This paper presents a novel approach to approximating tensor eigenvalues using randomized benchmarking techniques. Traditional methods for tensor eigenvalue computation, particularly for large-scale tensors, often suffer from computational bottlenecks and scalability issues. Our method addresses this challenge by exploiting the inherent redundancy in tensor representations through randomized sampling. We develop a framework that efficiently estimates tensor eigenvalues by iteratively applying randomized measurements and analyzing the resulting data. The core of the method lies in the strategic selection of measurement bases and the subsequent application of benchmarking techniques to minimize the variance of eigenvalue estimates. The resulting algorithm offers a scalable solution suitable for handling tensors with a significantly higher dimensionality compared to conventional methods. This approach demonstrates a promising avenue for accelerating tensor computations and unlocking new possibilities in various scientific and engineering domains. The theoretical analysis provides insights into the convergence properties of the algorithm, and the proposed method is demonstrated with a theoretical framework.
This paper presents a novel approach to approximating tensor eigenvalues using randomized benchmarking techniques. Traditional methods for tensor eigenvalue computation, particularly for large-scale tensors, often suffer from computational bottlenecks and scalability issues. Our method addresses this challenge by exploiting the inherent redundancy in tensor representations through randomized sampling. We develop a framework that efficiently estimates tensor eigenvalues by iteratively applying randomized measurements and analyzing the resulting data. The core of the method lies in the strategic selection of measurement bases and the subsequent application of benchmarking techniques to minimize the variance of eigenvalue estimates. The resulting algorithm offers a scalable solution suitable for handling tensors with a significantly higher dimensionality compared to conventional methods. This approach demonstrates a promising avenue for accelerating tensor computations and unlocking new possibilities in various scientific and engineering domains. The theoretical analysis provides insights into the convergence properties of the algorithm, and the proposed method is demonstrated with a theoretical framework.
Many problems in modern scientific computing are challenging because of a \emph{curse of dimension}, where their mathematical formulation involves objects whose dimension is \emph{exponential} in the nominal "size" of the problem. Tensor networks can provide a compact representation for exponentially large vectors and matrices that arise in applications, but these representations do not always lead to reliable algorithms. This paper develops and analyzes techniques for randomized dimension reduction of tensor network data. These techniques support a suite of efficient algorithms for provably solving exponential-scale linear algebra problems, including trace estimation and eigenvalue approximation. The paper includes several stylized illustrations from quantum many-body physics with ambient dimension up to 2200.
The accurate simulation of complex systems, encompassing fields like materials science, quantum chemistry, and finance, often presents a formidable computational challenge. Traditional methods frequently struggle to scale effectively, particularly when dealing with high-dimensionality or intricate correlations. This paper introduces a novel simulation approach leveraging tensor networks (TN) in conjunction with quantum-inspired sampling techniques. The core idea is to approximate the dynamics of these complex systems by exploiting the inherent structure of tensor networks and employing sampling strategies inspired by quantum measurement processes. We demonstrate that this hybrid approach offers a pathway towards scalable simulations, achieving high accuracy while mitigating the computational bottlenecks associated with conventional methods. The key innovations lie in the adaptive construction of TNs based on the system's correlation structure and the utilization of quantum-inspired sampling methods to efficiently capture the relevant dynamics. This work presents a theoretical framework and outlines a methodology for implementing such simulations, paving the way for advancements in the modeling and understanding of complex systems.
Quantum state estimation is a fundamental task in quantum information processing, crucial for applications such as quantum metrology, quantum sensing, and quantum process tomography. However, conventional methods for estimating the state of a quantum system often suffer from significant computational complexity, particularly for large systems or when high precision is required. This work introduces a novel approach to quantum state estimation that leverages Randomized Benchmarking (RB) techniques. By adapting RB protocols specifically designed for quantum systems, we achieve efficient and accurate state estimation with a reduced number of measurements compared to traditional methods. The core idea is to use a series of carefully designed, randomized measurements to extract information about the unknown quantum state. The resulting estimation process avoids the need for exhaustive state tomography, dramatically reducing the computational burden. We detail the mathematical framework underlying this approach, including the formulation of the RB protocol, the estimation algorithm, and an analysis of the estimation error. The results demonstrate the effectiveness and potential of this method for various quantum state estimation scenarios.
Optimizing hyperparameters in deep learning models remains a significant bottleneck in the development and deployment of effective neural networks. Traditional methods, such as grid search and random search, often struggle with the exponentially growing search space, leading to substantial computational costs and limited exploration of potentially optimal configurations. This paper proposes a novel approach to deep learning hyperparameter optimization that leverages the principles of quantum-inspired optimization, specifically employing tensor network methods, notably Matrix Product States (MPS), to efficiently approximate the complex optimization landscape. Our method utilizes quantum-inspired sampling techniques to guide the exploration, effectively mimicking the probabilistic nature of quantum measurements. We demonstrate the feasibility and potential advantages of this approach through theoretical analysis and a conceptual framework. The core of our method lies in representing the hyperparameter space as a tensor network, allowing for efficient calculations of the objective function (e.g., validation loss) without exhaustively evaluating every possible combination. This significantly reduces the computational burden compared to conventional methods. The paper outlines a detailed algorithm and discusses key considerations for implementation, including tensor network construction, sampling strategies, and convergence criteria. We propose a framework for iteratively refining the tensor network representation based on observed validation performance, ultimately converging towards a high-performing hyperparameter configuration. The theoretical underpinning draws upon concepts from quantum mechanics, particularly the idea of measurement and probability distribution, to drive the optimization process. We believe this quantum-inspired approach offers a promising avenue for accelerating deep learning hyperparameter tuning and achieving state-of-the-art performance.
This paper investigates the theoretical basis and computational applications of adaptive tensor networks. The core claim is to establish a theoretical framework that dynamically adjusts the structure of tensor networks based on input data and computational process, enabling more efficient matrix operations and numerical analysis. This is motivated by the limitations of current methods in handling complex calculations and offers a potential solution to computational bottlenecks. The paper details a novel algorithm which employs tensor network properties to calculate and dynamically restructure the tensor network, achieving enhanced performance through automated adjustment. The research explores the potential of this approach to address the challenges associated with traditional matrix multiplication and numerical methods. We present a detailed description of the proposed algorithm, its theoretical underpinnings, and preliminary results demonstrating its effectiveness. Finally, we discuss the implications of this work for the future of computational science and engineering.
This paper introduces a novel method for generating high-quality algorithmic randomness based on the principles of tensor networks. Traditional pseudo-random number generators (PRNGs) often exhibit predictable patterns and limitations when generating truly random outputs. Our approach leverages the inherent structure within tensor networks through contractions to efficiently sample from complex probability distributions. By exploiting correlations between variables represented within the tensor network, we can produce random numbers that are demonstrably less predictable than those generated by conventional PRNGs. This method offers a fundamentally different perspective on randomness generation, moving beyond deterministic algorithms and embracing the statistical properties inherent in multi-dimensional data representations. We demonstrate the potential of this technique to generate higher-quality randomness for a range of applications where unpredictability is paramount. The core claim of this work is that high-quality algorithmic randomness can be generated by leveraging the structure inherent in tensor networks. The core mechanism utilizes tensor network contractions to efficiently sample from complex probability distributions, exploiting correlations between variables to produce truly random outputs. ---