State Change, Complexity and Fractal in Quantum Systems

Masanori Ohya

Feng Bi, Chuan‐Feng Li

In the recent work of Kiss et al. [ Phys. Rev. Lett. 107 (2011) 100501 ], the evolvement of two-qubit quantum states in a measurement-based purification process is studied. As they pointed out, the purification results manifest sensitivity to the applied initial states. The convergence regions to different stable circles are depicted on a complex plane. Because of the result patterns' likeness to typical fractals, we make further study on the interesting patterns' connection to fractals. Finally, through a numerical method we conclude that the boundaries of different islands of the patterns are fractals, which possess a non-integral fractal dimension. Also, we show that the fractal dimension would vary with the change of the portion of the noise added to the initial states.

M. Ohya

Jincheng Zhang

Fractal Dynamics & Quantum Complexity The inherent complexity of quantum systems necessitates innovative approaches to control and manipulation. This research explores the potential of utilizing fractal dynamics to stabilize quantum systems, envisioning a 'quantum fractal' capable of self-organizing into complex patterns. This project focuses on the development of a novel algorithm, the 'Fractal Resonance' method, designed to harness the self-similarity inherent in fractal structures to achieve a state of quantum stability. The core mechanism involves a carefully orchestrated iterative process applied to quantum entanglement, promoting self-organization and stabilization through fractal self-similarity. This work addresses the limitations of current quantum computing by presenting a fundamentally different approach that leverages the principles of fractal geometry to enhance quantum coherence and robustness. The resulting 'quantum fractal' promises a pathway towards more stable and controllable quantum systems.

Gopal Chandra Santra, Alex Windey, Soumik Bandyopadhyay +2 more

Complex quantum systems -- composed of many, interacting particles -- are intrinsically difficult to model. When a quantum many-body system is subject to disorder, it can undergo transitions to regimes with varying non-ergodic and localized behavior, which can significantly reduce the number of relevant basis states. It remains an open question whether such transitions are also directly related to an abrupt change in the system's complexity. In this work, we study the transition from chaotic to integrable phases in several paradigmatic models, the power-law random banded matrix model, the Rosenzweig--Porter model, and a hybrid SYK+Ising model, comparing three complementary complexity markers -- fractal dimension, von Neumann entanglement entropy, and stabilizer Rényi entropy. For all three markers, finite-size scaling reveals sharp transitions between high- and low-complexity regimes, which, however, can occur at different critical points. As a consequence, while in the ergodic and localized regimes the markers align, they diverge significantly in the presence of an intermediate fractal phase. Additionally, our analysis reveals that the stabilizer Rényi entropy is more sensitive to underlying many-body symmetries, such as fermion parity and time reversal, than the other markers. As our results show, different markers capture complementary facets of complexity, making it necessary to combine them to obtain a comprehensive diagnosis of phase transitions. The divergence between different complexity markers also has significant consequences for the classical simulability of chaotic many-body systems.

Longzhao Lu, Xiangyang Yu