Tremendous recent progress in informational, computational, and quantum technologies has brought complex systems whose functionality is directly based on their NP/#P-complexity to the frontiers of modern science, research, and engineering. The examples are various quantum many-body systems for quantum computing aimed at demonstrating supremacy over classical computers in solving the problems that require an exponentially large number of operations. One such problem is boson sampling. Many other examples emerge in material science, condensed matter, and nuclear physics. In particular, they include various mesoscopic systems with unusual properties determined by the cooperative, topological, or critical phenomena. The basic models of the phase transitions in the statistical physics, such as the Ising and monomer-dimer models, directly address the NP/#P-complexity of the critical phenomena. The protocols and models employed in modern communications and cryptography, as well as the mathematical objects/models in the graph theory, theory of fractals and chaos, number theory, and combinatorics also have an intimate connection with the NP- and #P-complete problems of computational complexity theory. An example of the latter is a matrix permanent computing of which was the first counting problem proven to be #P-complete and yet corresponded to an easy, linear-time polynomial problem in accepting paths. Remarkably, the graph theory and the Markov chain Monte Carlo method recently provided a fully polynomial randomized approximation scheme (FPRAS) for numerical computation of the permanent of nonnegative matrices and the ferromagnetic Ising model. Machine learning and artificial intelligence techniques constitute an alternative, rapidly progressing way of addressing NP/#P-complex problems.
The modern stage in the development of the broad multidisciplinary area of research outlined above is characterized by a transition from the abstract general analysis and schemes to the invention, design, implementation, and application of the real NP/#P-complexity-based systems.
This Special Issue presents this recent advance in the theory of the NP/#P-complexity of nature, finding new models and systems that demonstrate or implement the NP/#P-complexity, developing analytic and computational methods for the analysis of such systems, as well as establishing connections and analogies between different complex systems. We welcome multidisciplinary NP/#P-complexity related papers—from papers devoted to the pure mathematics of the NP/#P-complex objects/models and computational/approximation methods and algorithms, including the ones based on the machine learning and artificial intelligence techniques, to papers addressing the particular NP/#P-complex features of the actual systems in physics, chemistry, biology, material science, engineering, communication technologies, cryptography, quantum and classical computing, statistics, social sciences and more.
Prof. Vitaly Kocharovsky
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