In the rapidly evolving field of quantum computing, tensor networks serve as an important tool due to their multifaceted utility. In this paper, we review the diverse applications of tensor networks and show that they are an important instrument for quantum computing. Specifically, we summarize the application of tensor networks in various domains of quantum computing, including simulation of quantum computation, quantum circuit synthesis, quantum error correction and mitigation, and quantum machine learning. Finally, we provide an outlook on the opportunities and the challenges of the tensor-network techniques.

Image classification, a pivotal task in multiple industries, faces computational challenges due to the burgeoning volume of visual data. This research addresses these challenges by introducing two quantum machine learning models that leverage the principles of quantum mechanics for effective computations. Our first model, a hybrid quantum neural network with parallel quantum circuits, enables the execution of computations even in the noisy intermediate-scale quantum era, where circuits with a large number of qubits are currently infeasible. This model demonstrated a record-breaking classification accuracy of 99.21% on the full MNIST dataset, surpassing the performance of known quantum-classical models, while having eight times fewer parameters than its classical counterpart. Also, the results of testing this hybrid model on a Medical MNIST (classification accuracy over 99%), and on CIFAR-10 (classification accuracy over 82%), can serve as evidence of the generalizability of the model and highlights the efficiency of quantum layers in distinguishing common features of input data. Our second model introduces a hybrid quantum neural network with a Quanvolutional layer, reducing image resolution via a convolution process. The model matches the performance of its classical counterpart, having four times fewer trainable parameters, and outperforms a classical model with equal weight parameters. These models represent advancements in quantum machine learning research and illuminate the path towards more accurate image classification systems.

We present the Quantum Memory Matrix (QMM) hypothesis, which addresses the longstanding Black Hole Information Paradox rooted in the apparent conflict between Quantum Mechanics (QM) and General Relativity (GR). This paradox raises the question of how information is preserved during black hole formation and evaporation, given that Hawking radiation appears to result in information loss, challenging unitarity in quantum mechanics. The QMM hypothesis proposes that space-time itself acts as a dynamic quantum information reservoir, with quantum imprints encoding information about quantum states and interactions directly into the fabric of space-time at the Planck scale. By defining a quantized model of space-time and mechanisms for information encoding and retrieval, QMM aims to conserve information in a manner consistent with unitarity during black hole processes. We develop a mathematical framework that includes space-time quantization, definitions of quantum imprints, and interactions that modify quantum state evolution within this structure. Explicit expressions for the interaction Hamiltonians are provided, demonstrating unitarity preservation in the combined system of quantum fields and the QMM. This hypothesis is compared with existing theories, including the holographic principle, black hole complementarity, and loop quantum gravity, noting its distinctions and examining its limitations. Finally, we discuss observable implications of QMM, suggesting pathways for experimental evaluation, such as potential deviations from thermality in Hawking radiation and their effects on gravitational wave signals. The QMM hypothesis aims to provide a pathway towards resolving the Black Hole Information Paradox while contributing to broader discussions in quantum gravity and cosmology.

The Berezinskii-Kosterlitz-Thouless (BKT) transition is the prototype of a phase transition driven by the formation and interaction of topological defects in two-dimensional (2D) systems. In typical models these are vortices: above a transition temperature $T_{\rm BKT}$ vortices are free, below this transition temperature they get confined. In this work we extend the concept of BKT transition to quantum systems in two dimensions. In particular, we demonstrate that a zero-temperature quantum BKT phase transition, driven by a coupling constant can occur in 2D models governed by an effective gauge field theory with a diverging dielectric constant. One particular example is that of a compact U(1) gauge theory with a diverging dielectric constant, where the quantum BKT transition is induced by non-relativistic, purely 2D magnetic monopoles, which can be viewed also as electric vortices. These quantum BKT transitions have the same diverging exponent $z$ as the quantum Griffiths transition but have nothing to do with disorder.

