Researchers unlock classical processes via spin-1/2 XXZ chains with varying Hurst exponents

The connection between quantum mechanics and classical stochastic processes continues to reveal surprising links, and new research demonstrates how quantum systems can generate classical behaviours with complex statistical properties. Zoltán Udvarnoki, Gábor Fáth, and Miklós Werner, from Eötvös Loránd University and the Wigner Research Centre for Physics, alongside Örs Legeza, investigate this relationship using the spin-1/2 XXZ chain, a fundamental model in quantum physics. Their work reveals that this system generates classical processes exhibiting fractional characteristics, meaning their behaviour displays long-range correlations and varying degrees of roughness or persistence. Importantly, the researchers demonstrate that specific symmetries within the quantum model directly influence the scaling of these classical processes, offering insights into how quantum properties translate to classical statistical behaviours and potentially providing a new framework for simulating complex systems.

Entangled quantum mechanical states in one dimension represent and simulate classical stochastic processes possessing nontrivial statistical properties. Long-range quantum correlations translate into fractional processes, with their asymptotic Hurst exponents characterizing roughness and persistence. This work explores this analogy specifically within the spin-1/2 XXZ chain, investigating properties of four distinct classical two-state processes that this quantum system generates. These processes exhibit fractional characteristics with varying Hurst exponents, demonstrating a connection between quantum entanglement and classical stochasticity.

Tensor Networks for Long-Range Time Series

This compilation assembles research papers, mathematical functions, and computational techniques related to time series analysis, quantum computing, tensor networks, and statistical physics. The core themes revolve around understanding and modeling time series data exhibiting long-range dependence, where past values persistently influence future values. Techniques like fractional differencing and the Hurst exponent are central to this analysis. Tensor networks, particularly matrix product states, serve as a powerful tool for representing and simulating quantum many-body systems, and are increasingly applied to classical machine learning and time series modeling.

Matrix product states efficiently represent high-dimensional data with reduced computational cost. The document also explores quantum-inspired algorithms and the potential of actual quantum computers for tasks like time series generation and option pricing. Concepts from statistical physics, like correlation functions and critical phenomena, are applied to understand the behavior of time series and quantum systems. Furthermore, the use of tensor networks for generative modeling, creating new data resembling the training data, is a growing area of research. The Hurst exponent measures long-range dependence in a time series, with values greater than 0.

5 indicating persistence, less than 0. 5 indicating anti-persistence, and 0. 5 suggesting a random walk. Fractional Brownian motion and fractional Gaussian noise are stochastic processes used to model time series with long-range dependence. Key libraries and tools mentioned include opeNAMPS, an open-source library for non-Abelian matrix product states, and Differentiable Hierarchy, a framework for differentiable tensor networks.

Quantum Spins Generate Classical Fractional Randomness

Researchers have demonstrated a remarkable connection between one-dimensional quantum systems and classical stochastic processes, revealing how the behavior of quantum spins can be used to generate and simulate complex, seemingly random phenomena. The team discovered that specific properties of the spin-1/2 XXZ chain, a model of interacting quantum spins, directly correspond to characteristics of four different classical two-state processes, effectively translating quantum mechanics into a framework for understanding classical randomness. Experiments revealed that these generated processes exhibit fractional characteristics, meaning their statistical properties fall between those of completely predictable and entirely random behaviors, and are defined by varying Hurst exponents which quantify roughness and persistence. The research demonstrates that the continuous symmetries present in the XXZ chain give rise to logarithmic scaling in the generated classical processes, a significant finding for understanding the relationship between quantum symmetries and classical statistical properties.

Researchers leveraged advanced numerical methods to substantiate these findings and explore the short-range properties of the model. The team showed how to generate classical time series from quantum states, specifically the ground state of the XXZ model, and demonstrated that the fractal properties of these time series reflect the statistical properties of magnetic domains within the quantum system. This breakthrough delivers a new approach to generating classical stochastic processes, offering a powerful tool for modeling complex phenomena in fields ranging from finance and climate science to materials science and beyond.

Quantum Chain Generates Classical Fractional Processes

The research demonstrates a connection between one-dimensional physical systems and classical stochastic processes, specifically showing how the spin-1/2 XXZ chain can generate processes with fractional characteristics. Researchers investigated four classical two-state processes generated by this system, finding that they exhibit fractional behaviour with varying Hurst exponents, which characterise roughness and persistence. The study highlights that continuous symmetries within the XXZ chain lead to logarithmic scaling in these generated processes. This work establishes a framework for understanding how the properties of quantum systems can be linked to the statistical characteristics of classical stochastic processes. The findings suggest that the XXZ chain, due to its inherent symmetries, can produce processes with specific scaling properties. Future research could focus on exploring how modifications to the quantum system might allow for more precise control over the statistical properties of the generated classical processes, potentially leading to new methods for simulating complex stochastic phenomena.