Efficient quantum algorithm for preparing molecular-system-like states on a quantum computer

We present an efficient quantum algorithm for preparing a pure state on a quantum computer, where the quantum state corresponds to that of a molecular system with a given number $m$ of electrons occupying a given number $n$ of spin orbitals. Each spin orbital is mapped to a qubit: the states $|1⟩$ and $|0⟩$ of the qubit represent, respectively, whether the spin orbital is occupied by an electron or not. To prepare a general state in the full Hilbert space of $n$ qubits, which is of dimension $2^n$, $O\left(2^n\right)$ controlled-NOT gates are needed, i.e., the number of gates scales exponentially with the number of qubits. We make use of the fact that the state to be prepared lies in a smaller Hilbert space, and we find an algorithm that requires at most $O\left(2^{m+1}{n}^m/m!\right)$ gates, i.e., scales polynomially with the number of qubits $n$, provided $n⪢m$. The algorithm is simulated numerically for the cases of the hydrogen molecule and the water molecule. The numerical simulations show that when additional symmetries of the system are considered, the number of gates to prepare the state can be drastically reduced, in the examples considered in this paper, by several orders of magnitude, from the above estimate.