An improved formalism for quantum computation based on geometric algebra—case study: Grover’s search algorithm

The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is desirable to represent it in the simplest and most intuitive formalism possible. We show firstly, that Clifford’s geometric algebra, provides a significantly simpler representation than the conventional bra-ket notation, and secondly, that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of the Grover search analogous to the precession of a spin- ${\frac{1}{2}}$ particle. Using this formalism we efficiently solve the exact search problem, as well as easily representing more general search situations. We do not claim the development of an improved algorithm, but show in a tutorial paper that geometric algebra provides extremely compact and elegant expressions with improved clarity for the Grover search algorithm. Being a key algorithm in quantum computing and one of the most studied, it forms an ideal basis for a tutorial on how to elucidate quantum operations in terms of geometric algebra—this is then of interest in extending the applicability of geometric algebra to more complicated problems in fields of quantum computing, quantum decision theory, and quantum information.     

Dynamical Solution to the Quantum Measurement Problem, Causality, and Paradoxes of the Quantum Century

A history and drama of the development of quantum theory is outlined starting from the discovery of the Plank's constant exactly 100 years ago. It is shown that before the rise of quantum mechanics 75 years ago, the quantum theory had appeared first in the form of the statistics of quantum thermal noise and quantum spontaneous jumps which have never been explained by quantum mechanics. Moreover, the only reasonable probabilistic interpretation of quantum theory put forward by Max Born was in fact in irreconcilable contradiction with traditional mechanical reality and causality. This led to numerous quantum paradoxes; some of them, related to the great inventors of quantum theory such as Einstein and Schrödinger, are reconsidered in the paper. The development of quantum measurement theory, initiated by von Neumann, indicated a possibility for the resolution of this interpretational crisis by a divorce of the algebra of dynamical generators and a subalgebra of the actual observables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of past observables with the potential future. This rule together with self-compatibility of measurements ensuring the consitency of histories is called the nondemolition principle. The application of these rules in the form of dynamical commutation relations leads to the derivation of the von Neumann projection postulate, as well as to more general reductions, instantaneous, spontaneous, and even continuous in time. This gives a quantum probabilistic solution in the form of dynamical filtering equations to the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting from Schrödinger and Bohr. The simplest Markovian quantum stochastic model for time-continuous measurements involves a boundary-value problem in second quantization for input "offer" waves in one extra dimension, and a reduction of the algebra of "actual" observables to an Abelian subalgebra for the output waves.     

GHZ States,Almost-Complex Structure and Yang–Baxter Equation

Recent research suggests that there are natural connections between quantum information theory and the Yang–Baxter equation. In this paper, in terms of the almost-complex structure and with the help of its algebra, we define the Bell matrix to yield all the Greenberger–Horne–Zeilinger (GHZ) states from the product basis, prove it to form a unitary braid representation and presents a new type of solution of the quantum Yang–Baxter equation. We also study Yang–Baxterization, Hamiltonian, projectors, diagonalization, noncommutative geometry, quantum algebra and FRT dual algebra associated with this generalized Bell matrix.      

On Ideal Measurements and Their Corresponding Instruments on von Neumann Algebras

We study some features of quantum measurement in the framework of the theory of instruments — a mathematical model for measurement theory. This investigation is carried in an algebraic formalism in which the observables are represented by elements of a von Neumann algebra and the states are linear normal positive functionals on this algebra. We give a characterization of ideal instruments and study their connections with weakly and strongly repeatable instruments under the assumption that the associated observables are projection-valued measures. We also show that the observables of ideal weakly repeatable instruments must be projection-valued measures.     

Feynman graph generation and calculations in the Hopf algebra of Feynman graphs

Two programs for the computation of perturbative expansions of quantum field theory amplitudes are provided. feyngen can be used to generate Feynman graphs for Yang–Mills, QED and φ^k${\phi }^k$ theories. Using dedicated graph theoretic tools feyngen can generate graphs of comparatively high loop orders. feyncop implements the Hopf algebra of those Feynman graphs which incorporates the renormalization procedure necessary to calculate finite results in perturbation theory of the underlying quantum field theory. feyngen is validated by comparison to explicit calculations of zero dimensional quantum field theories and feyncop is validated using a combinatorial identity on the Hopf algebra of graphs.     

Symbolic expansion of transcendental functions

Higher transcendental function occur frequently in the calculation of Feynman integrals in quantum field theory. Their expansion in a small parameter is a non-trivial task. We report on a computer program which allows the systematic expansion of certain classes of functions. The algorithms are based on the Hopf algebra of nested sums. The program is written in C++ and uses the GiNaC library.     

