Stefan Woerner

Fabian L. Müller, Stefan Woerner, John Lygeros

Flexible energy resources can help balance the power grid by providing different types of ancillary services. However, the balancing potential of most types of resources is restricted by physical constraints such as the size of their energy buffer, limits on power-ramp rates, or control delays. Using the example of Secondary Frequency Regulation, this paper shows how the flexibility of various resources can be exploited more efficiently by considering multiple resources with complementary physical properties and controlling them in a coordinated way. To this end, optimal adjustable control policies are computed based on robust optimization. Our problem formulation takes into account power ramp-rate constraints explicitly, and accurately models the different timescales and lead times of the energy and reserve markets. Simulations demonstrate that aggregations of select resources can offer significantly more regulation capacity than the resources could provide individually.

Fabian L. Müller, Stefan Woerner, John Lygeros

The energetic flexibility of electric energy resources can be exploited when trading on wholesale energy and ancillary service markets. This paper considers the problem of a Balance Responsible Party to maximize its profit from trading on energy markets while simultaneously offering Secondary Frequency Reserves to the System Operator. However, the accurate provision of regulation power can be compromised by power ramp-rate limitations of the resources providing the service. To avoid this shortcoming, we take power ramp-rate constraints into account explicitly when computing optimal adjustable energy trading policies that are robust against uncertain activation of reserves. The approach proposed is applicable in real market settings because it models all the different timescales of the day-ahead, intra-day, and reserve markets. Finally, the effect of different market settings and energy trading policies on the amount of available Secondary Frequency Reserves is investigated.

Gian Gentinetta, David Sutter, Christa Zoufal et al.

Quantum support vector machines have the potential to achieve a quantum speedup for solving certain machine learning problems. The key challenge for doing so is finding good quantum kernels for a given data set -- a task called kernel alignment. In this paper we study this problem using the Pegasos algorithm, which is an algorithm that uses stochastic gradient descent to solve the support vector machine optimization problem. We extend Pegasos to the quantum case and and demonstrate its effectiveness for kernel alignment. Unlike previous work which performs kernel alignment by training a QSVM within an outer optimization loop, we show that using Pegasos it is possible to simultaneously train the support vector machine and align the kernel. Our experiments show that this approach is capable of aligning quantum feature maps with high accuracy, and outperforms existing quantum kernel alignment techniques. Specifically, we demonstrate that Pegasos is particularly effective for non-stationary data, which is an important challenge in real-world applications.

Nicola Mariella, Albert Akhriev, Francesco Tacchino et al.

Optimal Transport (OT) has fueled machine learning (ML) across many domains. When paired data measurements $(\boldsymbolμ, \boldsymbolν)$ are coupled to covariates, a challenging conditional distribution learning setting arises. Existing approaches for learning a $\textit{global}$ transport map parameterized through a potentially unseen context utilize Neural OT and largely rely on Brenier's theorem. Here, we propose a first-of-its-kind quantum computing formulation for amortized optimization of contextualized transportation plans. We exploit a direct link between doubly stochastic matrices and unitary operators thus unravelling a natural connection between OT and quantum computation. We verify our method (QontOT) on synthetic and real data by predicting variations in cell type distributions conditioned on drug dosage. Importantly we conduct a 24-qubit hardware experiment on a task challenging for classical computers and report a performance that cannot be matched with our classical neural OT approach. In sum, this is a first step toward learning to predict contextualized transportation plans through quantum computing.

Gian Gentinetta, Arne Thomsen, David Sutter et al.

Quantum support vector machines employ quantum circuits to define the kernel function. It has been shown that this approach offers a provable exponential speedup compared to any known classical algorithm for certain data sets. The training of such models corresponds to solving a convex optimization problem either via its primal or dual formulation. Due to the probabilistic nature of quantum mechanics, the training algorithms are affected by statistical uncertainty, which has a major impact on their complexity. We show that the dual problem can be solved in $O(M^{4.67}/\varepsilon^2)$ quantum circuit evaluations, where $M$ denotes the size of the data set and $\varepsilon$ the solution accuracy compared to the ideal result from exact expectation values, which is only obtainable in theory. We prove under an empirically motivated assumption that the kernelized primal problem can alternatively be solved in $O(\min \{ M^2/\varepsilon^6, \, 1/\varepsilon^{10} \})$ evaluations by employing a generalization of a known classical algorithm called Pegasos. Accompanying empirical results demonstrate these analytical complexities to be essentially tight. In addition, we investigate a variational approximation to quantum support vector machines and show that their heuristic training achieves considerably better scaling in our experiments.

Amira Abbas, David Sutter, Alessio Figalli et al.

Making statements about the performance of trained models on tasks involving new data is one of the primary goals of machine learning, i.e., to understand the generalization power of a model. Various capacity measures try to capture this ability, but usually fall short in explaining important characteristics of models that we observe in practice. In this study, we propose the local effective dimension as a capacity measure which seems to correlate well with generalization error on standard data sets. Importantly, we prove that the local effective dimension bounds the generalization error and discuss the aptness of this capacity measure for machine learning models.

Amira Abbas, David Sutter, Christa Zoufal et al.

Fault-tolerant quantum computers offer the promise of dramatically improving machine learning through speed-ups in computation or improved model scalability. In the near-term, however, the benefits of quantum machine learning are not so clear. Understanding expressibility and trainability of quantum models-and quantum neural networks in particular-requires further investigation. In this work, we use tools from information geometry to define a notion of expressibility for quantum and classical models. The effective dimension, which depends on the Fisher information, is used to prove a novel generalisation bound and establish a robust measure of expressibility. We show that quantum neural networks are able to achieve a significantly better effective dimension than comparable classical neural networks. To then assess the trainability of quantum models, we connect the Fisher information spectrum to barren plateaus, the problem of vanishing gradients. Importantly, certain quantum neural networks can show resilience to this phenomenon and train faster than classical models due to their favourable optimisation landscapes, captured by a more evenly spread Fisher information spectrum. Our work is the first to demonstrate that well-designed quantum neural networks offer an advantage over classical neural networks through a higher effective dimension and faster training ability, which we verify on real quantum hardware.