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Gaussian process (GP) is a powerful modeling method with applications in machine learning for various engineering and non-engineering fields. Despite numerous benefits of modeling using GPs, the computational complexity associated with GPs demanding immense resources make their practical usage highly challenging. In this article, we develop a quantum version of multi-output Gaussian Process (QGP) by implementing a well-known quantum algorithm called HHL, to perform the Kernel matrix inversion within the Gaussian Process. To reduce the large circuit depth of HHL a circuit optimization technique called Approximate Quantum Compiling (AQC) has been implemented. We further showcase the application of QGP for a real-world problem to estimate line parameters of an electrical grid. Using AQC, up to 13-qubit HHL circuit has been implemented for a 32x32 kernel matrix inversion on IBM Quantum hardware for demonstrating QGP based line parameter estimation experimentally. Finally, we compare its performance against noise-less quantum simulators and classical computation results., Comment: 9 pages, 7 figures

The transition to 100% renewable energy requires new techniques for managing energy networks, such as dividing them into sensible subsets of prosumers called micro-grids. Doing so in an optimal manner is a difficult optimization problem, as it can be abstracted to the Coalition Structure Generation problem in Induced Subgraph Games, a NP-complete problem which requires dividing an undirected, complete, weighted graph into subgraphs in a way that maximizes the sum of their internal weights. Recently, Venkatesh et al. (arXiv:2212.11372) published a Quantum Annealing (QA)-based iterative algorithm called GCS-Q, which they claim to be the best currently existing solver for the problem in terms of runtime complexity. As this algorithm makes the application of QA to the problem seem promising, but is a greedy one, this work proposes several less greedy QA-based approaches and investigates whether any of them can outperform GCS-Q in terms of solution quality. While we find that this is not the case yet on D-Wave hardware, most of them do when using the classical QBSolv software as a solver. Especially an algorithm we call 4-split iterative R-QUBO shows potential here, finding all optima in our dataset while scaling favorably with the problem size in terms of runtime. Thus, it appears to be interesting for future research on quantum approaches to the problem, assuming QA hardware will become more noise-resilient over time., Comment: 7 pages, 4 figures (8 if counting subfigures). To be published in the proceedings of the 2024 IEEE International Conference on Quantum Computing and Engineering (QCE)

Innovative solutions and developments are being inspected to tackle rising electrical power demand to be supplied by clean forms of energy. The integration of renewable energy generations, varying nature loads, importance of active role of distribution system and consumer participation in grid operation has changed the landscape of classical power grids. Implementation of smarter applications to plan, monitor, operate the grid safely are deemed paramount for efficient, secure and reliable functioning of grid. Although sophisticated computations to process gigantic volume of data to produce useful information in a time critical manner is the paradigm of future grid operations, it brings along the burden of computational complexity. Advancements in quantum technologies holds promising solution for dealing with demanding computational complexity of power system related applications. In this article, we lay out clear motivations for seeking quantum solutions for solving computational burden challenges associated with power system applications. Next we present an overview of quantum solutions for various power system related applications available in current literature and suggest future topics for research. We further highlight challenges with existing quantum solutions for exploiting full quantum capabilities. Additionally, this paper serves as a bridge for power engineers to the quantum world by outlining essential quantum computation fundamentals for enabling smoother transition to future of power system computations.

Electric Vehicles (EVs) are emerging as battery energy storage systems (BESSs) of increasing importance for different power grid services. However, the unique characteristics of EVs makes them more difficult to operate than dedicated BESSs. In this work, we apply a data-driven learning approach to leverage EVs as a BESS to provide capacity-related services to the grid. The approach uses machine learning to predict how to charge and discharge EVs while satisfying their operational constraints. As a paradigm application, we use flexible energy commercialization in the wholesale markets, but the approach can be applied to a broader range of capacity-related grid services. We evaluate the proposed approach numerically and show that when the number of EVs is large, we can obtain comparable objective values to CPLEX and approximate dynamic programming, but with shorter run times. These reduced run times are important because they allow us to (re)optimize frequently to adapt to the time-varying system conditions.

