Transformer models have revolutionized sequential learning across various domains, yet their self-attention mechanism incurs quadratic computational cost, posing limitations for real-time and resource-constrained tasks. To address this, we propose Quantum Adaptive Self-Attention (QASA), a novel hybrid architecture that enhances classical Transformer models with a quantum attention mechanism. QASA replaces dot-product attention with a parameterized quantum circuit (PQC) that adaptively captures inter-token relationships in the quantum Hilbert space. Additionally, a residual quantum projection module is introduced before the feedforward network to further refine temporal features. Our design retains classical efficiency in earlier layers while injecting quantum expressiveness in the final encoder block, ensuring compatibility with current NISQ hardware. Experiments on synthetic time-series tasks demonstrate that QASA achieves faster convergence and superior generalization compared to both standard Transformers and reduced classical variants. Preliminary complexity analysis suggests potential quantum advantages in gradient computation, opening new avenues for efficient quantum deep learning models.

Quantum Recurrent Neural Networks with Encoder-Decoder for Time-Dependent Partial Differential Equations Yuan Chen 1, Abdul Khaliq 1,2, and Khaled M. Furati 3 1 Computational and Data Science Program, Middle Tennessee State University, Murfreesboro, 37132, TN, USA 2 Department of Mathematical Science, Middle Tennessee State University, Murfreesboro, 37132, TN, USA 3 Department of Mathematics, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia Nonlinear time-dependent partial differential equations are essential in modeling complex phenomena across diverse fields, yet they pose significant challenges due to their computational complexity, especially in higher dimensions. This study explores Quantum Recurrent Neural Networks within an encoder-decoder framework, integrating V ariational Quantum Circuits into Gated Recurrent Units and Long Short-T erm Memory networks. W e evaluate the algorithms on the Hamilton-Jacobi-Bellman equation, Burgers' equation, the Gray-Scott reaction-diffusion system, and the three dimensional Michaelis-Menten reaction-diffusion equation. The results demonstrate the superior performance of the quantum-based algorithms in capturing nonlinear dynamics, handling high-dimensional spaces, and providing stable solutions, highlighting their potential as an innovative tool in solving challenging and complex systems. 1 Introduction Partial differential equations (PDEs) are fundamental mathematical tools for modeling diverse phenomena in many fields such as physics, biology, chemistry, and economics. However, for many complex and high-dimensional PDEs, analytical solutions are often unattainable due to Yuan Chen: yc3y@mtmail.mtsu.edu To address this, numerical methods such as the finite-difference method (FDM) [1], finite-element method (FEM) [2], and finite-volume method (FVM) [3] have been developed to approximate solutions. These techniques have been effective in a variety of applications but face limitations in computational complexity, stability, and scalability, especially when applied to non-linear or high-dimensional problems.

We present a series of algorithms in tensor networks for anomaly detection in datasets, by using data compression in a Tensor Train representation. These algorithms consist of preserving the structure of normal data in compression and deleting the structure of anomalous data. The algorithms can be applied to any tensor network representation. We test the effectiveness of the methods with digits and Olivetti faces datasets and a cybersecurity dataset to determine cyber-attacks.

The problem of efficient quantum state learning, also called shadow tomography, aims to comprehend an unknown $d$-dimensional quantum state through POVMs. Yet, these states are rarely static; they evolve due to factors such as measurements, environmental noise, or inherent Hamiltonian state transitions. This paper leverages techniques from adaptive online learning to keep pace with such state changes. The key metrics considered for learning in these mutable environments are enhanced notions of regret, specifically adaptive and dynamic regret. We present adaptive and dynamic regret bounds for online shadow tomography, which are polynomial in the number of qubits and sublinear in the number of measurements. To support our theoretical findings, we include numerical experiments that validate our proposed models.

In this paper we present a study of the applicability and feasibility of quantum-inspired algorithms and techniques in tensor networks for industrial environments and contexts, with a compilation of the available literature and an analysis of the use cases that may be affected by such methods. In addition, we explore the limitations of such techniques in order to determine their potential scalability.

