Quantum Adaptive Dynamic Chaos Integral Algorithm (QADCI) presents a novel approach to integral calculation, leveraging quantum self-adaptational chaos integral structures to dynamically adjust integral parameters and enhance performance, particularly in complex integration scenarios. This algorithm aims to address limitations of traditional integral methods by optimizing the integral process through a self-correcting mechanism. The core design hinges on the intrinsic properties of chaotic integrals, enabling automated parameter refinement to achieve superior results. The paper details the algorithm's architecture, the quantum self-adaptational process, and its demonstrated performance improvements across diverse integral types. This research explores the potential of quantum mechanics to fundamentally alter the way integrals are computed, offering a promising avenue for more efficient and accurate integral analysis.
Studies of integrable quantum many-body systems have a long history with an impressive record of success. However, quite surprisingly, an unambiguous definition of quantum integrability remains a matter of an ongoing debate. We contribute to this debate by presenting several observation motivated by a particular quantum model of
N spins
1/2. Remarkably, although this system is nonintegrable, it possesses a large number of mutually commuting local binary integrals of motion. We address the question of the number of independent integrals of motion. Strikingly, we find that, according to a common definition of independence of quantum integrals of motion (QIMs), this number equals to the number of spins,
N. A common wisdom would then suggest that the system is completely integrable, which is not the case. To resolve this conundrum, we dwell upon the notion of independence of QIMs. We argue that the common definition should be amended in order to match the intuitive notion of independence. We suggest a new definition, which, in particular, implies that some of the
N QIMs of the considered model are dependent. Still, even according to this strict definition,
N/2 integrals of motion remain independent. This challenges the intuition on the integrable quantum many-body systems as well as some of the most elaborated definitions of quantum integrability.