In this project summary paper, we summarize the key results and use-cases explored in the German Federal Ministry of Education and Research (BMBF) funded project "Q-GRID" which aims to assess potential quantum utility optimization applications in the electrical grid. The project focuses on two layers of optimization problems relevant to decentralized energy generation and transmission as well as novel energy transportation/exchange methods such as Peer-2-Peer energy trading and microgrid formation. For select energy grid optimization problems, we demonstrate exponential classical optimizer runtime scaling even for small problem instances, and present initial findings that variational quantum algorithms such as QAOA and hybrid quantum annealing solvers may provide more favourable runtime scaling to obtain similar solution quality. These initial results suggest that quantum computing may be a key enabling technology in the future energy transition insofar that they may be able to solve business problems which are already challenging at small problem instance sizes., Comment: 12 pages. arXiv admin note: text overlap with arXiv:2309.05502

Demand Side Response (DSR) is a strategy that enables consumers to actively participate in managing electricity demand. It aims to alleviate strain on the grid during high demand and promote a more balanced and efficient use of (renewable) electricity resources. We implement DSR through discount scheduling, which involves offering discrete price incentives to consumers to adjust their electricity consumption patterns to times when their local energy mix consists of more renewable energy. Since we tailor the discounts to individual customers' consumption, the Discount Scheduling Problem (DSP) becomes a large combinatorial optimization task. Consequently, we adopt a hybrid quantum computing approach, using D-Wave's Leap Hybrid Cloud. We benchmark Leap against Gurobi, a classical Mixed Integer optimizer in terms of solution quality at fixed runtime and fairness in terms of discount allocation. Furthermore, we propose a large-scale decomposition algorithm/heuristic for the DSP, applied with either quantum or classical computers running the subroutines, which significantly reduces the problem size while maintaining solution quality. Using synthetic data generated from real-world data, we observe that the classical decomposition method obtains the best overall \newp{solution quality for problem sizes up to 3200 consumers, however, the hybrid quantum approach provides more evenly distributed discounts across consumers.

We generalize the Approximate Quantum Compiling algorithm into a new method for CNOT-depth reduction, which is apt to process wide target quantum circuits. Combining this method with state-of-the-art techniques for error mitigation and circuit compiling, we present a 10-qubit experimental demonstration of Iterative Amplitude Estimation on a quantum computer. The target application is the derivation of the Expected Value of contract portfolios in the energy industry. In parallel, we also introduce a new variant of the Quantum Amplitude Estimation algorithm which we call Dynamic Amplitude Estimation, as it is based on the dynamic circuit capability of quantum devices. The algorithm achieves a reduction in the circuit width in the order of the binary precision compared to the typical implementation of Quantum Amplitude Estimation, while simultaneously decreasing the number of quantum-classical iterations (again in the order of the binary precision) compared to the Iterative Amplitude Estimation. The calculation of the Expected Value, VaR and CVaR of contract portfolios on quantum hardware provides a proof of principle of the new algorithm., Comment: 25 pages, 11 figures, 5 tables

Computing nonlinear functions over multilinear forms is a general problem with applications in risk analysis. For instance in the domain of energy economics, accurate and timely risk management demands for efficient simulation of millions of scenarios, largely benefiting from computational speedups. We develop a novel hybrid quantum-classical algorithm based on polynomial approximation of nonlinear functions, computed through Quantum Hadamard Products, and we rigorously assess the conditions for its end-to-end speedup for different implementation variants against classical algorithms. In our setting, a quadratic quantum speedup, up to polylogarithmic factors, can be proven only when forms are bilinear and approximating polynomials have second degree, if efficient loading unitaries are available for the input data sets. We also enhance the bidirectional encoding, that allows tuning the balance between circuit depth and width, proposing an improved version that can be exploited for the calculation of inner products. Lastly, we exploit the dynamic circuit capabilities, recently introduced on IBM Quantum devices, to reduce the average depth of the Quantum Hadamard Product circuit. A proof of principle is implemented and validated on IBM Quantum systems.