Variational Quantum Algorithms (VQAs) are often viewed as the best hope for near-term quantum advantage. However, recent studies have shown that noise can severely limit the trainability of VQAs, e.g., by exponentially flattening the cost landscape and suppressing the magnitudes of cost gradients. Error Mitigation (EM) shows promise in reducing the impact of noise on near-term devices. Thus, it is natural to ask whether EM can improve the trainability of VQAs. In this work, we first show that, for a broad class of EM strategies, exponential cost concentration cannot be resolved without committing exponential resources elsewhere. This class of strategies includes as special cases Zero Noise Extrapolation, Virtual Distillation, Probabilistic Error Cancellation, and Clifford Data Regression. Second, we perform analytical and numerical analysis of these EM protocols, and we find that some of them (e.g., Virtual Distillation) can make it harder to resolve cost function values compared to running no EM at all. As a positive result, we do find numerical evidence that Clifford Data Regression (CDR) can aid the training process in certain settings where cost concentration is not too severe. Our results show that care should be taken in applying EM protocols as they can either worsen or not improve trainability. On the other hand, our positive results for CDR highlight the possibility of engineering error mitigation methods to improve trainability.

Finding practically relevant problems where quantum computation offers a speedup compared to the best known classical algorithms is one of the central challenges in the field. Quantifying a speedup requires a provable convergence rate of the quantum algorithms, which restricts us to studying algorithms that can be analyzed rigorously. The impressive recent progress on building quantum computers gives us a new possibility: We can use heuristic quantum algorithms that can be run on current devices to demonstrate the speedup empirically. This however requires a hardware friendly implementation, i.e., a moderate number of qubits and shallow circuits. In recent years, more and more evidence has been found supporting machine learning tasks as good candidates for demonstrating quantum advantage [1-4]. In particular, the so-called supervised learning setting, where in the simplest case the goal is to learn a binary classifier of classical data, received much attention. The reasons are manifold: (i) The algorithms only require classical access to data.

Quantum machine learning has the potential for a transformative impact across industry sectors and in particular in finance. In our work we look at the problem of hedging where deep reinforcement learning offers a powerful framework for real markets. We develop quantum reinforcement learning methods based on policy-search and distributional actor-critic algorithms that use quantum neural network architectures with orthogonal and compound layers for the policy and value functions. We prove that the quantum neural networks we use are trainable, and we perform extensive simulations that show that quantum models can reduce the number of trainable parameters while achieving comparable performance and that the distributional approach obtains better performance than other standard approaches, both classical and quantum. We successfully implement the proposed models on a trapped-ion quantum processor, utilizing circuits with up to $16$ qubits, and observe performance that agrees well with noiseless simulation. Our quantum techniques are general and can be applied to other reinforcement learning problems beyond hedging.

We present a novel method for initializing layers of tensorized neural networks in a way that avoids the explosion of the parameters of the matrix it emulates. The method is intended for layers with a high number of nodes in which there is a connection to the input or output of all or most of the nodes. The core of this method is the use of the Frobenius norm of this layer in an iterative partial form, so that it has to be finite and within a certain range. This norm is efficient to compute, fully or partially for most cases of interest. We apply the method to different layers and check its performance. We create a Python function to run it on an arbitrary layer, available in a Jupyter Notebook in the i3BQuantum repository: https://github.com/i3BQuantumTeam/Q4Real/blob/e07c827651ef16bcf74590ab965ea3985143f891/Quantum-Inspired%20Variational%20Methods/Normalization_process.ipynb

Neural network approaches to approximate the ground state of quantum hamiltonians require the numerical solution of a highly nonlinear optimization problem. We introduce a statistical learning approach that makes the optimization trivial by using kernel methods. Our scheme is an approximate realization of the power method, where supervised learning is used to learn the next step of the power iteration. We show that the ground state properties of arbitrary gapped quantum hamiltonians can be reached with polynomial resources under the assumption that the supervised learning is efficient. Using kernel ridge regression, we provide numerical evidence that the learning assumption is verified by applying our scheme to find the ground states of several prototypical interacting many-body quantum systems, both in one and two dimensions, showing the flexibility of our approach.