Quantum Harmonic Oscillator and Non-Markovian Decoherence: A Research Synthesis — E8 Intelligence Research

FINDING: The search results are dominated by pedagogical material on the quantum harmonic oscillator (QHO) and general decoherence, with one substantive research paper on non-Markovian open quantum systems (PRX 14, 011010). The Feynman-Vernon influence functional is not directly extracted from these sources; the core mathematical content is the QHO spectrum and the scaling of simulation complexity in non-Markovian systems. MATH: - Quantum harmonic oscillator: \( \hat{H} = \hbar\omega(\hat{a}^\dagger\hat{a} + \tfrac{1}{2}) \), eigenvalues \( E_n = \hbar\omega(n+\tfrac{1}{2}) \), \( n \in \mathbb{Z}_{\ge 0} \). - Creation/annihilation algebra: \( [\hat{a}, \hat{a}^\dagger] = 1 \). - PRX paper (sublinear scaling): The key result is that non-Markovian open quantum system simulations can be performed with sublinear scaling in the number of bath modes — specifically, the computational cost scales as \( O(N^\alpha) \) with \( \alpha < 1 \) (exact exponent not given in abstract, but the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com