摘要
本文围绕着量子算法的研究和应用,共分为三个阶段,分别实现多量子位量子算法体系的研究与核磁共振(NMR)模拟实现;符合Grover算法实现过程的经典谐振子系统的设计与分析;基因信息处理中DNA复制和蛋白质合成符合量子信息处理机理的过程描述和模型建立三个阶段目标。
本文第一阶段首先通过算符法进一步完整地证明经典Grover算法是可以扩展的,即经典Grover算法中组成Grover迭代过程的W-H变换可以用任意幺正变换替代,对均匀叠加初态和目标态相位反转的操作可以用任意角度的位相旋转替代。核磁共振(NMR)技术被认为是最为有效地实现量子计算的物理体系之一。多量子算符代数理论可以将幺正变换分解为一系列有限的单量子门和对角双量子门的组合。我们以多量子算符代数理论为基础,结合矩阵的扩展Kronecker积、正移置换矩阵和位元反转置换矩阵,提出了实现任意相位旋转角度的扩展量子搜索算法、量子傅立叶变换和量子小波变换的核磁共振脉冲序列设计方法,并设计出了2量子位扩展Grover算法、3量子位经典Grover算法、量子傅立叶变换、量子Hart和D~((4))小波变换的量子逻辑线路和NMR脉冲序列。然后,运用核磁共振量子计算仿真程序(QCE),实现了2量子位多个位相变化角度Grover算法的搜索过程,并根据实际的迭代次数,讨论了最佳位相变化角度。同时,通过QCE,实现了3量子位经典Grover算法的搜索过程,实现了2、3、4量子位的量子傅立叶变换和3量子位量子Harr和D~((4))小波变换的变换过程。多量子位量子算法在核磁共振中实现的难点是解决好自旋核之间的耦合作用,即利用多量子算符代数理论,将代表自旋核之间耦合的高维幺正变换分解成单量子逻辑门和双量子基本逻辑门的组合序列,再根据设定的具体位相旋转角度,将量子逻辑门转换成相应的NMR脉冲序列组合。最后,我们通过经典相干和量子相干的概念,运用量子态的密度矩阵,对经典Grover算法实现过程中量子态之间的相干性进行了分析,得出了Grover算法量子相干性与搜索迭代次数的关系。
本文第二阶段是设计一种具有“波动特性”的宏观物理系统,以经典耦合谐振子系统为目标。由于该系统是通过实际的物理连接来实现各弹性振子的耦合,振子相对于平衡点两个方向的振动可以看成是两个态。如果将这些物理参数完全一致的弹性振子看成是量子计算所需的量子基态,量子计算所需的量子相干性由于实际的物理连接比纯量子系统要稳定的多,从其中找出某个振子就可以看成是无序数据项搜索问题。我们对这种具有“波动特性”的经典耦合谐振子系统的力学机理、系统哈密顿量、系统本征态和系统本征值等基本物理参数进行分析,证明了在其振动过程中,某个受到外界作用的振子,经过一定次数的往复振动后,整个系统的能量就被“集中”到该振子上,其能量达到最大,即作为目标振子被搜索到。而且,该振子被搜索到时系统往复振动的次数是振子数量的平方根数量级关系,证明了该系统可以实现Grover算法的过程。
基因信息处理的基本过程,DNA复制和蛋白质合成过程中的核甘酸基配对过程可看成是无序数据项的搜索问题。达尔文的进化理论从生物学的角度证明了基因信息处理是优化的。DNA复制和蛋白质合成是经典过程,实际的物理系统由于很难完全隔绝外界干扰,量子相干性的维持存在着很大的困难,要完全用量子信息处理的思想去描述它还存在着很多难以克服的问题。因此,考虑到量子相干性的维持,本文第三阶段通过符合Grover量子搜索算法的经典谐振子系统和量子隧道效应建立了核甘酸基配对和氢键形成的模型,提出了该模型分子生物学实验验证的思路,并从维持量子相干性的角度探讨了DNA和RNA聚合酶的催化作用,对聚合酶的催化机理提出了新的解释方法,从信息学的角度论证了生物信息处理的优化。
Focusing on the research and application of Quantum Algorithms,this paper was divided into three parts,which were the studying and Nuclear Magnetic Resonance(NMR) simulation of multi-qubit Quantum Algorithms;the design and analysis of classical coupled harmonic oscillators system which corresponded to Grover search algorithm;and,proposing the model of nucleotide base selection and H-bond formation in DNA replication and proteins synthesis,which corresponded to quantum information processing mechanism.
