Differential geometric treewidth estimation in adiabatic quantum computation

The D-Wave adiabatic quantum computing platform is designed to solve a particular class of problems—the Quadratic Unconstrained Binary Optimization (QUBO) problems. Due to the particular “Chimera” physical architecture of the D-Wave chip, the logical problem graph at hand needs an extra process called minor embedding in order to be solvable on the D-Wave architecture. The latter problem is itself NP-hard. In this paper, we propose a novel polynomial-time approximation to the closely related treewidth based on the differential geometric concept of Ollivier–Ricci curvature. The latter runs in polynomial time and thus could significantly reduce the overall complexity of determining whether a QUBO problem is minor embeddable, and thus solvable on the D-Wave architecture.     

通用的辅助量子计算

设计了一个通用的辅助量子计算协议。该协议的客户端Alice仅拥有经典计算机或有限的量子技术,这些资源不足以让Alice做通用量子计算,因此Alice需要把她的量子计算任务委派给远程的量子服务器Bob。Bob拥有充分成熟的量子计算机,并会诚实地帮助Alice执行委派的量子计算任务,但他却得不到Alice的任何输入、输出信息。该协议只要求Alice能发送量子态和执行Pauli门操作,协议具有通用性、半盲性、正确性和可验证性。     

An improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects

In this paper we propose a new recursive algorithm for computing the staircase form of a matrix pencil, and implicitly its Kronecker structure. The algorithm compares favorably to existing ones in terms of elegance, versatility, and complexity. In particular, the algorithm without any modification yields the structural invariants associated with a generalized state-space system and its system pencil. Two related geometric aspects are also discussed: we show that an appropriate choice of a set of nested spaces related to the pencil leads directly to the staircase form; we extend the notion of deflating subspace to the singular pencil case.     

Mathematical models of quantum computation

In this paper, we introduce two mathematical models of realistic quantum computation. First, we develop a theory of bulk quantum computation such as NMR (Nuclear Magnetic Resonance) quantum computation. For this purpose, we define bulk quantum Turing machine (BQTM for short) as a model of bulk quantum computation. Then, we define complexity classes EBQP, BBQP and ZBQP as counterparts of the quantum complexity classes EQP, BQP and ZQP, respectively, and show that EBQP=EQP, BBQP=BQP and ZBQP=ZQP. This implies that BQTMs are polynomially related to ordinary QTMs as long as they are used to solve decision problems. We also show that these two types of QTMs are also polynomially related when they solve a function problem which has a unique solution. Furthermore, we show that BQTMs can solve certain instances of NP-complete problems efficiently. On the other hand, in the theory of quantum computation, only feed-forward quantum circuits are investigated, because a quantum circuit represents a sequence of applications of time evolution operators. But, if a quantum computer is a physical device where the gates are interactions controlled by a current computer such as laser pulses on trapped ions, NMR and most implementation proposals, it is natural to describe quantum circuits as ones that have feedback loops if we want to visualize the total amount of the necessary hardware. For this purpose, we introduce a quantum recurrent circuit model, which is a quantum circuit with feedback loops. LetC be a quantum recurrent circuit which solves the satisfiability problem for a blackbox Boolean function includingn variables with probability at least 1/2. And lets be the size ofC (i.e. the number of the gates inC) andt be the number of iterations that is needed forC to solve the satisfiability problem. Then, we show that, for those quantum recurrent circuits, the minimum value ofmax(s, t) isO(n ²2 ^n/3). Tetsuro Nishino, D.Sc.: He is presently an Associate Professor in the Department of Information and Communication Engineering, The University of Electro-Communications. He received the B.S., M.S. and D.Sc degrees in mathematics from Waseda University, in 1982, 1984 and 1991 respectively. From 1984 to 1987, he joined Tokyo Research Laboratory, IBM Japan. From 1987 to 1992, he was a Research Associate of Tokyo Denki University, and from 1992 to 1994, he was an Associate Professor of Japan Advanced Institute of Science and Technology, Hokuriku. His main interests are circuit complexity theory, computational learning theory and quantum complexity theory.     