Predicting solar panel power output is crucial for advancing the transition to renewable energy but is complicated by the variable and non-linear nature of solar energy. This is influenced by numerous meteorological factors, geographical positioning, and photovoltaic cell properties, posing significant challenges to forecasting accuracy and grid stability. Our study introduces a suite of solutions centered around hybrid quantum neural networks designed to tackle these complexities. The first proposed model, the Hybrid Quantum Long Short-Term Memory, surpasses all tested models by achieving mean absolute errors and mean squared errors that are more than 40% lower. The second proposed model, the Hybrid Quantum Sequence-to-Sequence neural network, once trained, predicts photovoltaic power with 16% lower mean absolute error for arbitrary time intervals without the need for prior meteorological data, highlighting its versatility. Moreover, our hybrid models perform better even when trained on limited datasets, underlining their potential utility in data-scarce scenarios. These findings represent progress towards resolving time series prediction challenges in energy forecasting through hybrid quantum models, showcasing the transformative potential of quantum machine learning in catalyzing the renewable energy transition.

Image recognition is one of the primary applications of machine learning algorithms. Nevertheless, machine learning models used in modern image recognition systems consist of millions of parameters that usually require significant computational time to be adjusted. Moreover, adjustment of model hyperparameters leads to additional overhead. Because of this, new developments in machine learning models and hyperparameter optimization techniques are required. This paper presents a quantum-inspired hyperparameter optimization technique and a hybrid quantum-classical machine learning model for supervised learning. We benchmark our hyperparameter optimization method over standard black-box objective functions and observe performance improvements in the form of reduced expected run times and fitness in response to the growth in the size of the search space. We test our approaches in a car image classification task and demonstrate a full-scale implementation of the hybrid quantum ResNet model with the tensor train hyperparameter optimization. Our tests show a qualitative and quantitative advantage over the corresponding standard classical tabular grid search approach used with a deep neural network ResNet34. A classification accuracy of 0.97 was obtained by the hybrid model after 18 iterations, whereas the classical model achieved an accuracy of 0.92 after 75 iterations.

Recent advances in the manipulation of complex oxide layers, particularly the fabrication of atomically thin cuprate superconducting films via molecular beam epitaxy, have revealed new ways in which nanoscale engineering can govern superconductivity and its interwoven electronic orders. In parallel, the creation of twisted cuprate heterostructures through cryogenic stacking techniques marks a pivotal step forward, exploiting cuprate superconductors to deepen our understanding of exotic quantum states and propel next-generation quantum technologies. This review explores over three decades of research in the emerging field of cuprate twistronics, examining both experimental breakthroughs and theoretical progress. It also highlights the methodologies poised to surmount the outstanding challenges in leveraging these complex quantum materials, underscoring their potential to expand the frontiers of quantum science and technology.

We extend the Quantum Memory Matrix (QMM) framework, originally developed to reconcile quantum mechanics and general relativity by treating space-time as a dynamic information reservoir, to incorporate the full suite of Standard Model gauge interactions. In this discretized, Planck-scale formulation, each space-time cell possesses a finite-dimensional Hilbert space that acts as a local memory, or quantum imprint, for matter and gauge field configurations. We focus on embedding non-Abelian SU(3)c (quantum chromodynamics) and SU(2)L x U(1)Y (electroweak interactions) into QMM by constructing gauge-invariant imprint operators for quarks, gluons, electroweak bosons, and the Higgs mechanism. This unified approach naturally enforces unitarity by allowing black hole horizons, or any high-curvature region, to store and later retrieve quantum information about color and electroweak charges, thereby preserving subtle non-thermal correlations in evaporation processes. Moreover, the discretized nature of QMM imposes a Planck-scale cutoff, potentially taming UV divergences and modifying running couplings at trans-Planckian energies. We outline major challenges, such as the precise formulation of non-Abelian imprint operators and the integration of QMM with loop quantum gravity, as well as possible observational strategies - ranging from rare decay channels to primordial black hole evaporation spectra - that could provide indirect probes of this discrete, memory-based view of quantum gravity and the Standard Model.

