Experimental implementation of an adiabatic quantum optimization algorithm

We report the realization of a nuclear magnetic resonance computer with three quantum bits that simulates an adiabatic quantum optimization algorithm. Adiabatic quantum algorithms offer new insight into how quantum resources can be used to solve hard problems. This experiment uses a particularly well-suited three quantum bit molecule and was made possible by introducing a technique that encodes general instances of the given optimization problem into an easily applicable Hamiltonian. Our results indicate an optimal run time of the adiabatic algorithm that agrees well with the prediction of a simple decoherence model.     

Implementation of ternary Shor's algorithm based on vibrational states of an ion in anharmonic potential

It is widely believed that Shor's factoring algorithm provides a driving force to boost the quantum computing research.However, a serious obstacle to its binary implementation is the large number of quantum gates. Non-binary quantum computing is an efficient way to reduce the required number of elemental gates. Here, we propose optimization schemes for Shor's algorithm implementation and take a ternary version for factorizing 21 as an example. The optimized factorization is achieved by a two-qutrit quantum circuit, which consists of only two single qutrit gates and one ternary controlled-NOT gate. This two-qutrit quantum circuit is then encoded into the nine lower vibrational states of an ion trapped in a weakly anharmonic potential. Optimal control theory(OCT) is employed to derive the manipulation electric field for transferring the encoded states. The ternary Shor's algorithm can be implemented in one single step. Numerical simulation results show that the accuracy of the state transformations is about 0.9919.     

Experimental quantum deletion in an NMR quantum information processor

We report an NMR experimental realization of a rapid quantum deletion algorithm that deletes marked states in an unsorted database.Unlike classical deletion,where search and deletion are equivalent,quantum deletion can be implemented with only a single query,which achieves exponential speed-up compared to the optimal classical analog.In the experimental realization,the GRAPE algorithm was used to obtain an optimized NMR pulse sequence,and the efficient method of maximum-likelihood has been used to reconstruct the experimental output state.     

Quantum computers and unstructured search: finding and counting items with an arbitrarily entangled initial state

Grover's quantum algorithm for an unstructured search problem and the COUNT algorithm by Brassard et al. are generalized to the case when the initial state is arbitrarily and maximally entangled. This ansatz might be relevant with quantum subroutines, when the computational qubits and the environment are coupled, and in general when the control over the quantum system is partial.     

Experimental application of decoherence-free subspaces in an optical quantum-computing algorithm

For a practical quantum computer to operate, it is essential to properly manage decoherence. One important technique for doing this is the use of "decoherence-free subspaces" (DFSs), which have recently been demonstrated. Here we present the first use of DFSs to improve the performance of a quantum algorithm. An optical implementation of the Deutsch-Jozsa algorithm can be made insensitive to a particular class of phase noise by encoding information in the appropriate subspaces; we observe a reduction of the error rate from 35% to 7%, essentially its value in the absence of noise.     

Determining the parity of a permutation using an experimental NMR qutrit

We present the NMR implementation of a recently proposed quantum algorithm to find the parity of a permutation. In the usual qubit model of quantum computation, it is widely believed that computational speedup requires the presence of entanglement and thus cannot be achieved by a single qubit. On the other hand, a qutrit is qualitatively more quantum than a qubit because of the existence of quantum contextuality and a single qutrit can be used for computing. We use the deuterium nucleus oriented in a liquid crystal as the experimental qutrit. This is the first experimental exploitation of a single qutrit to carry out a computational task.     

Eigenstates of an operating quantum computer: hypersensitivity to static imperfections

We study the properties of eigenstates of an operating quantum computer which simulates the dynamical evolution in the regime of quantum chaos. Even if the quantum algorithm is polynomial in number of qubits n_q, it is shown that the ideal eigenstates become mixed and strongly modified by static imperfections above a certain threshold which drops exponentially with n_q. Above this threshold the quantum eigenstate entropy grows linearly with n_q but the computation remains reliable during a time scale which is polynomial in the imperfection strength and in n_q. Received 7 March 2002/ Received in final form 3 May 2002 Published online 19 July 2002     

