"Jeder Arbeiter ist seines Lohnes wert" : rechtliche Schranken der Lohnfestlegung

We may consider labour as boon or bane ‒ man’s existence is not conceivable without labour. The expulsion from the Garden of Eden can be understood as a punishment for the consumption of the forbidden fruit from the tree of knowledge, but it can likewise be interpreted as the perfection of God’s creation. Hence only beyond Eden, God’s creature becomes man. He or she becomes human by cultivat-ing the earth, by working. Labour is not only a necessary evil in order to secure existence, nor does it serve as a means for self-preservation, but also for self-fulfillment. Labour is the epitome of the ability of self-being and thus of man’s liberty

Grover Mixers for QAOA: Shifting Complexity from Mixer Design to State Preparation

We propose GM-QAOA, a variation of the Quantum Alternating Operator Ansatz (QAOA) that uses Grover-like selective phase shift mixing operators. GM-QAOA works on any NP optimization problem for which it is possible to efficiently prepare an equal superposition of all feasible solutions; it is designed to perform particularly well for constraint optimization problems, where not all possible variable assignments are feasible solutions. GM-QAOA has the following features: (i) It is not susceptible to Hamiltonian Simulation error (such as Trotterization errors) as its operators can be implemented exactly using standard gate sets and (ii) Solutions with the same objective value are always sampled with the same amplitude. We illustrate the potential of GM-QAOA on several optimization problem classes: for permutation-based optimization problems such as the Traveling Salesperson Problem, we present an efficient algorithm to prepare a superposition of all possible permutations of $n$ numbers, defined on $O(n^2)$ qubits; for the hard constraint $k$-Vertex-Cover problem, and for an application to Discrete Portfolio Rebalancing, we show that GM-QAOA outperforms existing QAOA approaches

Collective fast delivery by energy-efficient agents

We consider k mobile agents initially located at distinct nodes of an undirected graph (on n nodes, with edge lengths) that have to deliver a single item from a given source node s to a given target node t. The agents can move along the edges of the graph, starting at time 0 with respect to the following: Each agent i has a weight w_i that defines the rate of energy consumption while travelling a distance in the graph, and a velocity v_i with which it can move. We are interested in schedules (operating the k agents) that result in a small delivery time T (time when the package arrives at t), and small total energy consumption E. Concretely, we ask for a schedule that: either (i) Minimizes T, (ii) Minimizes lexicographically (T,E) (prioritizing fast delivery), or (iii) Minimizes epsilon*T + (1-epsilon)*E, for a given epsilon, 0<epsilon<1. We show that (i) is solvable in polynomial time, and show that (ii) is polynomial-time solvable for uniform velocities and solvable in time O(n + k log k) for arbitrary velocities on paths, but in general is NP-hard even on planar graphs. As a corollary of our hardness result, (iii) is NP-hard, too. We show that there is a 3-approximation algorithm for (iii) using a single agent.Comment: In an extended abstract of this paper [MFCS 2018], we erroneously claimed the single agent approach for variant (iii) to have approximation ratio

QAOA-based Fair Sampling on NISQ Devices

We study the status of fair sampling on Noisy Intermediate Scale Quantum (NISQ) devices, in particular the IBM Q family of backends. Using the recently introduced Grover Mixer-QAOA algorithm for discrete optimization, we generate fair sampling circuits to solve six problems of varying difficulty, each with several optimal solutions, which we then run on ten different backends available on the IBM Q system. For a given circuit evaluated on a specific set of qubits, we evaluate: how frequently the qubits return an optimal solution to the problem, the fairness with which the qubits sample from all optimal solutions, and the reported hardware error rate of the qubits. To quantify fairness, we define a novel metric based on Pearson's $\chi^2$ test. We find that fairness is relatively high for circuits with small and large error rates, but drops for circuits with medium error rates. This indicates that structured errors dominate in this regime, while unstructured errors, which are random and thus inherently fair, dominate in noisier qubits and longer circuits. Our results provide a simple, intuitive means of quantifying fairness in quantum circuits, and show that reducing structured errors is necessary to improve fair sampling on NISQ hardware

Threshold-Based Quantum Optimization

We propose and study Th-QAOA (pronounced Threshold QAOA), a variation of the Quantum Alternating Operator Ansatz (QAOA) that replaces the standard phase separator operator, which encodes the objective function, with a threshold function that returns a value $1$ for solutions with an objective value above the threshold and a $0$ otherwise. We vary the threshold value to arrive at a quantum optimization algorithm. We focus on a combination with the Grover Mixer operator; the resulting GM-Th-QAOA can be viewed as a generalization of Grover's quantum search algorithm and its minimum/maximum finding cousin to approximate optimization. Our main findings include: (i) we show semi-formally that the optimum parameter values of GM-Th-QAOA (angles and threshold value) can be found with $O(\log(p) \times \log M)$ iterations of the classical outer loop, where $p$ is the number of QAOA rounds and $M$ is an upper bound on the solution value (often the number of vertices or edges in an input graph), thus eliminating the notorious outer-loop parameter finding issue of other QAOA algorithms; (ii) GM-Th-QAOA can be simulated classically with little effort up to 100 qubits through a set of tricks that cut down memory requirements; (iii) somewhat surprisingly, GM-Th-QAOA outperforms its non-thresholded counterparts in terms of approximation ratios achieved. This third result holds across a range of optimization problems (MaxCut, Max k-VertexCover, Max k-DensestSubgraph, MaxBisection) and various experimental design parameters, such as different input edge densities and constraint sizes