Finding the distribution of the velocities and pressures of a fluid by solving the Navier-Stokes equations is a principal task in the chemical, energy, and pharmaceutical industries, as well as in mechanical engineering and the design of pipeline systems. With existing solvers, such as OpenFOAM and Ansys, simulations of fluid dynamics in intricate geometries are computationally expensive and require re-simulation whenever the geometric parameters or the initial and boundary conditions are altered. Physics-informed neural networks are a promising tool for simulating fluid flows in complex geometries, as they can adapt to changes in the geometry and mesh definitions, allowing for generalization across fluid parameters and transfer learning across different shapes. We present a hybrid quantum physics-informed neural network that simulates laminar fluid flows in 3D Y-shaped mixers. Our approach combines the expressive power of a quantum model with the flexibility of a physics-informed neural network, resulting in a 21% higher accuracy compared to a purely classical neural network. Our findings highlight the potential of machine learning approaches, and in particular hybrid quantum physics-informed neural network, for complex shape optimization tasks in computational fluid dynamics. By improving the accuracy of fluid simulations in complex geometries, our research using hybrid quantum models contributes to the development of more efficient and reliable fluid dynamics solvers.

We propose a fundamental duality between the geometric properties of spacetime and the informational content of quantum fields. Specifically, we establish that the curvature of spacetime is directly related to the entanglement entropy of quantum states, with geometric invariants mapping to informational measures. This framework modifies Einstein's field equations by introducing an informational stress-energy tensor derived from quantum entanglement entropy. Our findings have implications for black hole thermodynamics, cosmology, and quantum gravity, suggesting that quantum information fundamentally shapes the structure of spacetime. We incorporate this informational stress-energy tensor into Einstein's field equations, leading to modified spacetime geometry, particularly in regimes of strong gravitational fields, such as near black holes. We compute corrections to Newton's constant (G) due to entanglement entropy contributions from various quantum fields and explore the consequences for black hole thermodynamics and cosmology. These corrections include explicit dependence on fundamental constants (h-bar, c, and k_B), ensuring dimensional consistency in our calculations. Our results indicate that quantum information plays a crucial role in gravitational dynamics, providing new insights into the nature of spacetime and potential solutions to long-standing challenges in quantum gravity.

Cancer is one of the leading causes of death worldwide. It is caused by a variety of genetic mutations, which makes every instance of the disease unique. Since chemotherapy can have extremely severe side effects, each patient requires a personalized treatment plan. Finding the dosages that maximize the beneficial effects of the drugs and minimize their adverse side effects is vital. Deep neural networks automate and improve drug selection. However, they require a lot of data to be trained on. Therefore, there is a need for machine-learning approaches that require less data. Hybrid quantum neural networks were shown to provide a potential advantage in problems where training data availability is limited. We propose a novel hybrid quantum neural network for drug response prediction, based on a combination of convolutional, graph convolutional, and deep quantum neural layers of 8 qubits with 363 layers. We test our model on the reduced Genomics of Drug Sensitivity in Cancer dataset and show that the hybrid quantum model outperforms its classical analog by 15% in predicting IC50 drug effectiveness values. The proposed hybrid quantum machine learning model is a step towards deep quantum data-efficient algorithms with thousands of quantum gates for solving problems in personalized medicine, where data collection is a challenge.

Primordial black holes (PBHs) remain one of the most intriguing candidates for dark matter and a unique probe of physics at extreme curvatures. Here, we examine their formation in a bounce cosmology when the post-crunch universe inherits a highly inhomogeneous distribution of imprint entropy from the Quantum Memory Matrix (QMM). Within QMM, every Planck-scale cell stores quantum information about infalling matter; the surviving entropy field S(x) contributes an effective dust component T^QMM_{\mu\nu} = lambda * [ (nabla_mu S)(nabla_nu S) - (1/2) * g_{\mu\nu} * (nabla S)^2 + ... ] that deepens curvature wherever S is large. We show that (i) reasonable bounce temperatures and a QMM coupling lambda ~ O(1) naturally amplify these "information wells" until the density contrast exceeds the critical value delta_c ~ 0.3; (ii) the resulting PBH mass spectrum spans 10^{-16} to 10^3 solar masses, matching current microlensing and PTA windows; and (iii) the same mechanism links PBH abundance to earlier QMM explanations of dark matter and the cosmic matter-antimatter imbalance. Observable signatures include a mild blue tilt in small-scale power, characteristic mu-distortions, and an enhanced integrated Sachs-Wolfe signal - all of which will be tested by upcoming CMB, PTA, and lensing surveys.

