IM criterion and, thus, further increases the optimization efficiency in the general case.