An efficient quantum algorithm for preparing states of molecular systems
Hefeng Wang, Sahel Ashhab, Franco Nori · arXiv (Cornell University) · 2009
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 $2^{n}$%, $O(2^{n})$ CNOT gates are needed, i.e. the number of gates scales \emph{% 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 $O((2 n)^{m}/{m!})$ CNOT gates, i.e. scales \emph{polynomially} with the number of qubits $n$, provided $n\gg m$. The algorithm is simulated numerically to prepare the electronic states of the hydrogen molecule and the water molecule. We show that when additional symmetries of the system are considered, the number of gates to prepare the state can be reduced dramatically.