Birman–Wenzl–Murakami algebra, topological parameter and Berry phase

In this paper, a 3?×?3-matrix representation of Birman?CWenzl?CMurakami (BWM) algebra has been presented. Based on which, unitary matrices A(??, ?? ₁, ?? ₂) and B(??, ?? ₁, ?? ₂) are generated via Yang?CBaxterization approach. A Hamiltonian is constructed from the unitary B(??, ??) matrix. Then we study Berry phase of the Yang?CBaxter system, and obtain the relationship between topological parameter and Berry phase.     

Temperley–Lieb algebra,Yang-Baxterization and universal gate

A method of constructing n ² × n ² matrix realization of Temperley–Lieb algebras is presented. The single loop of these realizations are \({d=\sqrt{n}}\). In particular, a 9 × 9-matrix realization with single loop \({d=\sqrt{3}}\) is discussed. A unitary Yang–Baxter \({\breve{R}\theta,q_{1},q_{2})}\) matrix is obtained via the Yang-Baxterization process. The entanglement properties and geometric properties (i.e., Berry Phase) of this Yang–Baxter system are explored.     

Yang–Baxterizations,Universal Quantum Gates and Hamiltonians

The unitary braiding operators describing topological entanglements can be viewed as universal quantum gates for quantum computation. With the help of the Brylinski’s theorem, the unitary solutions of the quantum Yang–Baxter equation can be also related to universal quantum gates. This paper derives the unitary solutions of the quantum Yang–Baxter equation via Yang–Baxterization from the solutions of the braid relation. We study Yang–Baxterizations of the non-standard and standard representations of the six-vertex model and the complete solutions of the non-vanishing eight-vertex model. We construct Hamiltonians responsible for the time-evolution of the unitary braiding operators which lead to the Schrödinger equations.     

Quantum Arthur–Merlin games

This paper studies quantum Arthur–Merlin games, which are Arthur–Merlin games in which Arthur and Merlin can perform quantum computations and Merlin can send Arthur quantum information. As in the classical case, messages from Arthur to Merlin are restricted to be strings of uniformly generated random bits. It is proved that for one-message quantum Arthur–Merlin games, which correspond to the complexity class QMA, completeness and soundness errors can be reduced exponentially without increasing the length of Merlins message. Previous constructions for reducing error required a polynomial increase in the length of Merlins message. Applications of this fact include a proof that logarithmic length quantum certificates yield no increase in power over BQP and a simple proof that Other facts that are proved include the equivalence of three (or more) message quantum Arthur–Merlin games with ordinary quantum interactive proof systems and some basic properties concerning two-message quantum Arthur–Merlin games.     

N-qubit quantum teleportation, information splitting and superdense coding through the composite GHZ–Bell channel

We introduce a general odd qubit entangled system composed of GHZ and Bell pairs and explicate its usefulness for quantum teleportation, information splitting and superdense coding. After demonstrating the superdense coding protocol on the five qubit system, we prove that ‘2N + 1’ classical bits can be sent by sending ‘N + 1’ quantum bits using this channel. It is found that the five-qubit system is also ideal for arbitrary one qubit and two qubit teleportation and quantum information splitting (QIS). For the single qubit QIS, three different protocols are feasible, whereas for the two qubit QIS, only one protocol exists. Protocols for the arbitrary N-qubit state teleportation and quantum information splitting are then illustrated.     

Quantum steering and entanglement in three-mode triangle Bose–Hubbard system

We consider the possibility of generation steerable states in Bose–Hubbard system composed of three interacting wells in the form of a triangle. We show that although our system still fulfills the monogamy relations, the presence of additional coupling which transforms a chain of wells onto triangle gives a variety of new possibilities for the generation of steerable quantum states. Deriving analytical formulas for the parameters describing steering and bipartite entanglement, we show that interplay between two couplings influences quantum correlations of various types. We compare the time evolution of steering parameters to those describing bipartite entanglement and find the relations between the appearance of maximal entanglement and disappearance of steering effect.     

Classical and Quantum Complexity of the Sturm–Liouville Eigenvalue Problem

We study the approximation of the smallest eigenvalue of a Sturm–Liouville problem in the classical and quantum settings. We consider a univariate Sturm–Liouville eigenvalue problem with a nonnegative function q from the class C² ([0,1]) and study the minimal number n() of function evaluations or queries that are necessary to compute an -approximation of the smallest eigenvalue. We prove that n()=(^–1/2) in the (deterministic) worst case setting, and n()=(^–2/5) in the randomized setting. The quantum setting offers a polynomial speedup with bit queries and an exponential speedup with power queries. Bit queries are similar to the oracle calls used in Grovers algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix exp((1/2) iM), where M is an n× n matrix obtained from the standard discretization of the Sturm–Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is polylog in n is an open issue. In particular, we show how to compute an -approximation with probability (3/4) using n()=(^–1/3) bit queries. For power queries, we use the phase estimation algorithm as a basic tool and present the algorithm that solves the problem using n()=(log ^–1) power queries, log ^2–1 quantum operations, and (3/2) log ^–1 quantum bits. We also prove that the minimal number of qubits needed for this problem (regardless of the kind of queries used) is at least roughly (1/2) log ^–1. The lower bound on the number of quantum queries is proven in Bessen (in preparation). We derive a formula that relates the Sturm–Liouville eigenvalue problem to a weighted integration problem. Many computational problems may be recast as this weighted integration problem, which allows us to solve them with a polylog number of power queries. Examples include Grovers search, the approximation of the Boolean mean, NP-complete problems, and many multivariate integration problems. In this paper we only provide the relationship formula. The implications are covered in a forthcoming paper (in preparation).PACS: 03.67.Lx, 02.60.-x.     

