  Simulating the dynamics of many-body quantum systems is believed to be one of
the first fields that quantum computers can show a quantum advantage over
classical computers. Noisy intermediate-scale quantum (NISQ) algorithms aim at
effectively using the currently available quantum hardware. For quantum
simulation, various types of NISQ algorithms have been proposed with individual
advantages as well as challenges. In this work, we propose a new algorithm,
truncated Taylor quantum simulator (TTQS), that shares the advantages of
existing algorithms and alleviates some of the shortcomings. Our algorithm does
not have any classical-quantum feedback loop and bypasses the barren plateau
problem by construction. The classical part in our hybrid quantum-classical
algorithm corresponds to a quadratically constrained quadratic program (QCQP)
with a single quadratic equality constraint, which admits a semidefinite
relaxation. The QCQP based classical optimization was recently introduced as
the classical step in quantum assisted eigensolver (QAE), a NISQ algorithm for
the Hamiltonian ground state problem. Thus, our work provides a conceptual
unification between the NISQ algorithms for the Hamiltonian ground state
problem and the Hamiltonian simulation. We recover differential equation-based
NISQ algorithms for Hamiltonian simulation such as quantum assisted simulator
(QAS) and variational quantum simulator (VQS) as particular cases of our
algorithm. We test our algorithm on some toy examples on current cloud quantum
computers. We also provide a systematic approach to improve the accuracy of our
algorithm.