The solution of computational fluid dynamics problems is one of the most computationally hard tasks, especially in the case of complex geometries and turbulent flow regimes. We propose to use Tensor Train (TT) methods, which possess logarithmic complexity in problem size and have great similarities with quantum algorithms in the structure of data representation. We develop the Tensor train Finite Element Method -- TetraFEM -- and the explicit numerical scheme for the solution of the incompressible Navier-Stokes equation via Tensor Trains. We test this approach on the simulation of liquids mixing in a T-shape mixer, which, to our knowledge, was done for the first time using tensor methods in such non-trivial geometries. As expected, we achieve exponential compression in memory of all FEM matrices and demonstrate an exponential speed-up compared to the conventional FEM implementation on dense meshes. In addition, we discuss the possibility of extending this method to a quantum computer to solve more complex problems. This paper is based on work we conducted for Evonik Industries AG.

Accurate prediction and stabilization of blast furnace temperatures are crucial for optimizing the efficiency and productivity of steel production. Traditional methods often struggle with the complex and non-linear nature of the temperature fluctuations within blast furnaces. This paper proposes a novel approach that combines hybrid quantum machine learning with pulverized coal injection control to address these challenges. By integrating classical machine learning techniques with quantum computing algorithms, we aim to enhance predictive accuracy and achieve more stable temperature control. For this we utilized a unique prediction-based optimization method. Our method leverages quantum-enhanced feature space exploration and the robustness of classical regression models to forecast temperature variations and optimize pulverized coal injection values. Our results demonstrate a significant improvement in prediction accuracy over 25 percent and our solution improved temperature stability to +-7.6 degrees of target range from the earlier variance of +-50 degrees, highlighting the potential of hybrid quantum machine learning models in industrial steel production applications.

An essential component of many sophisticated metaheuristics for solving combinatorial optimization problems is some variation of a local search routine that iteratively searches for a better solution within a chosen set of immediate neighbors. The size $l$ of this set is limited due to the computational costs required to run the method on classical processing units. We present a qubit-efficient variational quantum algorithm that implements a quantum version of local search with only $\lceil \log_2 l \rceil$ qubits and, therefore, can potentially work with classically intractable neighborhood sizes when realized on near-term quantum computers. Increasing the amount of quantum resources employed in the algorithm allows for a larger neighborhood size, improving the quality of obtained solutions. This trade-off is crucial for present and near-term quantum devices characterized by a limited number of logical qubits. Numerically simulating our algorithm, we successfully solved the largest graph coloring instance that was tackled by a quantum method. This achievement highlights the algorithm's potential for solving large-scale combinatorial optimization problems on near-term quantum devices.

The primary objective of this paper is to conduct a comparative analysis between two Machine Learning approaches: Tensor Networks (TN) and Variational Quantum Classifiers (VQC). While both approaches share similarities in their representation of the Hilbert space using a logarithmic number of parameters, they diverge in the manifolds they cover. Thus, the aim is to evaluate and compare the expressibility and trainability of these approaches. By conducting this comparison, we can gain insights into potential areas where quantum advantage may be found. Our findings indicate that VQC exhibits advantages in terms of speed and accuracy when dealing with data, characterized by a small number of features. However, for high-dimensional data, TN surpasses VQC in overall classification accuracy. We believe that this disparity is primarily attributed to challenges encountered during the training of quantum circuits. We want to stress that in this article, we focus on only one particular task and do not conduct thorough averaging of the results. Consequently, we recommend considering the results of this article as a unique case without excessive generalization.

