  The quantum open-system simulation is an important category of quantum
simulation. By simulating the thermalisation process at the zero temperature,
we can solve the ground-state problem of quantum systems. To realise the
open-system evolution on the quantum computer, we need to encode the
environment using qubits. However, usually the environment is much larger than
the system, i.e. numerous qubits are required if the environment is directly
encoded. In this paper, we propose a way to simulate open-system dynamics by
reproducing reservoir correlation functions using a minimised Hilbert space. In
this way, we only need a small number of qubits to represent the environment.
To simulate the $n$-th-order expansion of the time-convolutionless master
equation by reproducing up to $n$-time correlation functions, the number of
qubits representing the environment is $\sim \lfloor \frac{n}{2} \rfloor
\log_2(N_\omega N_\beta)$. Here, $N_\omega$ is the number of frequencies in the
discretised environment spectrum, and $N_\beta$ is the number of terms in the
system-environment interaction. By reproducing two-time correlation functions,
i.e. taking $n = 2$, we can simulate the Markovian quantum master equation. In
our algorithm, the environment on the quantum computer could be even smaller
than the system.
