  Designing noisy-resilience quantum algorithms is indispensable for practical
applications on Noisy Intermediate-Scale Quantum~(NISQ) devices. Here we
propose a quantum approximate optimization algorithm~(QAOA) with a very shallow
circuit, called loop-QAOA, to avoid issues of noises at intermediate depths,
while still can be able to exploit the power of quantum computing. The key
point is to use outputs of shallow-circuit QAOA as a bias to update the problem
Hamiltonian that encodes the solution as the ground state. By iterating a loop
between updating the problem Hamiltonian and optimizing the parameterized
quantum circuit, the loop-QAOA can gradually transform the problem Hamiltonian
to one easy for solving. We demonstrate the loop-QAOA on Max-Cut problems both
with and without noises. Compared with the conventional QAOA whose performance
will decrease due to noises, the performance of the loop-QAOA can still get
better with an increase in the number of loops. The insight of exploiting
outputs from shallow circuits as bias may be applied for other quantum
algorithms.