Conflict-Free Chromatic Art Gallery Coverage

We consider a chromatic variant of the art gallery problem, where each guard is assigned one of k distinct colors. A placement of such colored guards is conflict-free if each point of the polygon is seen by some guard whose color appears exactly once among the guards visible to that point. What is the smallest number k(n) of colors that ensure a conflict-free covering of all n-vertex polygons? We call this the conflict-free chromatic art gallery problem. Our main result shows that k(n) is O(logn) for orthogonal and for monotone polygons, and O(log2 n) for arbitrary simple polygons. By contrast, if all guards visible from each point must have distinct colors, then k(n) is Ω(n) for arbitrary simple polygons, as shown by Erickson and LaValle (Robotics: Science and Systems, vol.VII, pp.81-88, 2012). The problem is motivated by applications in distributed robotics and wireless sensor networks but is also of interest from a theoretical point of view

Predicting Expressibility of Parameterized Quantum Circuits using Graph Neural Network

Parameterized Quantum Circuits (PQCs) are essential to quantum machine learning and optimization algorithms. The expressibility of PQCs, which measures their ability to represent a wide range of quantum states, is a critical factor influencing their efficacy in solving quantum problems. However, the existing technique for computing expressibility relies on statistically estimating it through classical simulations, which requires many samples. In this work, we propose a novel method based on Graph Neural Networks (GNNs) for predicting the expressibility of PQCs. By leveraging the graph-based representation of PQCs, our GNN-based model captures intricate relationships between circuit parameters and their resulting expressibility. We train the GNN model on a comprehensive dataset of PQCs annotated with their expressibility values. Experimental evaluation on a four thousand random PQC dataset and IBM Qiskit's hardware efficient ansatz sets demonstrates the superior performance of our approach, achieving a root mean square error (RMSE) of 0.03 and 0.06, respectively

The New Swiss Glacier Inventory SGI2016: From a Topographical to a Glaciological Dataset

Glaciers in Switzerland are shrinking rapidly in response to ongoing climate change. Repeated glacier inventories are key to monitor such changes at the regional scale. Here we present the new Swiss Glacier Inventory 2016 (SGI2016) that has been acquired based on sub-meter resolution aerial imagery and digital elevation models, bringing together topographical and glaciological approaches and knowledge. We define the process, workflow and required glaciological adaptations to compile a highly detailed inventory based on the digital Swiss Topographic Landscape model. The SGI2016 provides glacier outlines (areas), supraglacial debris cover and ice divides for all Swiss glaciers referring to the years 2013–2018. The SGI2016 maps 1,400 individual glacier entities with a total surface area of 961 ± 22 km2, whereof 11% (104 km2) are debris-covered. It constitutes the so far most detailed cartographic representation of glacier extent in Switzerland. Interpretation in the context of topographic parameters indicates that glaciers with moderate inclination and low median elevation tend to have highest fractions of supraglacial debris. Glacier-specific area changes since 1973 show the largest relative changes for small and low-elevation glaciers. The analysis further indicates a tendency for glaciers with a high share of supraglacial debris to show larger relative area changes. Between 1973 and 2016, an area change rate of –0.6% a−1 is found. Based on operational data sets and the presented methodology, the Swiss Glacier Inventory will be updated in 6- yr time intervals, leading to a high consistency in future glacier change assessments

Lower Bounds on Number of QAOA Rounds Required for Guaranteed Approximation Ratios

The quantum alternating operator ansatz (QAOA) is a heuristic hybrid quantum-classical algorithm for finding high-quality approximate solutions to combinatorial optimization problems, such as Maximum Satisfiability. While QAOA is well-studied, theoretical results as to its runtime or approximation ratio guarantees are still relatively sparse. We provide some of the first lower bounds for the number of rounds (the dominant component of QAOA runtimes) required for QAOA. For our main result, (i) we leverage a connection between quantum annealing times and the angles of QAOA to derive a lower bound on the number of rounds of QAOA with respect to the guaranteed approximation ratio. We apply and calculate this bound with Grover-style mixing unitaries and (ii) show that this type of QAOA requires at least a polynomial number of rounds to guarantee any constant approximation ratios for most problems. We also (iii) show that the bound depends only on the statistical values of the objective functions, and when the problem can be modeled as a $k$-local Hamiltonian, can be easily estimated from the coefficients of the Hamiltonians. For the conventional transverse field mixer, (iv) our framework gives a trivial lower bound to all bounded occurrence local cost problems and all strictly $k$-local cost Hamiltonians matching known results that constant approximation ratio is obtainable with constant round QAOA for a few optimization problems from these classes. Using our novel proof framework, (v) we recover the Grover lower bound for unstructured search and -- with small modification -- show that our bound applies to any QAOA-style search protocol that starts in the ground state of the mixing unitaries.Comment: 24 pages, comments welcome, v3: correct some phrasing; results stay unchange