Developed as a workhorse for classical simulations of quantum algorithms and quantum many-body systems, Tensor Network methods have entered the scientific mainstream in quantum physics. Among various types of tensor networks, Tensor Trains (commonly know as Matrix Product States in the quantum computing community) have already found applications in machine learning. These methods often rely on a powerful linear algebra tool called the Singular Value Decomposition (SVD). Several conditional GAN architectures for image denoising incorporate SVD as a single-cut decomposition step applied to generator feature maps. In this work we introduce TT-Net, which replaces the per-channel SVD denoising block with a two-cut tensor-train decomposition capable of accessing cross-channel information directly, a capability absent from contemporary alternatives. In a controlled comparison differing only in this decomposition mechanism, TT-Net outperforms SVD-Net on PSNR and SSIM across all three noise types tested (Gaussian, motion blur, and salt-and-pepper), supporting the hypothesis that cross-channel access improves denoising quality. Training-dynamics analysis further shows that TT-Net's adversarial loss term consistently saturates to a stagnant state across all three noise types, more so than SVD-Net's, while reconstruction quality continues to improve regardless, raising an open question about the adversarial component's contribution that this work identifies but does not resolve. Furthermore, for Gaussian noise our method outperforms both the EigenGAN and the state of the art Pix2pix method which does not assume any linear algebra decompositions and does not retain any linear algebra information. Our manuscript shows how quantum inspired tools can be used as practical real world feature filters for deep learning applications.
Developed as a workhorse for classical simulations of quantum algorithms and quantum many-body systems, Tensor Network methods have entered the scientific mainstream in quantum physics. Among various types of tensor networks, Tensor Trains (commonly know as Matrix Product States in the quantum computing community) have already found applications in machine learning. These methods often rely on a powerful linear algebra tool called the Singular Value Decomposition (SVD). Several conditional GAN architectures for image denoising incorporate SVD as a single-cut decomposition step applied to generator feature maps. In this work we introduce TT-Net, which replaces the per-channel SVD denoising block with a two-cut tensor-train decomposition capable of accessing cross-channel information directly, a capability absent from contemporary alternatives. In a controlled comparison differing only in this decomposition mechanism, TT-Net outperforms SVD-Net on PSNR and SSIM across all three noise types tested (Gaussian, motion blur, and salt-and-pepper), supporting the hypothesis that cross-channel access improves denoising quality. Training-dynamics analysis further shows that TT-Net's adversarial loss term consistently saturates to a stagnant state across all three noise types, more so than SVD-Net's, while reconstruction quality continues to improve regardless, raising an open question about the adversarial component's contribution that this work identifies but does not resolve. Furthermore, for Gaussian noise our method outperforms both the EigenGAN and the state of the art Pix2pix method which does not assume any linear algebra decompositions and does not retain any linear algebra information. Our manuscript shows how quantum inspired tools can be used as practical real world feature filters for deep learning applications.
Synthetic Aperture Radar (SAR) imaging is crucial in remote sensing due to its ability to produce high-resolution images regardless of weather conditions or daylight. However, SAR images often suffer from various types of noise, especially speckle noise, which degrades image quality and complicates data analysis. Traditional noise reduction techniques face challenges in balancing noise suppression with the preservation of image details. Recent deep learning approaches, such as U-Net architectures, have made strides in addressing these issues. Simultaneously, quantum computing has emerged as a promising field that can potentially enhance computational methods in image processing. We propose a hybrid classical-quantum U-Net framework that integrates quantum-classical algorithms with tensor networks for improved noise reduction in SAR images. Leveraging the representational capacity of tensor networks and the computational strengths of quantum algorithms, our method aims to surpass the limitations of classical techniques in managing complex noise patterns. In our architecture, certain layers of the U-Net are implemented using quantum circuits specifically optimized for noise suppression. The inclusion of tensor networks enables efficient handling of high-dimensional data within the hybrid model. The idea is to perform extensive experiments on standard SAR datasets to demonstrate that our hybrid U-Net outperforms traditional denoising methods and purely classical deep learning models in both noise reduction and detail preservation. Quantitative assessments using metrics like Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) would be used for comparison. Moreover, we will evaluate the computational efficiency of our approach, underscoring the potential of quantum computing to expedite complex image processing tasks. This work aims to highlight the feasibility and benefits of integrating quantum computing into practical remote sensing applications. Our hybrid quantum-classical U-Net architecture with tensor networks paves the way for advanced noise reduction techniques in SAR imaging and potentially other domains. Future research will focus on optimizing quantum circuit designs for specific noise types and exploring the scalability of our approach alongside advancements in quantum hardware.
