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.

Figure 1

Quantum circuit for the controlled-$\tilde{H}$ gate.

Figure 2

Quantum circuit for transforming a Bell-type state $|\beta ⟩=a|10⟩+b|01⟩$) to the initial state $|00⟩$. So, $|00⟩={\tilde{H}}^1\cdot \text{CNOT}\cdot X^1\cdot |\beta ⟩$.

Figure 3

Quantum circuit for transforming the three-qubit state $|{\Psi }_T⟩=\frac{1}{\sqrt{3}}|001⟩+\frac{1}{\sqrt{2}}|100⟩+\frac{1}{\sqrt{6}}|010⟩$ to the state ${|0⟩}^{\otimes 3}=|000⟩$.

Figure 4

Quantum circuit for transforming a general state of the one-electron system $|{\Psi }_T\left(n,1\right)⟩$ to the initial state ${|0⟩}^{\otimes n}$.

Figure 5

Quantum circuit for the decomposition of the unitary operator $Q\left(n-1,1\right)$.

Figure 6

(Color online) Quantum circuit for transforming a general state $|{\Psi }_T\left(n,m\right)⟩$ of the $m$-electron system to the initial state ${|0⟩}^{\otimes n}$.

Figure 7

Top quantum circuit for the recursive procedure to transform the state $|{\Psi }_T\left(n,1\right)⟩$ to the initial state ${|0⟩}^{\otimes n}$. It has two onefold control operations and one twofold operation in the middle. The bottom circuit shows its simplified version, without twofold operations.

Figure 8

Quantum circuit for two controlled unitary operators $\text{C-}U\left(p,q\right)$ and $\text{C-}U\left(p,q-1\right)$ and their simplified versions. The two unitary operators in the top circuit are expanded in the second circuit, which is simplified in the bottom circuit.