Neural quantum states for non-Markovian open quantum dynamics in bosonic environments

Condensed-phase quantum dynamics are strongly shaped by vibrational, solvent, and protein environments, yet their nonperturbative, non-Markovian description requires a rapidly growing high-dimensional representation as environmental complexity increases. Here, we extend neural quantum states (NQS) to the dissipaton-embedded quantum master equation (DQME) for bosonic environments using a complex-valued restricted Boltzmann machine adapted to nonbinary bosonic occupations. An explicit filtering scheme further concentrates the neural representation on dynamically relevant dissipaton configurations. The resulting NQS-DQME approach reproduces representative quantum dissipative dynamics while effectively compressing the reduced density tensor (RDT). For an anharmonic model of condensed-phase hydrogen transfer, the method also reproduces weak-coupling tunneling dynamics. At stronger system-bath coupling, however, errors become localized in a small subset of RDT elements, showing that a small global variational residual does not necessarily ensure accurate chemically relevant observables. This limitation identifies observable-level error control as a key challenge for reliable long-time NQS propagation.