Hybrid Quantum Transition Algorithms for Scalable Simulation of Complex Networks Based on Lindblad Dynamics

Quantum transition simulations on complex networks capture superposition, interference, and decoherence beyond classical stochastic models. This review provides a unified, mathematically rigorous account of the theory, algorithms, and cross-domain applications of quantum-transition-based network simulation. Starting from the Schrödinger equation and density matrix formalism, we develop the Lindblad master equation for open system dynamics and extend it to hybrid quantum–classical stochastic differential equations and hierarchical equations of motion for non-Markovian environments. Three quantum walk classes are formalized—continuous-time, discrete-time, and quantum stochastic walks—showing how decoherence and network topology govern the transition from ballistic quantum spreading to classical diffusion. Four simulation algorithms are compared: exact diagonalization [O(D4) memory, O(D6) diagonalization cost, D ≤ 20 states]; Krylov subspace [O(D2m), D∼100 states]; quantum trajectory Monte Carlo [O(DM), D∼103 states]; and tensor networks [L∼104 sites, where L is the number of sites and D = dL is the Hilbert space dimension with local dimension d]. Seven metrics—fidelity, coherence, purity, participation ratio, mean first-passage time, von Neumann entropy, and spectral gap—are linked to experimental observables. Applications span physical systems (excitation transport), biological systems (environment-assisted quantum transport in the FMO complex), and socio-technical systems (opinion polarization). Key limitations and four emerging directions of machine learning-assisted optimization, non-Markovian hierarchical equations of motion, simulation-guided network design, and hybrid classical–quantum computation are identified.