Non-ideal teleportation of tripartite entanglement: Einstein–Podolsky–Rosen versus Greenberger–Horne–Zeilinger schemes

Channels composed by Einstein–Podolsky–Rosen (EPR) pairs are capable of teleporting arbitrary multipartite states. The question arises whether EPR channels are also optimal against imperfections. In particular, the teleportation of Greenberger–Horne–Zeilinger states (GHZ) requires three EPR states as the channel and full measurements in the Bell basis. We show that, by using two GHZ states as the channel, it is possible to transport any unknown three-qubit state of the form \(c_0|000\rangle +c_1|111\rangle \). The teleportation is made through measurements in the GHZ basis, and, to obtain deterministic results, in most of the investigated scenarios, four out of the eight elements of the basis need to be unambiguously distinguished. Most importantly, we show that when both systematic errors and noise are considered, the fidelity of the teleportation protocol is higher when a GHZ channel is used in comparison with that of a channel composed by EPR pairs.     

Quantum measurement and the Aharonov–Bohm effect with superposed magnetic fluxes

We consider the magnetic flux in a quantum mechanical superposition of two values and find that the Aharonov–Bohm effect interference pattern contains information about the nature of the superposition, allowing information about the state of the flux to be extracted without disturbance. The information is obtained without transfer of energy or momentum and by accumulated nonlocal interactions of the vector potential $\varvec{A}$ with many charged particles forming the interference pattern, rather than with a single particle. We suggest an experimental test using already experimentally realized superposed currents in a superconducting ring and discuss broader implications.     

Quantum computing and communications – Introduction and challenges

Computer engineers are continuously seeking new solutions to increase available processing speed, achievable transmission rates, and efficiency in order to satisfy users’ expectations. While multi-core systems, computing clouds, and other parallel processing techniques dominate current technology trends, elementary particles governed by quantum mechanics have been borrowed from the physicists’ laboratory and applied to computer engineering in the efforts to solve sophisticated computing and communications problems. In this paper, we review the quantum mechanical background of quantum computing from an engineering point of view and describe the possibilities offered by quantum-assisted and quantum-based computing and communications. In addition to the currently available solutions, the corresponding challenges will also be surveyed.     

Quantum holonomies for displaced Landau–Aharonov–Casher states

We construct displaced Fock states for a Landau–Aharonov–Casher system for neutral particles. Abelian and non-Abelian geometric phases can be obtained in an adiabatic cyclic evolution using this displaced states. Moreover, we show that a possible logical base related to the angular momenta of the neutral particle with permanent magnetic dipole moment can be defined, and then quantum holonomies for specific paths can be built and used to implement one-qubit quantum gates.     

The research seminar on computer algebra in 2009–2010

An annual report on meetings of the scientific research seminar on computer algebra is presented.     

The research seminar on computer algebra in 2007–2008

An annual report on meetings of the scientific research seminar on computer algebra is presented.     

The research seminar on computer algebra in 2006–2007

An annual report on meetings of the scientific research seminar on computer algebra is presented.     

Quantum phase transition,quantum fidelity and fidelity susceptibility in the Yang–Baxter system

In this paper, we investigate the ground-state fidelity and fidelity susceptibility in the many-body Yang–Baxter system and analyze their connections with quantum phase transition. The Yang–Baxter system was perturbed by a twist of \( e^{i\varphi } \) at each bond, where the parameter \( \varphi \) originates from the q-deformation of the braiding operator U with \(q = e^{-i\varphi }\) (Jimbo in Yang–Baxter equations in integrable systems, World Scientific, Singapore, 1990), and \( \varphi \) has a physical significance of magnetic flux (Badurek et al. in Phys. Rev. D 14:1177, 1976). We test the ground-state fidelity related by a small parameter variation \(\varphi \) which is a different term from the one used for driving the system toward a quantum phase transition. It shows that ground-state fidelity develops a sharp drop at the transition. The drop gets sharper as system size N increases. It has been verified that a sufficiently small value of \(\varphi \) used has no effect on the location of the critical point, but affects the value of \( F(g_{c},\varphi ) \). The smaller the twist \(\varphi \), the more the value of \( F(g_{c},\varphi ) \) is close to 0. In order to avoid the effect of the finite value of \( \varphi \), we also calculate the fidelity susceptibility. Our results demonstrate that in the Yang–Baxter system, the quantum phase transition can be well characterized by the ground-state fidelity and fidelity susceptibility in a special way.     