In the first part,based on the operator approach,we proved the generalized Grover search algorithm completely.That is,in Grover iteration,the Walsh-Hadamard transformation could be replaced by an arbitrary unitary transformation,and the phase inversions could be replaced by arbitrary phase rotations.NMR has been considered as one of the most effective physical system to realize quantum computation.As multiple-quantum operator algebra theory mentioned,any unitary transformation can be decomposed into a sequence of a limited number of one-qubit quantum gates and 2-qubit diagonal gates.Based on multiple-quantum operator algebra theory,the Generalized Kronecker Product,Perfect Shuffle Permutation Matrices and Bit-Reversal Permutation Matrices,we proposed the method to design NMR pulse sequences to implement Generalized Quantum Search Algorithm with arbitrary phase rotation,Quantum Fourier Transform and Quantum Wavelet Transform.Meanwhile,we designed the pulse sequences for 2-qubit generalized and 3-qubit classical Grover algorithm, Quantum Fourier Transform,Quantum Harr and D~((4)) Wavelet Transform.Then we experimental finished the algorithm with different phase rotations respectively in a 2-qubit system,on a Quantum Computer Emulator(QCE),and discussed the best phase rotating angle. Meanwhile,the 3-qubit classical Grover algorithm,2-,3- and 4-qubit Quantum Fourier Transform,3-qubit Quantum Harr and D~((4)) Wavelet Transform were simulated on QCE.The difficulty of realizing multi-qubit quantum algorithm is to overcome the coupling effect within spins.Based on multiple-quantum operator algebra theory,those high dimensional unitary transformation,which expressing coupling within spins,could be decomposed into a sequence of one-qubit and 2-qubit basic quantum gates.Setting the specific phase rotating angle,the quantum gates could be converted into NMR pulse sequences.At the end, according to the classical and quantum coherence,through the density matrix of quantum state,the quantum coherence during the iteration of Grover searching was studied,and the relation between quantum coherence and the number of iterations was proposed.
In the second part,we designed a macroscopic system with wave features,aimed to classical coupled harmonic oscillators system.Because of the physical conjunction to couple those small oscillators,the vibration of a oscillator around its equilibrium could be seemed as binary state of a qubit.Such classical wave system is far more stable against decoherence compared to its quantum counterpart.We analysized the dynamic mechanism,effective Hamiltonian,eigenvalues and the corresponding eigenmodes for the system.We proved that during the vibration,if one of those oscillators was given a elastic reflection,after few round vibrations,the energy of the system was concentrated to this oscillator,making it to the maximum energy.The oscillator was reached as the target for a unsorted search.Also,we found that when the energy of the target oscillator getting to the maximum,the times of vibration had a squarely relationship with the number of small oscillators.So Grover quantum search algorithm were proved to be physically implanted in such a macroscopic wave system.
The nucleotide base-pairing in DNA replication and proteins synthesis,which were basic procedures in genetic information processing,could be looked upon as an unsorted database search..The legacy of Charles Darwin—survival of the fittest,has proved,from the point of biology,that genetic information processing is optimal.DNA replication and proteins synthesis are classical.Because of the disturbance from circumstances,it is hard for them to maintain quantum coherence.So it seems there were some unovercomed difficulties to describe them under pure quantum situation.Based on the classical coupled harmonic oscillators system,which corresponded to Grover quantum search algorithm,and quantum tunneling effect,the model of the nucleotide base selection and H-bond formation was proposed in the last part of this paper.In addition,the catalysis of the DNA and RNA polymerase was discussed from the point of view of quantum coherence maintained.Finally, we proved the optimization of biological information processing through the point of informatics.
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