Complex Quantum Systems

C.-D. Yang, Shiang-Yi Han

The complex-valued quantum mechanics considers quantum motion on the complex plane instead of on the real axis, and studies the variations of a particle complex position, momentum and energy along a complex trajectory. On the basis of quantum Hamilton-Jacobi formalism in the complex space, we point out that having complex-valued motion is a universal property of quantum systems, because every quantum system is actually accompanied with an intrinsic complex Hamiltonian originating from the equation. It is revealed that the conventional real-valued quantum mechanics is a special case of the complex-valued quantum mechanics in that the eigen-structures of real and complex quantum systems, such as their eigenvalues, eigenfunctions and eigen-trajectories, are invariant under linear complex mapping. In other words, there is indeed no distinction between Hermitian systems, PT-symmetric systems, and non PT-symmetric systems when viewed from a complex domain. Their eigen-structures can be made coincident through linear transformation of complex coordinates.

V. M. Akulin

Pravir Malik

This paper explores the intersection of Complex Adaptive Systems (CAS) and whole-systems quantum computing, focusing on how CAS principles can enhance quantum computing methods and vice versa. CAS, characterized by complexity, adaptability, self-organization, and emergence, amongst other properties, are prevalent in various domains, including ecology, economics, and social systems. Whole-systems quantum computing, particularly through quantum annealing, manipulates the global quantum state of a system, utilizing entanglement and superposition to solve optimization problems. The paper outlines a methodology for modeling complex adaptive problems using quantum annealing, alongside case studies illustrating CAS theory’s application in quantum computation. By examining representative quantum computing companies the paper investigates their management of entropy-a key summary metric in CAS-and correlations with quantum supremacy. The integration of CAS dynamics into quantum computing is posited to aid in designing more advanced Hamiltonians and quantum algorithms, enhancing the capability to tackle complex, real-world problems more efficiently. The paper calls for further research into several promising avenues, such as the development of sophisticated Hamiltonians that encode complex interdependencies, the validation of CAS models on quantum hardware, and the potential for systematizing quantum supremacy through a virtuous cycle of CAS-informed quantum algorithm development.

Yuri Campbell, José Roberto Castilho Piqueira

The complexity measures role has become much clearer in recent years as they help to better understand complex systems dynamical behavior. Even though the large number of measures proposed to tackle this issue for classical systems, for quantum systems only Kolmogorov’s algorithm complexity extensions have been proposed. Hence, the present approach makes use of a new and mathematically well-established complexity measure for classical systems and extends it to assess quantum states complexity as well. Then the proposed extension is applied to a mixed state constructed with a W-state together with controlled white noise, showing a convex behavior of quantum state complexity. Thus, this reinforces the differences from previous known quantum complexities.