Quantum correlations and decoherence dynamics for a qutrit–qutrit system under random telegraph noise

We analyze the effect of a classical random telegraph noise on the dynamics of quantum correlations and decoherence between two non-interacting spin-qutrit particles, initially entangled, and coupled either to independent sources or to a common source of noise. Both Markovian and non-Markovian environments are considered. For the Markov regime, as the noise switching rate decreases, a monotonic decay of the initial quantum correlations is found and the loss of coherence increases monotonically with time up to the saturation value. For the non-Markov regime, evident oscillations of correlations and decoherence are observed due to the noise regime, but correlations, however, avoid sudden death phenomena. The oscillatory behavior is more and more prominent as the noise switching rate decreases in this regime, thus enhancing robustness of correlations. Similarly to the qubits case, independent environments coupling is more effective than a common environment coupling in preserving quantum correlations and coherence of the system for a Markovian noise; meanwhile, the opposite is found for the non-Markovian one.     

Quantum discord in spin systems with dipole–dipole interaction

The behavior of total quantum correlations (discord) in dimers consisting of dipolar-coupled spins 1/2 are studied. We found that the discord $Q=0$ at absolute zero temperature. As the temperature $T$ increases, the quantum correlations in the system increase at first from zero to its maximum and then decrease to zero according to the asymptotic law $T^{-2}$ . It is also shown that in absence of external magnetic field $B$ , the classical correlations $C$ at $T\rightarrow 0$ are, vice versa, maximal. Our calculations predict that in crystalline gypsum $\hbox {CaSO}_{4}\cdot \hbox {2H}_{2}{\hbox {O}}$ the value of natural $(B=0)$ quantum discord between nuclear spins of hydrogen atoms is maximal at the temperature of 0.644  $\upmu $ K, and for 1,2-dichloroethane $\hbox {H}_{2}$ ClC– $\hbox {CH}_{2}{\hbox {Cl}}$ the discord achieves the largest value at $T=0.517~\upmu $ K. In both cases, the discord equals $Q\approx 0.083$  bit/dimer what is $8.3\,\%$ of its upper limit in two-qubit systems. We estimate also that for gypsum at room temperature $Q\sim 10^{-18}$  bit/dimer, and for 1,2-dichloroethane at $T=90$  K the discord is $Q\sim 10^{-17}$  bit per a dimer.     

Quantum algorithm to solve function inversion with time–space trade-off

In general, it is a difficult problem to solve the inverse of any function. With the inverse implication operation, we present a quantum algorithm for solving the inversion of function via using time–space trade-off in this paper. The details are as follows. Let function \(f(x)=y\) have k solutions, where \(x\in \{0, 1\}^{n}, y\in \{0, 1\}^{m}\) for any integers n, m. We show that an iterative algorithm can be used to solve the inverse of function f(x) with successful probability \(1-\left( 1-\frac{k}{2^{n}}\right) ^{L}\) for \(L\in Z^{+}\). The space complexity of proposed quantum iterative algorithm is O(Ln), where L is the number of iterations. The paper concludes that, via using time–space trade-off strategy, we improve the successful probability of algorithm.     

Finding the Schwarzschild metric from gravity’s Exterior and Interior Effacement algebra

In the author’s previous publications, a recursive linear algebraicmethod was introduced for obtaining (sans gravitational radiation) the full potential expansions for the gravitational metric field components and the Lagrangian for the general N-body system. Two apparent properties of gravity—Exterior Effacement and Interior Effacement—were defined and enforced to obtain the recursive algebra, especially for the motion-independent potential expansions of the general N-body situation. The linear algebraic equations of this method permit determination of the potential coefficients at any order n ^? of the expansions in terms of the lower order coefficients. To illustrate the capabilities of this algebraic method by enforcing exterior and interior effacement, and focusing on only a needed few potential series of the full motion-independent potential expansions, the complete exterior metric field for a single, spherically symmetric mass source is here obtained—the Schwarzschild metric field of general relativity (the Eddington PPN parameter γ = 1) as well as its generalization if the isotropic spatial metric potential’s linearized form is ?g _SS (γ, r) = 1 + 2γ Gm/c ² r +.... with γ ≠ 1 are obtained.     

Quantum synchronization of chaotic oscillator behaviors among coupled BEC–optomechanical systems

We consider and theoretically analyze a Bose-Einstein condensate (BEC) trapped inside an optomechanical system consisting of single-mode optical cavity with a moving end mirror. The BEC is formally analogous to a mirror driven by radiation pressure with strong nonlinear coupling. Such a nonlinear enhancement can make the oscillator display chaotic behavior. By establishing proper oscillator couplings, we find that this chaotic motion can be synchronized with other oscillators, even an oscillator network. We also discuss the scheme feasibility by analyzing recent experiment parameters. Our results provide a promising platform for the quantum signal transmission and quantum logic control, and they are of potential applications in quantum information processing and quantum networks.