绝热量子搜索算法中的纠缠与能量分析

Entanglement Versus Energy in Adiabatic Quantum Search Algorithms

为了进一步研究量子纠缠与量子计算速度及能量的关系,通过计算von Neumann纠缠熵,分析了时间复杂度分别为O(N^(1/2))和O(1)的绝热量子搜索算法的量子纠缠度随时间的变化关系,并对两者进行了比较.实验结果表明,量子纠缠对绝热量子计算的运行时间具有明显的影响,较大的纠缠可以导致更短的运行时间,反之亦然.同时对纠缠与能量的关系给出了一般性解释,即注入能量导致系统的纠缠增大,并因此缩短算法的运行时间.此外还分析了纠缠与量子系统初态的关系.实验表明系统初态形式不同,其纠缠度也不一样.初态为等幅叠加态的算法涉及的纠缠度明显大于初态为非等幅叠加态的算法.

This paper is concerned with the role of entanglement in speeding up of quantum algorithms as well as its effect of energy on entanglement. To this end, based on the von Neumann entropy, the evolution of entanglement is investigated with respect to time in adiabatic quantum search algorithms with complexity O(N^(1/2) ) and O(1), respectively, and the comparison between them are made accordingly. The analysis results support the point that entanglement impact significantly on the computational efficiency of both of the algorithms. That is, the greater amount entanglement is, the higher quantum computation speed is, and vice versa. Furthermore, the correlations between entanglement and energy are discussed. It is shown that the entanglement degree becomes larger when input of energy into a quantum system increases, and thus the computational time is reduced accordingly. In addition, the effect of the initial states of the system on entanglement is discussed. It is concluded that entanglement behave differently with respect to one initial state from another. In particular, when algorithms are equally weighted superposition initial states, their entanglement degree becomes larger than those with non-equally weighted superposition.