Quantum teleportation and Birman–Murakami–Wenzl algebra

In this paper, we investigate the relationship of quantum teleportation in quantum information science and the Birman–Murakami–Wenzl (BMW) algebra in low-dimensional topology. For simplicity, we focus on the two spin-1/2 representation of the BMW algebra, which is generated by both the Temperley–Lieb projector and the Yang–Baxter gate. We describe quantum teleportation using the Temperley–Lieb projector and the Yang–Baxter gate, respectively, and study teleportation-based quantum computation using the Yang–Baxter gate. On the other hand, we exploit the extended Temperley–Lieb diagrammatical approach to clearly show that the tangle relations of the BMW algebra have a natural interpretation of quantum teleportation. Inspired by this interpretation, we construct a general representation of the tangle relations of the BMW algebra and obtain interesting representations of the BMW algebra. Therefore, our research sheds a light on a link between quantum information science and low-dimensional topology.     

多量子位Grover量子搜索算法的NMR仿真实现

核磁共振（NMR）技术目前是能有效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。本文以核磁共振和多量子算符代数理论为基础,提出了实现多量子位Grover量子搜索算法的核磁共振脉冲序列设计方法,并在量子计算仿真程序上进行了3量子位的Grover量子搜索算法的实验验证。     

基于Unsharp量子逻辑的自动机和文法理论

初步建立了具有某种分配律的扩展格序效应代数和格序QMV代数这两种unsharp量子结构上的自动机与文法理论的基本框架。引入了ε-值正则文法的概念,证明了任意ε-值自动机识别的语言等价于某种ε-值正则文法所生成的语言;反之,任意[ε]-值正则文法所生成的语言等价于某种ε-值自动机识别的语言。讨论了ε-值正则语言在和、连接及反转运算下的封闭性质。     

Information states in stochastic control and filtering: a Liealgebraic theoretic approach

The purpose of the paper is two-fold: (i) to introduce the sufficient statistic algebra which is responsible for propagating the sufficient statistics, or information state, in the optimal control of stochastic systems and (ii) to apply certain Lie algebraic methods and gauge transformations, widely used in nonlinear control theory and quantum physics, to derive new results concerning finite dimensional controllers. This enhances our understanding of the role played by the sufficient statistics. The sufficient statistic algebra enables us to determine a priori whether there exist finite-dimensional controllers; it also enables us to classify all finite-dimensional controllers. Relations to minimax dynamic games are delineated     

量子搜索算法体系的仿真实现及研究

核磁共振（NMR）技术被认为是最为有效的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。本文以核磁共振和多量子算符代数理论为基础，提出了实现任意相位旋转角度的一般化量子搜索算法的核磁共振脉冲序列设计方法，并在量子计算仿真程序上进行了双量子位的不同相位旋转角度的量子搜索算法的实验验证。     

Symbolic implementation of arbitrary-order perturbation theory using computer algebra: Application to vibrational–rotational analysis of diatomic molecules

Theoretical details necessary to calculate arbitrary-order correction terms to vibrational–rotational energies and wave functions in Rayleigh–Schrödinger perturbation theory are presented. Since manual derivation of high-order perturbation formulae is not feasible due to the lengthy algebra involved, the commercial computer algebra software Mathematica® is employed to perform the symbolic manipulations necessary to derive the requisite correction formulae in terms of universal constants, molecular constants, and quantum numbers. Correction terms through sixth order for ¹∑ diatomic molecules are derived and then evaluated for H₂, HD, N₂, CO, and HF. It is thus possible, with the aid of computer-generated algebra, to apply arbitrarily high-order perturbation theory successfully to the problem of intramolecular nuclear motion.     

Algebraic Perturbation Theory for Hydrogen Atom in Weak Electric Fields

An algorithm for symbolic calculation of eigenvalues and eigenfunctions of a hydrogen atom in weak electric fields is suggested. A perturbation theory scheme is constructed that is based on an irreducible infinite-dimensional representation of algebra so(4, 2) of the group of dynamical symmetry for the hydrogen atom [1]. The scheme implementation does not rely on the assumption that the independent variables of the perturbation operator can be separated, and fractional powers of parabolic quantum numbers are not used in the recurrent relations determining the operation of algebra generators on the corresponding basis of the irreducible representation [2]. A seventh-order correction to the energy spectrum of the hydrogen atom in a uniform electric field is given. The algorithm suggested is implemented in REDUCE 3.6 [4].     

多量子位量子小波变换逻辑线路及仿真实现

由于量子计算相比于经典计算的突出优越性,量子小波变换的实现对于小波变换的理论完善和实际应用具有重要的意义,而逻辑线路是该变换实现的基础。应用多量子算符代数理论设计了3量子位Haar和D^（4）小波变换的逻辑线路,进而将逻辑线路转化成核磁共振系统可以实现的脉冲序列,并在量子计算仿真器（QCE）上进行了模拟实现,验证了逻辑线路的合理性。     

Symbolic evaluation of dimensionally regularized Feynman diagrams

A modification of the symbolic and algebraic manipulation program REDUCE is reported which allows the treatment of vector and gamma algebra in an arbitrary number of dimensions. The number of dimensions serves as a cut-off for divergences in the Feynman diagrams one has to evaluate in a theory like quantum gravity.     