State-of-the-art noisy intermediate-scale quantum devices (NISQ), although imperfect, enable computational tasks that are manifestly beyond the capabilities of modern classical supercomputers. However, present quantum computations are restricted to exploring specific simplified protocols, whereas the implementation of full-scale quantum algorithms aimed at solving concrete large scale problems arising in data analysis and numerical modelling remains a challenge. Here we introduce and implement a hybrid quantum algorithm for solving linear systems of equations with exponential speedup, utilizing quantum phase estimation, one of the exemplary core protocols for quantum computing. We introduce theoretically classes of linear systems that are suitable for current generation quantum machines and solve experimentally a $2^{17}$-dimensional problem on superconducting IBMQ devices, a record for linear system solution on quantum computers. The considered large-scale algorithm shows superiority over conventional solutions, demonstrates advantages of quantum data processing via phase estimation and holds high promise for meeting practically relevant challenges.

The question "What is real?" can be traced back to the shadows in Plato's cave. Two thousand years later, Rene Descartes lacked knowledge about arguing against an evil deceiver feeding us the illusion of sensation. Descartes' epistemological concept later led to various theories of sensory experiences. The concept of "illusionism", proposing that even the very conscious experience we have is an illusion, is not only a red-pill scenario found in the 1999 science fiction movie "The Matrix" but is also a philosophical concept promoted by modern tinkers, most prominently by Daniel Dennett. Reflection upon a possible simulation and our perceived reality was beautifully visualized in "The Matrix", bringing the old ideas of Descartes to coffee houses around the world. Irish philosopher Bishop Berkeley was the father of what was later coined as "subjective idealism", basically stating that "what you perceive is real". With the advent of quantum technologies based on the control of individual fundamental particles, the question of whether our universe is a simulation isn't just intriguing. Our ever-advancing understanding of fundamental physical processes will likely lead us to build quantum computers utilizing quantum effects for simulating nature quantum-mechanically in all complexity, as famously envisioned by Richard Feynman. In this article, we outline constraints on the limits of computability and predictability in/of the universe, which we then use to design experiments allowing for first conclusions as to whether we participate in a simulation chain. Eventually, in a simulation in which the computer simulating a universe is governed by the same physical laws as the simulation, the exhaustion of computational resources will halt all simulations down the simulation chain unless an external programmer intervenes, which we may be able to observe.

Quantum machine learning has become an area of growing interest but has certain theoretical and hardware-specific limitations. Notably, the problem of vanishing gradients, or barren plateaus, renders the training impossible for circuits with high qubit counts, imposing a limit on the number of qubits that data scientists can use for solving problems. Independently, angle-embedded supervised quantum neural networks were shown to produce truncated Fourier series with a degree directly dependent on two factors: the depth of the encoding and the number of parallel qubits the encoding applied to. The degree of the Fourier series limits the model expressivity. This work introduces two new architectures whose Fourier degrees grow exponentially: the sequential and parallel exponential quantum machine learning architectures. This is done by efficiently using the available Hilbert space when encoding, increasing the expressivity of the quantum encoding. Therefore, the exponential growth allows staying at the low-qubit limit to create highly expressive circuits avoiding barren plateaus. Practically, parallel exponential architecture was shown to outperform the existing linear architectures by reducing their final mean square error value by up to 44.7% in a one-dimensional test problem. Furthermore, the feasibility of this technique was also shown on a trapped ion quantum processing unit.

In the current quantum computing paradigm, significant focus is placed on the reduction or mitigation of quantum decoherence. When designing new quantum processing units, the general objective is to reduce the amount of noise qubits are subject to, and in algorithm design, a large effort is underway to provide scalable error correction or mitigation techniques. Yet some previous work has indicated that certain classes of quantum algorithms, such as quantum machine learning, may, in fact, be intrinsically robust to or even benefit from the presence of a small amount of noise. Here, we demonstrate that noise levels in quantum hardware can be effectively tuned to enhance the ability of quantum neural networks to generalize data, acting akin to regularisation in classical neural networks. As an example, we consider a medical regression task, where, by tuning the noise level in the circuit, we improved the mean squared error loss by 8%.