Completeness of classical spin models and universal quantum computation

We study mappings between distinct classical spin systems that leave the partition function invariant. As recently shown in [Phys. Rev. Lett. 100, 110501 (2008)], the partition function of the 2D square lattice Ising model in the presence of an inhomogeneous magnetic field, can specialize to the partition function of any Ising system on an arbitrary graph. In this sense the 2D Ising model is said to be "complete". However, in order to obtain the above result, the coupling strengths on the 2D lattice must assume complex values, and thus do not allow for a physical interpretation. Here we show how a complete model with real -and, hence, "physical"- couplings can be obtained if the 3D Ising model is considered. We furthermore show how to map general q-state systems with possibly many-body interactions to the 2D Ising model with complex parameters, and give completeness results for these models with real parameters. We also demonstrate that the computational overhead in these constructions is in all relevant cases polynomial. These results are proved by invoking a recently found cross-connection between statistical mechanics and quantum information theory, where partition functions are expressed as quantum mechanical amplitudes. Within this framework, there exists a natural correspondence between many-body quantum states that allow universal quantum computation via local measurements only, and complete classical spin systems.Comment: 43 pages, 28 figure

On Sylow normalizers of finite groups

Electronic version of an article published as Journal of Algebra and Its Applications Vol. 13, No. 3 (2014) 1350116 (20 pages). DOI 10.1142/S0219498813501168. © [copyright World Scientific Publishing Company] http://www.worldscientific.com/[EN] The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup- closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.The second and third authors have been supported by Proyecto MTM2010-19938C03-02, Ministerio de Econom ia y Competitividad, Spain. The first author would like to thank the Universitat de Valencia and the Universitat Politecnica de Valencia for their warm hospitality during the preparation of this paper. He has been also supported by RFBR Project 13-01-00469.Kazarin, L.; Martínez Pastor, A.; Perez Ramos, MD. (2014). On Sylow normalizers of finite groups. Journal of Algebra and Its Applications. 13(3):1-20. https://doi.org/10.1142/S0219498813501168S12013

Modelling the unfolding pathway of biomolecules: theoretical approach and experimental prospect

We analyse the unfolding pathway of biomolecules comprising several independent modules in pulling experiments. In a recently proposed model, a critical velocity $v_{c}$ has been predicted, such that for pulling speeds $v>v_{c}$ it is the module at the pulled end that opens first, whereas for $v<v_{c}$ it is the weakest. Here, we introduce a variant of the model that is closer to the experimental setup, and discuss the robustness of the emergence of the critical velocity and of its dependence on the model parameters. We also propose a possible experiment to test the theoretical predictions of the model, which seems feasible with state-of-art molecular engineering techniques.Comment: Accepted contribution for the Springer Book "Coupled Mathematical Models for Physical and Biological Nanoscale Systems and Their Applications" (proceedings of the BIRS CMM16 Workshop held in Banff, Canada, August 2016), 16 pages, 6 figure

Quantum geometry and quantum algorithms

Motivated by algorithmic problems arising in quantum field theories whose dynamical variables are geometric in nature, we provide a quantum algorithm that efficiently approximates the colored Jones polynomial. The construction is based on the complete solution of Chern-Simons topological quantum field theory and its connection to Wess-Zumino-Witten conformal field theory. The colored Jones polynomial is expressed as the expectation value of the evolution of the q-deformed spin-network quantum automaton. A quantum circuit is constructed capable of simulating the automaton and hence of computing such expectation value. The latter is efficiently approximated using a standard sampling procedure in quantum computation.Comment: Submitted to J. Phys. A: Math-Gen, for the special issue ``The Quantum Universe'' in honor of G. C. Ghirard

