  Digital quantum computers are potentially an ideal platform for simulating
open quantum many-body systems beyond the digital classical computers. Many
studies have focused on obtaining the ground state by simulating time dynamics
or variational approaches of closed quantum systems. However, dynamics of open
quantum systems has not been given much attention with a reason being their
non-unitary dynamics not natural to simulate on a set of unitary gate
operations in quantum computing. Here we study prototypical models in open
quantum systems with Trotterisations for the modified stochastic
Schr{\"o}dinger equation (MSSE). Minimising the leading error in MSSE enables
to optimise the quantum circuits, and we run the optimised circuits with the
noiseless \textit{quantum assembly language (QASM) simulator} and the noisy IBM
Quantum devices. The \textit{QASM simulator} enables to study the reachable
system size that is comparable to the limits of classical computers. The
results show that the nonequilibrium critical phenomena in open quantum systems
are successfully obtained with high precision. Furthermore, we run the
algorithm on the IBM Quantum devices, showing that the current machine is
challenging, to give quantitatively accurate time dynamics due to the noise.
Despite errors, the results by IBM devices qualitatively follow the trend of
critical behaviour and include a possibility to demonstrate quantum advantage
when the noise is reduced. We discuss how much noise should be reduced for a
certain fidelity using the noise model, which will be crucial to demonstrate
quantum advantage from future quantum devices.