The rapid advancement of quantum-inspired classical algorithms, particularly Quantics Tensor Trains (QTT), has demonstrated that exponentially large vectors can be manipulated efficiently via Matrix Product States (MPS). Conventionally, the compression of these tensor networks relies exclusively on Singular Value Decomposition (SVD)-based truncation. While SVD is optimal for minimizing the Frobenius norm error, it remains structurally blind to exact algebraic correlations, such as reversible logic gates or Clifford symmetries, that do not require numerical approximation to factor out. In this work, we propose a hybrid compression protocol that integrates the topological rigor of ZX-Calculus with the numerical power of SVD. We construct an explicit isomorphism between Rank-3 MPS tensors and ZX-diagrams, allowing us to subject QTT representations to formal diagrammatic rewriting rules prior to numerical truncation. We establish that this Topological Preconditioning can algebraically collapse the effective bond dimension (χ) and T-count of the underlying tensor network without information loss. By benchmarking the algorithmic implementation against hardware-accelerated SVD on discretized functions and stabilizer states, we demonstrate that ZX-driven algebraic erasure fundamentally bypasses standard [Formula: see text] bottlenecks, yielding up to a computational speedup. Finally, we formalize this advantage by defining the structural complexity class [Formula: see text], proving that for systems characterized by algebraic symmetry, this work reflects our ongoing effort to deepen understanding of Categorical Quantum Mechanics, aiming to contribute to the optimization of high-dimensional linear algebra.
Abstract Tensors are increasingly ubiquitous in various areas of applied mathematics and computing, and tensor decompositions are of practical significance and benefit many applications in data completion, image processing, computer vision, collaborative filtering, etc. Recently, Kilmer and Martin propose a new tensor factorization strategy, tensor singular value decomposition (t-svd), which extends the matrix singular value decomposition to tensors. However, computing t-svd for high dimensional tensors costs much computation and thus cannot efficiently handle large scale datasets. Motivated by advantage of quantum computation, in this paper, we present a quantum algorithm of t-svd for third-order tensors and then extend it to order-p tensors. We prove that our quantum t-svd algorithm for a third-order N dimensional tensor runs in time N polylog ( N ) if we do not recover classical information from the quantum output state. Moreover, we apply our quantum t-svd algorithm to context-aware multidimensional recommendation systems, where we just need to extract partial classical information from the quantum output state, thus achieving low time complexity.
Tremendous progress has been witnessed in artificial intelligence where neural network backed deep learning systems have been used, with applications in almost every domain. As a representative deep learning framework, Generative Adversarial Network (GAN) has been widely used for generating artificial images, text-to-image or image augmentation across areas of science, arts and video games. However, GANs are computationally expensive, sometimes computationally prohibitive. Furthermore, training GANs may suffer from convergence failure and modal collapse. Aiming at the acceleration of use cases for practical quantum computers, we propose QuGAN, a quantum GAN architecture that provides stable convergence, quantum-states based gradients and significantly reduced parameter sets. The QuGAN architecture runs both the discriminator and the generator purely on quantum state fidelity and utilizes the swap test on qubits to calculate the values of quantum-based loss functions. Built on quantum layers, QuGAN achieves similar performance with a 94.98% reduction on the parameter set when compared to classical GANs. With the same number of parameters, additionally, QuGAN outperforms state-of-the-art quantum based GANs in the literature providing a 48.33% improvement in system performance compared to others attaining less than 0.5% in terms of similarity between generated distributions and original data sets. QuGAN code is released at https://github.com/yingmao/Quantum-Generative-Adversarial-Network
Tremendous progress has been witnessed in artificial intelligence, where neural network backed deep learning systems have been used, with applications in almost every domain. As a representative deep learning framework, Generative Adversarial Network (GAN) has been widely used for generating artificial images, text-to-image or image augmentation across areas of science, arts and video games. However, GANs are computationally expensive, sometimes computationally prohibitive. Furthermore, training GANs may suffer from convergence failure and modal collapse. Aiming at the acceleration of use cases for practical quantum computers, we propose QuGAN, a quantum GAN architecture that provides stable convergence, quantum-states based gradients and significantly reduced parameter sets. The QuGANarchitecture runs both the discriminator and the generator purely on quantum state fidelity and utilizes the swap test on qubits to calculate the values of quantum-based loss functions. Built on quantum layers, QuGAN achieves similar performance with a 94.98% reduction on the parameter set when compared to classical GANs. With the same number of parameters, addition-ally, QuGAN outperforms state-of-the-art quantum based GANsin the literature providing a 48.33% improvement in system performance compared to others attaining less than 0.5% in terms of similarity between generated distributions and original data sets.
Tensor Networks are a relatively new machine learning approach. The architectures proposed initially are inspired by approaches from quantum many-body physics simulations. One common layout is the matrix product state (MPS) also known as a tensor train optimized with gradient descent techniques. We introduce a global normalization condition, so that the MPS represents a quantum state. We investigate two optimization methods that find the locally optimal tensors and compare them regarding their effectiveness. One is based on gradient descent and the other on an adaptation of DMRG.