自旋量子系统的可控李代数计算

动态系统可控性同是经典和量子控制中研究的一个基本问题,本文研究了单自旋和双自旋量子系统的可控李代数的计算.首先基于量子系统可控的等价性条件,通过单量子系统Hamiltonian算符的李括号运算,给出了与基系数相关的系统可控的充要条件.然后利用Cartan分解方法构造了李代数su(4)的矩阵基,同时根据可控性基本定理提出了Hamiltonian算符多重李括号的计算方法及系统的可控性判据.     

A geometrical approach to quantum holonomic computing algorithms

The article continues a presentation of modern quantum mathematics backgrounds started in [Quantum Mathematics and its Applications. Part 1. Automatyka, vol. 6, AGH Publisher, Krakow, 2002, No. 1, pp. 234–2412; Quantum Mathematics: Holonomic Computing Algorithms and Their Applications. Part 2. Automatyka, vol. 7, No. 1, 2004]. A general approach to quantum holonomic computing based on geometric Lie-algebraic structures on Grassmann manifolds and related with them Lax type flows is proposed. Making use of the differential geometric techniques like momentum mapping reduction, central extension and connection theory on Stiefel bundles it is shown that the associated holonomy groups properly realizing quantum computations can be effectively found concerning diverse practical problems. Two examples demonstrating two-form curvature calculations important for describing the corresponding holonomy Lie algebra are presented in detail.     

Controllability of quantum walks on graphs

In this paper, we consider discrete time quantum walks on graphs with coin, focusing on the decentralized model, where the coin operation is allowed to change with the vertex of the graph. When the coin operations can be modified at every time step, these systems can be looked at as control systems and techniques of geometric control theory can be applied. In particular, the set of states that one can achieve can be described by studying controllability. Extending previous results, we give a characterization of the set of reachable states in terms of an appropriate Lie algebra. Controllability is verified when any unitary operation between two states can be implemented as a result of the evolution of the quantum walk. We prove general results and criteria relating controllability to the combinatorial and topological properties of the walk. In particular, controllability is verified if and only if the underlying graph is not a bipartite graph and therefore it depends only on the graph and not on the particular quantum walk defined on it. We also provide explicit algorithms for control and quantify the number of steps needed for an arbitrary state transfer. The results of the paper are of interest in quantum information theory where quantum walks are used and analyzed in the development of quantum algorithms.     

Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences

Starting from a generic generally covariant quantum theory, we introduce a logarithmic correction to the quantum wave equation. We demonstrate the emergence of evolution time from the group of automorphisms of the von Neumann algebra governed by this nonlinear correction. It turns out that such a parametrization of time is essentially energy-dependent and becomes global only asymptotically, as the energies become very small as compared to the effective quantum gravity scale. A similar thing happens to Lorentz invariance: in the resulting theory it becomes an asymptotic low-energy phenomenon. We show how the logarithmic nonlinearity deforms the vacuum wave dispersion relations and explains certain features of the astrophysical data coming from the recent observations of high-energy cosmic rays. In general, the estimates imply that, ceteris paribus, particles with higher energy propagate slower than those with lower energy, therefore, for a high-energy particle the mean free path, lifetime in a high-energy state and thus the travel distance from the source can be significantly larger than one would expect from the conventional theory. In addition, we discuss the possibility and conditions of transluminal phenomena in the physical vacuum such as Cherenkov-type shock waves.     

Renormalization Automated by Hopf Algebra

It was recently shown that the renormalization of quantum field theory is organized by the Hopf algebra of decorated rooted trees, whose coproduct identifies the divergences requiring subtraction and whose antipode achieves this. We automate this process in a few lines of recursive symbolic code, which deliver a finite renormalized expression for any Feynman diagram. We thus verify a representation of the operator product expansion, which generalizes Chen’s Lemma for iterated integrals. The subset of diagrams whose forest structure entails a unique primitive subdivergence provides a representation of the Hopf algebra H_Rof undecorated rooted trees. Our undecorated Hopf algebra program is designed to process the 24 213 878 BPHZ contributions to the renormalization of 7813 diagrams, with up to 12 loops. We consider 10 models, each in nine renormalization schemes. The two simplest models reveal a notable feature of the subalgebra of Connes and Moscovici, corresponding to the commutative part of the Hopf algebra H_Tof the diffeomorphism group: it assigns to Feynman diagrams those weights which remove zeta values from the counterterms of the minimal subtraction scheme. We devise a fast algorithm for these weights, whose squares are summed with a permutation factor, to give rational counterterms.