  Quantum simulation of non-Markovian open quantum dynamics is essential but
challenging for standard quantum computers due to their non-Hermitian nature,
leading to non-unitary evolution, and the limitations of available quantum
resources. Here we introduce a hybrid quantum-classical algorithm designed for
simulating dissipative dynamics in system with non-Markovian environment. Our
approach includes formulating a non-Markovian Stochastic Schr\"odinger equation
with complex frequency modes (cNMSSE) where the non-Markovianity is
characterized by the mode excitation. Following this, we utilize variational
quantum simulation to capture the non-unitary evolution within the cNMSSE
framework, leading to a substantial reduction in qubit requirements. To
demonstrate our approach, we investigated the spin-boson model and dynamic
quantum phase transitions (DQPT) within transverse field Ising model (TFIM).
Significantly, our findings reveal the enhanced DQPT in TFIM due to
non-Markovian behavior.