On the Sylow graph of a finite group

The final publication is available at Springer via http://dx.doi.org/10.1007/s11856-011-0138-xLet G be a finite group and Gp be a Sylow p-subgroup of G for a prime p in pi(G), the set of all prime divisors of the order of G. The automiser Ap(G) is defined to be the group NG(Gp)/GpCG(Gp). We define the Sylow graph gamma A(G) of the group G, with set of vertices pi(G), as follows: Two vertices p, q ¿ ¿(G) form an edge of ¿A(G) if either q ¿ ¿(Ap(G)) or p ¿ ¿(Aq(G)). The following result is obtained: Theorem: Let G be a finite almost simple group. Then the graph ¿A(G) is connected and has diameter at most 5. We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.The second and third authors have been supported by Proyecto MTM2007-68010-C03-03 and Proyecto MTM2010-19938-C03-02, Ministerio de Educacion y Ciencia and FEDER, Spain.Kazarin, SL.; Martínez Pastor, A.; Pérez-Ramos, M. (2011). On the Sylow graph of a finite group. Israel Journal of Mathematics. 186(1):251-271. doi:10.1007/s11856-011-0138-xS2512711861Z. Arad and D. Chillag, Finite groups containing a nilpotent Hall subgroup of even order, Houston Journal of Mathematics 7 (1981), 23–32.H. Azad, Semi-simple elements of order 3 in finite Chevalley groups, Journal of Algebra 56 (1979), 481–498.A. Ballester-Bolinches, A. Martínez-Pastor, M. C. Pedraza-Aguilera and M. D. Pérez-Ramos, On nilpotent-like fitting formations, in Groups St. Andrews 2001 in Oxford, (C. M. Campbell et al., eds.) London Mathematical Society Lecture Note Series 304, Cambridge University Press, 2003, pp. 31–38.M. Bianchi, A. Gillio Berta Mauri and P. Hauck, On finite groups with nilpotent Sylow normalizers, Archiv der Mathematik 47 (1986), 193–197.A. Borel, R. Carter, C.W. Curtis, N. Iwahori, T. A. Springer, R. Steinberg, Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes of Mathematics 131 Springer, Berlin, 1970.N. Bourbaki, Éléments de mathématique: Groupes et algèbres de Lie, Chapters IV, V, VI, Hermann, Paris, 1968.R. W. Carter, Simple groups of Lie type, Wiley, London, 1972.R. W. Carter, Conjugacy classes in the Weyl group, Compositio Mathematica 25 (1972), 1–59.R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley, London, 1985.A. D’Aniello, C. De Vivo and G. Giordano, On certain saturated formations of finite groups, in Proceedings Ischia Group Theory 2006, (T. Hawkes, P. Longobardy and M. Maj, eds.) World Scientific, Hackensack, NJ, 2007, pp. 22–32.A. D’Aniello, C. De Vivo and G. Giordano, Lattice formations and Sylow normalizers: a conjecture, Atti del Seminario Matematico e Fisico dell’ Università di Modena e Reggio Emilia 55 (2007), 107–112.A. D’Aniello, C. De Vivo, G. Giordano and M. D. Pérez-Ramos, Saturated formations closed under Sylow normalizers, Communications in Algebra 33 (2005), 2801–2808.K. Doerk, T. Hawkes, Finite soluble groups, Walter De Gruyter, Berlin-New York, 1992.G. Glauberman, Prime-power factor groups of finite groups II, Mathematische Zeitschrift 117 (1970), 46–56.D. Gorenstein, R. Lyons, The local 2-structure of groups of characteristic 2 type, Memoirs of the American Mathematical Society 42, No. 276, Providence, RI, 1983.R. M. Guralnick, G. Malle and G. Navarro, Self-normalizing Sylow subgroups, Proceedings of the American Mathematical Society 132 (2004), 973–979.F. Menegazzo, M. C. Tamburini, A property of Sylow p-normalizers in simple groups, Quaderni del seminario Matematico di Brescia, n. 45/02 (2002).R. Steinberg, Lectures on Chevalley Groups, Yale University, New Haven, Conn., 1968.E. Stensholt, An application of Steinberg’s construction of twisted groups, Pacific Journal of Mathematics 55 (1974), 595–618.E. Stensholt, Certain embeddings among finite groups of Lie type, Journal of Algebra 53 (1978), 136–187.K. Zsigmondy, Zur Theorie der Potenzreste, Monatshefte für Mathematik and Physik 3 (1892), 265–284

Classical Ising model test for quantum circuits

We exploit a recently constructed mapping between quantum circuits and graphs in order to prove that circuits corresponding to certain planar graphs can be efficiently simulated classically. The proof uses an expression for the Ising model partition function in terms of quadratically signed weight enumerators (QWGTs), which are polynomials that arise naturally in an expansion of quantum circuits in terms of rotations involving Pauli matrices. We combine this expression with a known efficient classical algorithm for the Ising partition function of any planar graph in the absence of an external magnetic field, and the Robertson-Seymour theorem from graph theory. We give as an example a set of quantum circuits with a small number of non-nearest neighbor gates which admit an efficient classical simulation.Comment: 17 pages, 2 figures. v2: main result strengthened by removing oracular settin

Particles and fields in fluid turbulence

The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.Comment: 165 pages, review article for Rev. Mod. Phy

Recent Developments in Understanding Two-dimensional Turbulence and the Nastrom-Gage Spectrum

Two-dimensional turbulence appears to be a more formidable problem than three-dimensional turbulence despite the numerical advantage of working with one less dimension. In the present paper we review recent numerical investigations of the phenomenology of two-dimensional turbulence as well as recent theoretical breakthroughs by various leading researchers. We also review efforts to reconcile the observed energy spectrum of the atmosphere (the spectrum) with the predictions of two-dimensional turbulence and quasi-geostrophic turbulence.Comment: Invited review; accepted by J. Low Temp. Phys.; Proceedings for Warwick Turbulence Symposium Workshop on Universal features in turbulence: from quantum to cosmological scales, 200

Binding of Superantigen Toxins into the CD28 Homodimer Interface Is Essential for Induction of Cytokine Genes That Mediate Lethal Shock

Bacterial superantigen toxins bind directly to the dimer interface of CD28, the principal co-stimulatory receptor, to induce a lethal cytokine storm, and peptides that prevent this binding can suppress superantigen lethality