【作者】 王洪福；

【导师】 张寿； 赵永芳；

【作者基本信息】 哈尔滨工业大学 ， 光学， 2010， 博士

【摘要】 量子计算是利用量子力学原理进行信息处理的新近发展起来的前沿学科,近20年来的量子计算理论研究表明,量子计算在很多方面比经典计算优越得多,特别是在量子系统的模拟、大数因子分解和无序数据库搜索等问题上尤为突出。对于在无序数据库中搜索若干特定目标时,Grover量子搜索算法可以对许多（虽不是全部）启发式搜索的经典算法起到实质性的二次加速作用。Grover量子搜索算法在搜索时忽略搜索元素的性质,而把注意力放在那些元素的指标上,因此具有很强的通用性。同时,它可以有效地破译DES密码体系,具有加速搜索密码系统密钥的潜在用途。本文主要围绕Grover量子搜索算法中的相位匹配问题、算法的物理实现以及应用等方面进行了研究,具体内容如下:量子纠缠是量子计算和量子信息处理中的重要资源,是产生量子加速的根本原因。基于线性光学和腔QED技术,分别提出制备量子多体纠缠态的物理方案。实验装置由简单的线性光学元件、原子–腔相互作用系统、单光子态、最大和非最大的双光子纠缠态、以及常规光子探测器所组成。在探测过程中,常规光子探测器的使用极大地降低了实验上对高质量探测器探测效率的需求,简化了实验的实现。利用数学计算方法,研究和讨论了Grover量子搜索算法中的相位匹配问题,并提出π/3.61相位搜索算法。在这个算法中,搜索成功的概率至少为94.11%。π/3.61相位搜索算法克服了Grover量子搜索算法中成功概率随目标项数目增多而急剧下降的弱点,同时也证明了Grover量子搜索算法能够鲁棒的反对噪声和一定的扰动。利用原子间偶极相互作用和原子–腔相互作用,提出在腔QED中实现两量子比特Grover量子搜索算法。在这个方案中,实现Grover量子搜索算法所需要的两量子比特条件相位门操作可以很容易地被实现,且不需要执行辅助的单量子比特操作;同时,由于使用了强经典场,这个方案对于腔场衰减和热场效应不敏感。基于Grover量子搜索算法,提出宇称（奇偶）确定算法和二次剩余算法。在宇称确定算法中,通过对满足条件f（x） = -1的元素进行计数,函数f（x）的宇称可以被确定。同时,讨论了不同情况下这个算法计算复杂度的上界和下界。在二次剩余算法中,算法的计算消耗主要集中在计算模M的二次剩余以及所需的迭代次数上。经典计算机上,求解二次剩余方程需要进行M/2次计算,利用量子算法则可以在O(M^1/2/2)步内以接近100%的成功概率求出二次剩余方程的解,实现了相对经典计算的二次加速。基于广义Grover量子搜索算法,提出应用量子计算机算法直接测量任意一个未知的两量子比特纯态系统的纠缠度。我们具体地构建了广义Grover迭代算子,通过对两体纯态系统的两个拷贝应用量子算法以及对辅助工作量子比特进行测量,可以得到系统纠缠度的一个较好的、近似的估计值。这个方案在实验上的实现对于更加复杂的、任意量子比特数目的、有限维数量子系统的纠缠测量将是一个重大的推进,同时能够展现出量子计算机强大的计算能力。基于腔QED技术,提出实现量子离散Fourier变换的有效的量子线路和物理方案。利用单原子–腔相互作用,提出物理方案实现N-比特量子离散Fourier变换。在这个方案中,通过发送原子通过一系列的经典场和腔场以及适当改变腔场的频率,得到一个可调的两量子比特条件相位门。在消相干时间范围内,所有的单比特和两比特量子门操作都能够被完成,有利于实现多比特量子Fourier变换。利用双原子–腔相互作用,提出物理方案实现N-比特量子离散Fourier变换。在这个方案中,基于CNOT门和SWCZ门操作（取代了原量子Fourier变换线路中复杂的受控-R_k门和SWAP门操作）和单量子比特门操作,设计了一个新的量子线路实现量子离散Fourier变换,并提出具体的原子–腔相互作用和原子–微波共振相互作用过程来实现这个量子线路。同时,我们提出具体的实验步骤并分析和讨论了这两个方案在实验实现上的可行性。更多还原

