  Various Hamiltonian simulation algorithms have been proposed to efficiently
study the dynamics of quantum systems on a quantum computer. The existing
algorithms generally approximate the time evolution operators, which may need a
deep quantum circuit that is beyond the capability of near-term noisy quantum
devices. Here, focusing on the time evolution of a fixed input quantum state,
we propose an adaptive approach to construct a low-depth time evolution
circuit. By introducing a measurable quantifier that characterizes the
simulation error, we use an adaptive strategy to learn the shallow quantum
circuit that minimizes that error. We numerically test the adaptive method with
electronic Hamiltonians of the $\mathrm{H_2O}$ and $\mathrm{H_4}$ molecules,
and the transverse field Ising model with random coefficients. Compared to the
first-order Suzuki-Trotter product formula, our method can significantly reduce
the circuit depth (specifically the number of two-qubit gates) by around two
orders while maintaining the simulation accuracy. We show applications of the
method in simulating many-body dynamics and solving energy spectra with the
quantum Krylov algorithm. Our work sheds light on practical Hamiltonian
simulation with noisy-intermediate-scale-quantum devices.
