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.     

Enumeration of perfect matchings of a type of Cartesian products of graphs

Let G be a graph and let Pm(G) denote the number of perfect matchings of G.We denote the path with m vertices by P_m and the Cartesian product of graphs G and H by G×H. In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175-188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results:1. Let T be a tree and let C_n denote the cycle with n vertices. Then Pm(C₄×T)=∏(2+α²), where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm(C₄×T) is always a square or double a square.2. Let T be a tree. Then Pm(P₄×T)=∏(1+3α²+α⁴), where the product ranges over all non-negative eigenvalues α of T.3. Let T be a tree with a perfect matching. Then Pm(P₃×T)=∏(2+α²), where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm(C₄×T)=[Pm(P₃×T)]².     

Reconstruction of the Point-Spread Function of the Human Eye Using a New Phase-Retrieval Algorithm

The double-pass method is thought to obtain the point spread function (PSF) in human eyes based on two techniques: the symmetric double-pass method using the same pupil size and the asymmetric double-pass method using a different pupil size. The symmetric double-pass method provides autocorrelation of the retinal PSF and, thus, the modulation transfer function. The asymmetric double-pass method provides low-frequency partial-phase information and the partial phase-retrieval algorithm is applied to obtain the complete-phase information, and to estimate the PSF. The partial phase-retrieval algorithm is based on the iteration method proposed by Fineup and Kowalczyk and requires a lengthy computation. In this study, we propose a new high-speed phase-retrieval algorithm based on the property that the real and imaginary parts of optical transfer functions (OTFs) continuously change in value.     

实时任务动态调度算法

任务调度算法是提高多任务系统效率的一种有效途径,特别是在时间紧迫环境下,实时任务调度算法更具有重要的应用价值.在研究建立实时任务调度模型方法的基础上,对模型进行理论分析,提出了实时任务调度的算法,并通过实例加以验证.     

A note on the computational complexity of graph vertex partition

A stable set of a graph is a vertex set in which any two vertices are not adjacent. It was proven in [A. Brandstädt, V.B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability, Discrete Appl. Math. 89 (1998) 59-73] that the following problem is NP-complete: Given a bipartite graph G, check whether G has a stable set S such thatG-Sis a tree. In this paper we prove the following problem is polynomially solvable: Given a graph G with maximum degree 3 and containing no vertices of degree 2, check whether G has a stable set S such thatG-Sis a tree. Thus we partly answer a question posed by the authors in the above paper. Moreover, we give some structural characterizations for a graph G with maximum degree 3 that has a stable set S such that G-S is a tree.     

Lorentz-Covariant Loop Quantum Gravity

We review the Lorentz-covariant approach to loop quantum gravity. This approach solves the Immirzi parameter problem occurring in the standard loop approach based on the SU(2) gauge group. We show that there exists a unique loop quantization preserving all the classical symmetries at the quantum level and that the results obtained with it, such as the area operator spectrum, are independent of the Immirzi parameter. The standard SU(2) approach violates the diffeomorphism invariance and is therefore an incorrect quantization of gravity.     

An approximate proximal-extragradient type method for monotone variational inequalities

Proximal point algorithms (PPA) are attractive methods for monotone variational inequalities. The approximate versions of PPA are more applicable in practice. A modified approximate proximal point algorithm (APPA) presented by Solodov and Svaiter [Math. Programming, Ser. B 88 (2000) 371–389] relaxes the inexactness criterion significantly. This paper presents an extended version of Solodov–Svaiter's APPA. Building the direction from current iterate to the new iterate obtained by Solodov–Svaiter's APPA, the proposed method improves the profit at each iteration by choosing the optimal step length along this direction. In addition, the inexactness restriction is relaxed further. Numerical example indicates the improvement of the proposed method.     

Compact directed acyclic word graphs for a sliding window

Suffix trees are a well-known and widely-studied data structure highly useful for string matching. The suffix tree of a string w can be constructed in O(n) time and space, where n denotes the length of w. Larsson achieved an efficient algorithm to maintain suffix trees for a sliding window. It contributes to prediction by partial matching (PPM) style statistical data compression scheme. Compact directed acyclic word graphs (CDAWGs) are a more space-economical data structure for indexing strings. In this paper we propose a linear-time algorithm to maintain CDAWGs for a sliding window.     

Indexing text using the Ziv–Lempel trie

Let a text of u characters over an alphabet of size σ be compressible to n phrases by the LZ78 algorithm. We show how to build a data structure based on the Ziv–Lempel trie, called the LZ-index, that takes 4nlog₂n(1+o(1)) bits of space (that is, 4 times the entropy of the text for ergodic sources) and reports the R occurrences of a pattern of length m in worst case time O(m³logσ+(m+R)logn). We present a practical implementation of the LZ-index, which is faster than current alternatives when we take into consideration the time to report the positions or text contexts of the occurrences found.     

The Computational Complexity of the Chow Form

We present a bounded probability algorithm for the computation of the Chowforms of the equidimensional components of an algebraic variety. In particular, this gives an alternative procedure for the effective equidimensional decomposition of the variety, since each equidimensional component is characterized by its Chow form. The expected complexity of the algorithm is polynomial in the size and the geometric degree of the input equation system defining the variety. Hence it improves (or meets in some special cases) the complexity of all previous algorithms for computing Chow forms. In addition to this, we clarify the probability and uniformity aspects, which constitutes a further contribution of the paper. The algorithm is based on elimination theory techniques, in line with the geometric resolution algorithm due to M. Giusti, J. Heintz, L. M. Pardo, and their collaborators. In fact, ours can be considered as an extension of their algorithm for zero-dimensional systems to the case of positive-dimensional varieties. The key element for dealing with positive-dimensional varieties is a new Poisson-type product formula. This formula allows us to compute the Chow form of an equidimensional variety from a suitable zero-dimensional fiber. As an application, we obtain an algorithm to compute a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As another application, we derive an algorithm for the computation of the (unique) solution of a generic overdetermined polynomial equation system.