  Universal quantum algorithms (UQA) implemented on fault-tolerant quantum
computers are expected to achieve an exponential speedup over classical
counterparts. However, the deep quantum circuits makes the UQA implausible in
the current era. With only the noisy intermediate-scale quantum (NISQ) devices
in hand, we introduce the quantum-assisted quantum algorithm, which reduces the
circuit depth of UQA via NISQ technology. Based on this framework, we present
two quantum-assisted quantum algorithms for simulating open quantum systems,
which utilize two parameterized quantum circuits to achieve a short-time
evolution. We propose a variational quantum state preparation method, as a
subroutine to prepare the ancillary state, for loading a classical vector into
a quantum state with a shallow quantum circuit and logarithmic number of
qubits. We demonstrate numerically our approaches for a two-level system with
an amplitude damping channel and an open version of the dissipative transverse
field Ising model on two sites.
