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arXiv 0902.1419quant-ph
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在量子计算机上制备类分子系统态的有效量子算法

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

Hefeng Wang, S. Ashhab, Franco Nori

备注：
11 pages, 8 figures, errors are corrected, Journal information added
发表信息：
Phys. Rev. A 79, 042335 (2009)
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摘要：

我们提出了一种在量子计算机上制备纯态的有效量子算法，其中量子态对应于具有给定数目$m$的电子占据给定数目$n$的自旋轨道的分子系统的量子态。每个自旋轨道被映射到一个量子位：量子位的状态$| 1 >$和$| 0>$分别表示自旋轨道是否被电子占据。为了在$n$量子位的全希尔伯特空间中准备维度为$2^{n}$ %的一般状态，需要$O(2^{n})$受控非门，即，门的数量与量子位的数量成比例地缩放\emph{exponentially}。我们利用要准备的状态位于更小的希尔伯特空间中的事实，并且我们找到最多需要$O(2^{m+1} n^{m}/{m!})$门的算法，即，用量子位数$n$（提供$n\gg m$）缩放\emph{polynomially}。对氢分子和水分子的情况进行了数值模拟。数值模拟表明，当考虑系统的额外对称性时，用于准备状态的门的数量可以大幅减少，在本文考虑的例子中，从上述估计中减少了几个数量级。

英文摘要：

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 $2^{n}$%, $O(2^{n})$ controlled-NOT 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 $O(2^{m+1} n^{m}/{m!})$ gates, i.e., scales \emph{polynomially} with the number of qubits $n$, provided $n\gg m$. 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.

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