Statistical geometric computation on tolerances for dimensioning

Dimensions are used to specify the distances between different features in geometric models. These dimensions will often be expressed as a range of allowable dimensions. When considering a group of toleranced dimensions, these ranges can be analyzed as either a worst-case bound on allowable ranges, or as a statistical measure of expected distribution. This paper presents a new geometric model for representing statistically-based tolerance regions. Methods for tolerance estimation and allocation on a geometric model are provided by generalizing root sum square (RSS) methods for compositing and cascading over tolerance zones. This gives us a geometric interpretation of a statistical analysis. Tolerance regions are determined by probabilities of variations of dimensions falling into the region. A dependency graph over dimensions can be represented by a topological graph on which the tolerance cascading and tolerance allocation can be processed. To illustrate applications of this geometric method, we provide examples of tolerance estimation and tolerance allocation on our model. The estimation examples utilize the compositing and cascading operations provided in the analysis method. The allocation examples present an automatic tolerance allocation procedure on the tolerance model. As opposed to existing methods, our allocation method allows us to specify not only a numerical objective of the optimization, but also a statistically-based objective for the geometric shape of the tolerance.     

Conceptual aspects of transformation invariance

A source of confusion and misconception, concerning invariant curve and surface algorithms, is investigated. The problem is shown to be due to posing the question of invariance in an complete way. It is seen that if due regard is given to the nature, and in particular the transformational behaviour, of the input data to such algorithms then the issue is resolved. The use of revised notation and a higher degree of rigour is advocated.     

Single-server blind quantum computation with quantum circuit model

Blind quantum computation (BQC) enables the client, who has few quantum technologies, to delegate her quantum computation to a server, who has strong quantum computabilities and learns nothing about the client’s quantum inputs, outputs and algorithms. In this article, we propose a single-server BQC protocol with quantum circuit model by replacing any quantum gate with the combination of rotation operators. The trap quantum circuits are introduced, together with the combination of rotation operators, such that the server is unknown about quantum algorithms. The client only needs to perform operations X and Z, while the server honestly performs rotation operators.     

Toward quantum computation: a five-qubit quantum processor

Quantum physics presents intriguing possibilities for achieving computational gains after conventional miniaturization reaches its limits. Accordingly, we describe a nuclear magnetic-resonance quantum computer demonstrating a quantum algorithm that exponentially outperforms classical algorithms     

An overview of quantum computation models: quantum automata

Quantum automata, as theoretical models of quantum computers, include quantum finite automata (QFA), quantum sequential machines (QSM), quantum pushdown automata (QPDA), quantum Turing machines (QTM), quantum cellular automata (QCA), and the others, for example, automata theory based on quantum logic (orthomodular lattice-valued automata). In this paper, we try to outline a basic progress in the research on these models, focusing on QFA, QSM, QPDA, QTM, and orthomodular lattice-valued automata. Also, other models closely relative to them are mentioned. In particular, based on the existing results in the literature, we finally address a number of problems to be studied in future.     

Tree search and quantum computation

Traditional tree search algorithms supply a blueprint for modeling problem solving behaviour. A diverse spectrum of problems can be formulated in terms of tree search. Quantum computation, namely Grover’s algorithm, has aroused a great deal of interest since it allows for a quadratic speedup to be obtained in search procedures. In this work we consider the impact of incorporating classical search concepts alongside Grover’s algorithm into a hybrid quantum search system. Some of the crucial points examined include: (1) the reverberations of contemplating the use of non-constant branching factors; (2) determining the consequences of incorporating an heuristic perspective into a quantum tree search model.     

辅助量子比特驱动型通用盲量子计算

应用量子隐形传态将Broadbent等人提出的通用盲量子计算(universal blind quantum computation)模型和辅助量子比特驱动型量子计算(ancilla-driven universal quantum computation)模型进行结合, 构造一个新的混合模型来进行计算。此外, 用计算寄存器对量子纠缠的操作来代替量子比特测量操作。因为后者仅限于两个量子比特, 所以代替后的计算优势十分明显。基于上述改进, 设计了实现辅助驱动型通用盲量子计算的协议。协议的实现, 能够使Anders等人的辅助驱动型量子计算增强计算能力, 并保证量子计算的正确性, 从而使得参与计算的任何一方都不能获得另一方的保密信息。     

An overview of quantum computation models: quantum automata

Quantum automata, as theoretical models of quantum computers, include quantum finite automata (QFA), quantum sequential machines (QSM), quantum pushdown automata (QPDA), quantum Turing machines (QTM), quantum cellular automata (QCA), and the others, for example, automata theory based on quantum logic (orthomodular lattice-valued automata). In this paper, we try to outline a basic progress in the research on these models, focusing on QFA, QSM, QPDA, QTM, and orthomodular lattice-valued automata. Also, other models closely relative to them are mentioned. In particular, based on the existing results in the literature, we finally address a number of problems to be studied in future.     