In this article we present the application of classical and quantum-classical hybrid anomaly detection schemes to explore exotic configuration with anomalous features. We consider the Anderson model as a prototype where we define two types of anomalies - a high conductance in presence of strong impurity and low conductance in presence of weak impurity - as a function of random impurity distribution. Such anomalous outcome constitutes an imperceptible fraction of the data set and is not a part of the training process. These exotic configurations, which can be a source of rich new physics, usually remain elusive to conventional classification or regression methods and can be tracked only with a suitable anomaly detection scheme. We also present a systematic study of the performance of the classical and the quantum-classical hybrid anomaly detection method and show that the inclusion of a quantum circuit significantly enhances the performance of anomaly detection which we quantify with suitable performance metrics. Our approach is quite generic in nature and can be used for any system that relies on a large number of parameters to find their new configurations which can hold exotic new features., Comment: 6 pages, 4 figures

In this article we demonstrate the applications of classical and quantum machine learning in quantum transport and spintronics. With the help of a two-terminal device with magnetic impurity we show how machine learning algorithms can predict the highly non-linear nature of conductance as well as the non-equilibrium spin response function for any random magnetic configuration. By mapping this quantum mechanical problem onto a classification problem, we are able to obtain much higher accuracy beyond the linear response regime compared to the prediction obtained with conventional regression methods. We finally describe the applicability of quantum machine learning which has the capability to handle a significantly large configuration space. Our approach is applicable for solid state devices as well as for molecular systems. These outcomes are crucial in predicting the behavior of large-scale systems where a quantum mechanical calculation is computationally challenging and therefore would play a crucial role in designing nano devices., Comment: 9 pages, 8 figures

We propose a Hybrid classical-quantum Autoencoder (HAE) model, which is a synergy of a classical autoencoder (AE) and a parametrized quantum circuit (PQC) that is inserted into its bottleneck. The PQC augments the latent space, on which a standard outlier detection method is applied to search for anomalous data points within a classical dataset. Using this model and applying it to both standard benchmarking datasets, and a specific use-case dataset which relates to predictive maintenance of gas power plants, we show that the addition of the PQC leads to a performance enhancement in terms of precision, recall, and F1 score. Furthermore, we probe different PQC Ans\"atze and analyse which PQC features make them effective for this task., Comment: 17 pages, 11 figures

In electronic structure calculations, the correlation energy is defined as the difference between the mean field and the exact solution of the non relativistic Schr\"odinger equation. Such an error in the different calculations is not directly observable as there is no simple quantum mechanical operator, apart from correlation functions, that correspond to such quantity. Here, we use the dimensional scaling approach, in which the electrons are localized at the large-dimensional scaled space, to describe a geometric picture of the electronic correlation. Both, the mean field, and the exact solutions at the large-D limit have distinct geometries. Thus, the difference might be used to describe the correlation effect. Moreover, correlations can be also described and quantified by the entanglement between the electrons, which is a strong correlation without a classical analog. Entanglement is directly observable and it is one of the most striking properties of quantum mechanics and bounded by the area law for local gapped Hamiltonians of interacting many-body systems. This study opens up the possibility of presenting a geometrical picture of the electron-electron correlations and might give a bound on the correlation energy. The results at the large-D limit and at D=3 indicate the feasibility of using the geometrical picture to get a bound on the electron-electron correlations., Comment: 27 pages, 13 figures

In this article we describe a technique to transfer data from classical domain to quantum domain. We consider a set of $N (=2^n)$ classical data in the form of a column matrix and prepare a $n$-qubit quantum state, whose components correspond to the $N$ classical data. To prepare this $n$-qubit quantum vector we use Schmidt decomposition and singular value decomposition techniques respectively and construct the corresponding family of quantum circuits. To strengthen our argument we also give specific examples by considering a set of 4 and 16 classical data and constructing the corresponding 2 and 4-qubit quantum vector respectively., Comment: 13 pages, 1 figure

We employ a simple and accurate dimensional interpolation formula for the shapes of random walks at $D=3$ and $D=2$ based on the analytically known solutions at both limits $D=\infty$ and $D=1$. The results obtained for the radii of gyration of an arbitrary shaped object are about $2\%$ error compared with accurate numerical results at $D = 3$ and $D = 2$. We also calculated the asphericity for a three-dimensional random walk using the dimensional interpolation formula. Result agrees very well with the numerically simulated result. The method is general and can be used to estimate other properties of random walks., Comment: 22 pages, 8 figures

Practical quantum computing applications to power grids are nonexistent at the moment. This paper investigates how a fundamental grid problem, namely DC power flow, can be solved using quantum computing. Power flow is the most widely used power system analysis technique, either as a stand-alone application or embedded in other applications; therefore, its fast and accurate solution is of utmost significance for grid operators. We base our studies on the Harrow-Hassidim-Lloyd (HHL) quantum algorithm, which has a proven theoretical speedup over classical algorithms in solving a system of linear equations. Practical studies on a quantum computer are conducted using the WSCC 9-bus system.