【Abstract】 Quantum computation, which processes information by using the principles of quan-tum mechanics, is a frontier discipline that is freshly developed. During the past twodecades, the study on the theory of quantum computation shows that quantum computa-tion can outperform classical computation in many aspects, especially for the problems ofquantum system simulation, large prime factorization, searching in a disorder database,etc.. For searching several marked items in a disorder database, Grover quantum search al-gorithm achieves quadratic speedup for several （although not all） heuristic classical searchalgorithm. Grover quantum search algorithm neglects the character of the searching ele-ments and concentrates on the indices of those elements, thus it has strong universalness.In the meantime, it can efficiently decode DES coding system and has potential purposefor speeding up the search of cryptographic key in coding system. In this paper, we focuson the phase matching problem, physical realization, and application of Grover quantumsearch algorithm. The main contents are presented as the followings:Quantum entanglement, which is the fundamental cause that produces quantumspeedup, is an important quantum resource in quantum information processing and lies atthe heart of quantum computation. Based on the linear optics and cavity QED technolo-gies, we present physical schemes for preparing quantum multipartite entangled states.The experimental setup comprises simple linear optical elements, maximally and non-maximally entangled two-photon polarization states, and conventional photon detectors.In the detection process, the use of conventional photon detectors greatly decreases thehigh-quality requirements of detectors in realistic experiments.We investigate the phase matching problem in Grover quantum search algorithm byusing the method of mathematical calculation, and presentπ/3.61 phase search algorithm.In this algorithm, the success probability for searching is at least 94.11%.π/3.61 phasesearch algorithm overcomes the weakness that the success probability of getting correctresults usually decreases ?eetly with the increase of marked items when Grover quantumsearch algorithm is applied to search an unordered database. In the meantime, this workalso indicates that Grover quantum search algorithm is considerably robust to certainkinds of perturbations and is robust against modest noise in the initialization procedure; we present a scheme for implementing two-qubit Grover quantum search based on theatom-atom dipole-dipole interaction and the atom-cavity interaction in cavity QED. In thisscheme, all the two-qubit conditional phase operations required to achieve the two-qubitGrover quantum search can be implemented directly without performing any auxiliarysingle-qubit rotation operations. Moreover, the scheme is insensitive to both the cavitydecay and the thermal field with the assistance of a strong classical field.We present parity determination algorithm and quadratic residue algorithm basedon Grover quantum search algorithm. In the parity determination algorithm, the parity offunction f（x） can be determined by counting exactly the number of satisfying f（x） = -1.We discuss the up and low bounds of the computational complexity of this algorithm. Inthe quadratic residue algorithm, the cost of the algorithm mainly depends on the calcula-tions of quadratic residue modulo M and the number of iterations. In classical computer,it needs M/2 calculations for solving quadratic residue equation. While in quantum com-puter, it needs only O(M^1/2) steps with the success probability of 100%, which realizesquadratic speedup compared with classical computation.We present a method to measure directly the concurrence of an arbitrary two-qubitpure state system based on generalized Grover quantum search algorithm. We constructthe explicit generalized Grover quantum iteration operator and the concurrence can be cal-culated by applying quantum algorithm to two available copies of the bipartite system, anda final measurement on the auxiliary working qubits gives a better estimation of the con-currence. The implementation of the scheme would be an important step toward quantuminformation processing and more complex entanglement measure of finite-dimensionalquantum system with an arbitrary number of qubits. In the meantime, the scheme woulddirectly show the power of quantum computer.We present quantum circuits and physical protocols for implementing quantum dis-crete Fourier transformation based on cavity QED. Firstly, we present a physical protocolfor the implementation of N-bit quantum discrete Fourier transformation by using sin-gle atom-cavity interaction. In the protocol we design a tunable two-qubit conditionalphase gate by sending the atoms through a series of classical fields and cavities and bychanging the frequency of the cavity appropriately. All the one-bit and two-bit quantumgate operations can be achieved before the decoherence sets in, which are very powerfulfor the implementation of N-bit quantum Fourier transformation. Secondly, we present a physical protocol for implementing N-bit quantum discrete Fourier transformation byusing single two-atom-cavity interaction. In this protocol, we give a new quantum cir-cuit for implementing the quantum discrete Fourier transformation by using CNOT andSWCZ gate operations instead of the controlled-R_k and SWAP gate operations, togetherwith several one-qubit operatons. We also present the explicit atom-cavity interaction andatom-microwave interaction processes for the implementation of the quantum circuit. Inthe meantime, we present the detailed experimental procedure and analyze the experi-mental feasibility of the two protocols.更多还原