  In this work, we give a quantum algorithm for solving the ground states of a
class of Hamiltonians. The mechanism of the exponential speedup that appeared
in our algorithm comes from dissipation in open quantum systems. To utilize the
dissipation, the central idea is to treat $n$-qubit density matrices $\rho$ as
$2n$-qubit pure states $|\rho\rangle$ by vectorization and normalization. By
doing so, the Lindblad master equation (LME) becomes a Schr\"odinger equation
with non-Hermitian Hamiltonian $L$. The steady-state $\rho_{ss}$ of the LME,
therefore, corresponds to the ground states $|\rho_{ss}\rangle$ of Hamiltonians
with the form $L^\dag L$. The runtime of the LME has no dependence on $\zeta$
the overlap between the initial state and the ground state compared with the
Heisenberg scaling $\mathcal{O}(\zeta^{-1})$ in other algorithms. For the input
part, given a Hamiltonian $H$, under plausible assumptions, we give a
polynomial-time classical procedure to judge and solve whether there exists $L$
such that $H-E_0=L^\dag L$. For the output part, we define the mission as
estimating expectation values of arbitrary operators with respect to the ground
state $|\rho_{ss}\rangle$, which can be done surprisingly by an efficient
measurement protocol on $\rho_{ss}$ with no need to prepare
$|\rho_{ss}\rangle$. We give several pieces of evidence on the quantum hardness
of really preparing $|\rho_{ss}\rangle$, which indicates a potential complexity
separation between our algorithm and those projection-based quantum algorithms
such as quantum phase estimation. Further, we show that the Hamiltonians that
can be efficiently solved by our algorithms contain classically hard instances
assuming $\text{P}\neq \text{BQP}$. Later, we discuss and analyze several
important aspects of the algorithm including generalizing to other types of
Hamiltonians and the "non-linear`` dynamics in the algorithm.
