### Abstract

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 2n, O (2n) 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 (2m+1 nm /m!) 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.

Original language | English |
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Article number | 042335 |

Journal | Physical Review A - Atomic, Molecular, and Optical Physics |

Volume | 79 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Apr 2009 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

**Efficient quantum algorithm for preparing molecular-system-like states on a quantum computer.** / Wang, Hefeng; Ashhab, Sahel; Nori, Franco.

Research output: Contribution to journal › Article

*Physical Review A - Atomic, Molecular, and Optical Physics*, vol. 79, no. 4, 042335. https://doi.org/10.1103/PhysRevA.79.042335

}

TY - JOUR

T1 - Efficient quantum algorithm for preparing molecular-system-like states on a quantum computer

AU - Wang, Hefeng

AU - Ashhab, Sahel

AU - Nori, Franco

PY - 2009/4/1

Y1 - 2009/4/1

N2 - 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 2n, O (2n) 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 (2m+1 nm /m!) 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.

AB - 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 2n, O (2n) 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 (2m+1 nm /m!) 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.

UR - http://www.scopus.com/inward/record.url?scp=66049119029&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=66049119029&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.79.042335

DO - 10.1103/PhysRevA.79.042335

M3 - Article

VL - 79

JO - Physical Review A - Atomic, Molecular, and Optical Physics

JF - Physical Review A - Atomic, Molecular, and Optical Physics

SN - 1050-2947

IS - 4

M1 - 042335

ER -