基于相位匹配的量子行走搜索算法及电路实现

量子行走是经典随机行走在量子力学框架下的对应, 理论上可以用来解决一类无序数据库的搜索问题. 因为携带信息的量子态的扩散速度与经典相比有二次方式的增长, 所以量子行走优于经典随机行走, 量子行走的特性值得加以利用. 量子行走作为一种新发现的物理现象的数学描述, 引发了一种新的思维方式, 孕育了一种新的理论计算模型. 最新研究表明, 量子行走本身也是一种通用计算模型, 可被视为设计量子算法的高级工具, 因此受到部分计算机理论科学领域学者的关注和研究. 对于多数问题求解方案的量子算法的设计, 理论上可以只在量子行走模型下进行考虑. 基于Grover算法的相位匹配条件, 本文提出了一个新的基于量子行走的搜索算法. 理论演算表明: 一般情况下本算法的时间复杂度与Grover算法相同, 但是当搜索的目标数目多于总数的1/3时, 本算法搜索成功的概率要大于Grover算法. 本文不但利用Grover算法中相位匹配条件构造了一个新的量子行走搜索算法, 而且在本研究室原有的量子电路设计研究成果的基础上给出了该算法的量子电路表述.     

Experimental implementation of local adiabatic evolution algorithms by an NMR quantum information processor

Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. In some cases, such as the adiabatic versions of Grover's search algorithm and Deutsch-Jozsa algorithm, applying the global adiabatic evolution yields a complexity similar to their classical algorithms. However, using the local adiabatic evolution, the algorithms given by J. Roland and N.J. Cerf for Grover's search [J. Roland, N.J. Cerf, Quantum search by local adiabatic evolution, Phys. Rev. A 65 (2002) 042308] and by Saurya Das, Randy Kobes, and Gabor Kunstatter for the Deutsch-Jozsa algorithm [S. Das, R. Kobes, G. Kunstatter, Adiabatic quantum computation and Deutsh's algorithm, Phys. Rev. A 65 (2002) 062301], yield a complexity of order N (where N=2(n) and n is the number of qubits). In this paper, we report the experimental implementation of these local adiabatic evolution algorithms on a 2-qubit quantum information processor, by Nuclear Magnetic Resonance.     

Exact Equivalence between Quantum Adiabatic Algorithm and Quantum Circuit Algorithm

We present a rigorous proof that quantum circuit algorithm can be transformed into quantum adiabatic algorithm with the exact same time complexity. This means that from a quantum circuit algorithm of L gates we can construct a quantum adiabatic algorithm with time complexity of O(L). Additionally, our construction shows that one may exponentially speed up some quantum adiabatic algorithms by properly choosing an evolution path.     

混合态数据库的Grover算法数学形式及其搜索成功率

Grover提出了容量为N的数据库量子搜索法.只需进行O(平方根N)次迭代就能以几乎为1的概率实现对目标的搜索.本文将文献[1]的Grover搜索法推广到混合态情形,给出了一个基于混合态的Grover搜索法,并分析了该搜索法成功的概率上界.进一步发现搜索法成功的概率完全依赖于所使用的初态(混合态).该结论为了解量子噪声对Grover搜索法的影响提供一定的理论依据.最后通过例子说明了如何实施基于混合态的Grover搜索法.     

清华物理系量子信息研究的新进展

在清华大学物理系成立60周年之际,我们对近年来清华大学物理系量子信息研究的主要进展情况作一介绍,包括量子搜索算法研究,核磁共振量子计算的实验研究,量子通讯的理论与实验研究.在量子搜索算法研究方面,我们提出了量子搜索算法的相位匹配,纠正了当时的一种错误观点,并且提出了一种成功率为100%的量子搜索算法,改进了Grover算法;在核磁共振量子计算实验方面,我们实现了2到7个量子比特的多种量子算法的实验演示;在量子通讯方面,我们提出了分布式传输的量子通讯的思想,应用于量子密钥分配、量子秘密共享、量子直接安全通讯等方面,构造了多个量子通讯的理论方案.在实验室,我们实现了2米距离的空间量子密码通讯的演示实验.     

A quantum search algorithm of two-dimensional convex hull

Despite the rapid development of quantum research in recent years, there is very little research in computational geometry. In this paper, to achieve the convex hull of a point set in a quantum system, a quantum convex hull algorithm based on the quantum maximum or minimum searching algorithm (QUSSMA) is proposed. Firstly, the novel enhanced quantum representation of digital images is employed to represent a group of point set, and then the QUSSMA algorithm and vector operation are used to search the convex hull of the point set. In addition, the algorithm is simulated and compared with the classical algorithm. It is concluded that the quantum algorithm accelerates the classical algorithm when the ${M}_{p}$ value of the convex hull point is under a certain condition.     