量子计算进展与展望

评述量子计算的历史、研究现状以及进一步发展的方向.着重论述量子算法的机理,对已知量子算法特征进行总结分析;归纳量子计算与经典智能计算的结合模式,比较其与传统智能计算的异同.在总结量子计算存在问题的基础上,探讨了今后的研究方向.     

A model of discrete quantum computation

We present a model of discrete quantum computing focused on a set of discrete quantum states. For this, we choose the set that is the most outstanding in terms of simplicity of the states: the set of Gaussian coordinate states, which includes all the quantum states whose coordinates in the computation base, except for a normalization factor \(\sqrt{2^{-k}}\), belong to the ring of Gaussian integers \(\mathbb {Z}[i]=\{a+bi\ |\ a,b\in \mathbb {Z}\}\). We also introduce a finite set of quantum gates that transforms discrete states into discrete states and generates all discrete quantum states, and the set of discrete quantum gates, as the quantum gates that leave the set of discrete states invariant. We prove that the quantum gates of the model generate the expected discrete states and the discrete quantum gates of 2-qubits and conjecture that they also generate the discrete quantum gates of n-qubits.     

基于LISA理论的概念模型相似度计算

在信息系统概念建模过程中,常常需要合并或重用已有模型形成复杂的模型,以更加精确地刻画物理世界模型,而概念模型间的相似度计算是实现这一目标的重要基础。通过引入LISA认知模型的系统相似性计算方法,将概念模型间相似度计算转化为关系,绑定,实体类和角色间相似度的综合计算。实验结果表明,这种计算方法综合考虑了各种相关因素,所得出的结果具有较高的可信度。     

Bayesian computation for geometric process in maintenance problems

Geometric process modeling is a useful tool to study repairable deteriorating systems in maintenance problems. This model has been used in a variety of situations such as the determination of the optimal replacement policy and the optimal inspection-repair-replacement policy for standby systems, and the analysis of data with trend. In this article, Bayesian inference for the geometric process with several popular life distributions, for instance, the exponential distribution and the lognormal distribution, are studied. The Gibbs sampler and the Metropolis algorithm are used to compute the Bayes estimators of the parameters in the geometric process. Simulation results are presented to illustrate the use of our procedures.     

Important aspects of geometric numerical integration

At the example of Hamiltonian differential equations, geometric properties of the flow are discussed that are only preserved by special numerical integrators (such as symplectic and/or symmetric methods). In the ‘non-stiff’ situation the long-time behaviour of these methods is well-understood and can be explained with the help of a backward error analysis. In the highly oscillatory (‘stiff’) case this theory breaks down. Using a modulated Fourier expansion, much insight can be gained for methods applied to problems where the high oscillations stem from a linear part of the vector field and where only one (or a few) high frequencies are present. This paper terminates with numerical experiments at space discretizations of the sine-Gordon equation, where a whole spectrum of frequencies is present.     

The problems of accuracy and robustness in geometric computation

Practical implementation of geometric operations remains error-prone, and the goal of implementing correct and robust systems for carrying out geometric computation remains elusive. The problem is variously characterized as a matter of achieving sufficient numerical precision, as a fundamental difficulty in dealing with interacting numeric and symbolic data, or as a problem of avoiding degenerate positions. The author examines these problems, surveys some of the approaches proposed, and assesses their potential for devising complete and efficient solutions. He restricts the analysis to objects with linear elements, since substantial problems already arise in this case. Three perturbation-free methods are considered: floating-point computation, limited-precision rational arithmetic, and purely symbolic representations. Some perturbation approaches are also examined, namely, representation and model, altering the symbolic data, and avoiding degeneracies