Quantum advantages in ground state preparation, combinatorial optimization, and quantum state preparation

We show that for any quantum Hamiltonian with an inverse-polynomial gap, the ground state can be prepared in a polynomial circuit depth to inverse-polynomial precision, if the system size is sufficiently large. The resulting circuit is composed of a polynomial number of Pauli rotations without ancilla qubit. Extending this result, we prove that for sufficiently large qubit number, any quantum state can be approximately prepared with a constant (polynomial) number of Pauli rotations to constant (inverse-polynomial) precision. Our theoretical findings reveal exponential quantum advantages in the prominent applications: ground state preparation, combinatorial optimization, and quantum state preparation.