Quantum feedback control of atomic motion in an optical cavity

We study quantum feedback cooling of atomic motion in an optical cavity. We design a feedback algorithm that can cool the atom to the ground state of the optical potential with high efficiency despite the nonlinear nature of this problem. An important ingredient is a simplified state-estimation algorithm, necessary for a real-time implementation of the feedback loop. We also describe the critical role of parity dynamics in the cooling process and present a simple theory that predicts the achievable steady-state atomic energies.     

量子算法核磁共振实现脉冲序列设计程序问题研究

从两量子位核磁共振量子处理器物理模型出发,利用Raedt小组提出的自旋-1/2代数理论,根据量子控制非门的定义及Grover量子算法原理,介绍了量子控制非门的4种不同脉冲序列及两量子位Grover量子算法的两种不同脉冲序列的设计过程,通过数值求解含时薛定谔方程模拟量子控制非门和两量子位Grover量子算法,等价于执行量子控制非门和两量子位Grover量子算法运算,演示和分析量子控制非门及两量子位Grover量子算法核磁共振脉冲序列设计呈现的量子程序问题.     

General Quantum Meet-in-the-Middle Search Algorithm Based on Target Solution of Fixed Weight

Similar to the classical meet-in-the-middle algorithm, the storage and computation complexity are the key factors that decide the efficiency of the quantum meet-in-the-middle algorithm. Aiming at the target vector of fixed weight, based on the quantum meet-in-the-middle algorithm, the algorithm for searching all n-product vectors with the same weight is presented, whose complexity is better than the exhaustive search algorithm. And the algorithm can reduce the storage complexity of the quantum meet-in-the-middle search algorithm. Then based on the algorithm and the knapsack vector of the Chor-Rivest public-key crypto of fixed weight d, we present a general quantum meet-in-the-middle search algorithm based on the target solution of fixed weight, whose computational complexity is ∑_j^d=(0(√C_n-k+1^d-j)+O(C_k^jlogC_k^j)) with ∑_i=0^dC_kⁱ memory cost. And the optimal value of k is given. Compared to the quantum meet-in-the-middle search algorithm for knapsack problem and the quantum algorithm for searching a target solution of fixed weight, the computational complexity of the algorithm is lower. And its storage complexity is smaller than the quantum meet-in-the-middle-algorithm.     

基于量子模距离的量子态聚类识别

针对量子系统的状态识别，定义了一种量子模距离作为量子态之间的相似性度量，提出了一种基于量子模距离的聚类算法，它既适用于对量子叠加态的识别，也适合对量子纠缠态的识别。在算法中，根据待识别的样本量子态求取聚类中心，分别计算各量子态到聚类中心的量子模距离，根据量子模距离对量子态进行聚类识别。算例说明了这种聚类识别方法的合理性和有效性。     

New quantum algorithm for studying NP-complete problems

Ordinary approach to quantum algorithm is based on quantum Turing machine or quantum circuits. It is known that this approach is not powerful enough to solve NP-complete problems. In this paper we study a new approach to quantum algorithm which is a combination of the ordinary quantum algorithm with a chaotic dynamical system. We consider the satisfiability problem as an example of NP-complete problems and argue that the problem, in principle, can be solved in polynomial time by using our new quantum algorithm.     

A Parallel Quantum Algorithm for the Satisfiability Problem

In this paper we present a classical parallel quantum algorithm for the satisfiability problem. We have exploited the classical parallelism of quantum algorithms developed in [G.L. Long and L. Xiao, Phys. Rev. A 69 （2004） 052303], so that additional acceleration can be gained by using classical parallelism. The quantum algorithm first estimates the number of solutions using the quantum counting algorithm, and then by using the quantum searching algorithm, the explicit solutions are found.     

Demonstration of essentiality of entanglement in a Deutsch-like quantum algorithm

Quantum algorithms can be used to efficiently solve certain classically intractable problems by exploiting quantum parallelism. However, the effectiveness of quantum entanglement in quantum computing remains a question of debate. This study presents a new quantum algorithm that shows entanglement could provide advantages over both classical algorithms and quantum algo- rithms without entanglement. Experiments are implemented to demonstrate the proposed algorithm using superconducting qubits. Results show the viability of the algorithm and suggest that entanglement is essential in obtaining quantum speedup for certain problems in quantum computing. The study provides reliable and clear guidance for developing useful quantum algorithms.