We employ a simple and mostly accurate dimensional interpolation formula using dimensional limits $D=1$ and $D=\infty$ to obtain $D=3$ ground-state energy of metallic hydrogen. We also present results describing the phase transitions for different symmetries of three-dimensional structure lattices. The interpolation formula not only predicts fairly accurate energies but also predicts a correct functional form of the energy as a function of the lattice parameters. That allows us to calculate different physical quantities such as the bulk modulus, Debye temperature, and critical transition temperature, from the gradient and the curvature of the energy curve as a function of the lattice parameters. These theoretical calculations suggest that metallic hydrogen is a likely candidate for high temperature superconductivity. The dimensional interpolation formula is robust and might be useful to obtain the energies of complex many-body systems., Comment: 21 pages, 10 figures

In this paper, we model and solve a fundamental power system problem, i.e., DC power flow, using a practical quantum computer. The Harrow-Hassidim-Lloyd (HHL) quantum algorithm is used to solve the DC power flow problem. The HHL algorithm for the solution of a system of linear equations (SLE) offers an exponential speedup over the classical computers. The speedup advantage is more significant when the size and the frequency of solving the power flow problem become more substantial. Verified quantum computing applications to power systems are merely nonexistent at this point. Through this paper, we plan to (1) provide a proof-of-concept that practical power system problems can be solved using quantum technology, (2) build a quantum-grid foundation by solving a fundamental power system problem with applications in many other critical grid problems, and (3) apply HHL to solve an SLE which has broad applications in many power system problems. A small 3-bus system is used for testing and demonstration purposes, considering the limitations of the available quantum computing hardware and software. The proposed method's merits and effectiveness are demonstrated using IBM open-source quantum computer and reported through proof-of-concept experimental demonstration using a 4-qubit quantum information processor., Comment: Journal Paper

In this article, we calculate the classical vortex solution of a spontaneously broken gauge theory interacting with gravity in (2+1)-dimension. We also calculate the conditions for the formation of a (2+1)-dimensional black hole due to magnetic vortex (a Nielsen-Olesen vortex). The semiclassical Hawking temperature for this black hole is calculated, where we see that the temperature of a BTZ black hole increases or decreases without changing the size of the horizon if we insert the magnetic vortex fields in the black hole. Finally, the first law of black hole thermodynamics is described for this particular solution, which shows that the additional work terms from the scalar and gauge fields compensate the change in the temperature relative to its usual value for the BTZ solution., Comment: 14 pages, no figures

We present a simple interpolation formula using dimensional limits $D=1$ and $D=\infty$ to obtain the $D=3$ ground-state energies of atoms and molecules. For atoms, these limits are linked by first-order perturbation terms of electron-electron interactions. This unorthodox approach is illustrated by ground-states for two, three, and four electron atoms, with modest effort to obtain fairly accurate results. Also, we treat the ground-state of H$_2$ over a wide range of the internuclear distance R, and compares well with the standard exact results from the Full Configuration Interaction method. Similar dimensional interpolations may be useful for complex many-body systems., Comment: 29 pages, 1 figure

Thermal correlation functions and the associated effective statistical potential are computed in two- and three-dimensional non-commutative space using an operator formulation that makes no reference to a star product. The corresponding results for the Moyal and Voros star products are then easily obtained by taking the corresponding overlap with Moyal and Voros bases. The forms of the correlation function and the effective potential are found to be the same, except that in the Voros case the thermal length undergoes a non-commutative deformation, ensuring that it has a lower bound of the order of $\sqrt{\theta}$. It is shown that in a suitable basis (called here quasi-commutative) in the multi-particle sector the thermal correlation function coincides with the commutative result both in the Moyal and Voros cases, with an appropriate non-commutative correction to the thermal length in the Voros case, and that the Pauli principle is restored., Comment: 31 pages, 7 figures