Connected Spin Systems Research Articles

Quantum accelerated approach to the thermal state of classical all-to-all connected spin systems with applications to pattern retrieval in the Hopfield neural network

We explore the question as to whether quantum effects can yield a speedup of the non-equilibrium evolution of spin systems towards a classical thermal state. In our approach we exploit the fact that the thermal state of a spin system can be mapped onto a node-free quantum state whose coefficients are given by thermal weights. This perspective permits the construction of a dissipative -- yet quantum -- dynamics which encodes in its stationary state the thermal state of the original problem. We show for the case of an all-to-all connected Ising spin model that an appropriate transformation of this dissipative dynamics allows to interpolate between a regime in which the order parameter obeys the classical equations of motion under Glauber dynamics, to a quantum regime with an accelerated approach to stationarity. We show that this effect enables in principle a speedup of pattern retrieval in a Hopfield neural network.

SU(3) semiclassical representation of quantum dynamics of interacting spins.

We present a formalism for simulating quantum dynamics of lattice spin-1 systems by first introducing local hidden variables and then doing semiclassical (truncated Wigner) approximation in the extended phase space. In this way, we exactly take into account the local on-site Hamiltonian and approximately treat spin-spin interactions. In particular, we represent each spin with eight classical SU(3) variables. Three of them represent the usual spin components and five others are hidden variables representing local spin-spin correlations. We compare our formalism with exact quantum dynamics of fully connected spin systems and find very good agreement. As an application, we discuss quench dynamics of a Bose-Hubbard model near the superfluid-insulator transition for a 3D lattice system consisting of 1000 sites.

Thermal entanglement in fully connected spin systems and its random-phase-approximation description

We examine the thermal pairwise entanglement in a symmetric system of $n$ spins fully connected through anisotropic $XYZ$-type couplings embedded in a transverse magnetic field. We consider both the exact evaluation together with that obtained with the static $\text{path}+\text{random}$ phase approximation (RPA) and the ensuing mean $\text{field}+\text{RPA}$. The latter is shown to provide an accurate analytic description of both the parallel and antiparallel thermal concurrence in large systems. We also analyze the limit temperature for pairwise entanglement, which is shown to increase for large fields and to decrease logarithmically with increasing $n$. Special finite-size effects are also discussed.

ABACUS, a direct method for protein NMR structure computation via assembly of fragments

The ABACUS algorithm obtains the protein NMR structure from unassigned NOESY distance restraints. ABACUS works as an integrated approach that uses the complete set of available NMR experimental information in parallel and yields spin system typing, NOE spin pair identities, sequence specific resonance assignments, and protein structure, all at once. The protocol starts from unassigned molecular fragments (including single amino acid spin systems) derived from triple-resonance (1)H/(13)C/(15)N NMR experiments. Identifications of connected spin systems and NOEs precede the full sequence specific resonance assignments. The latter are obtained iteratively via Monte Carlo-Metropolis and/or probabilistic sequence selections, molecular dynamics structure computation and BACUS filtering (A. Grishaev and M. Llinás, J Biomol NMR 2004;28:1-10). ABACUS starts from scratch, without the requirement of an initial approximate structure, and improves iteratively the NOE identities in a self-consistent fashion. The procedure was run as a blind test on data recorded on mth1743, a 70-amino acid genomic protein from M. thermoautotrophicum. It converges to a structure in ca. 15 cycles of computation on a 3-GHz processor PC. The calculated structures are very similar to the ones obtained via conventional methods (1.22 A backbone RMSD). The success of ABACUS on mth1743 further validates BACUS as a NOESY identification protocol.

Use of graph theory for secondary structure recognition and sequential assignment in heteronuclear (13C, 15N) NMR spectra: application to HU protein from Bacillus stearothermophilus.

A computer-assisted procedure, based upon a branch of mathematics known as graph theory, has been developed to recognize secondary structure elements in proteins from their corresponding nuclear Overhauser effect spectroscopy (NOESY)-type spectra and to carry out their sequential assignment. In the method, NOE connectivity templates characteristic of regular secondary structures are identified in the spectra. Resonance assignment is then achieved by connecting these NOE patterns of secondary structure together, and thereby matching connected spin systems to specific parts of the primary sequence. The range of NOE-graph templates of secondary structure motifs, incorporating alpha-helices and beta-strand motifs, has been examined for reliability and extent of secondary structure identification in a data base composed of the high resolution structures of 20 proteins. The analysis identified several robust NOE-graph templates and supports the implementation of an ordered search strategy. The method, known as SERENDIPITY, has been applied to the analysis of nuclear Overhauser effect data from a three-dimensional time-shared nuclear Overhauser effect spectroscopy (13C, 15N) heteronuclear single quantum correlation spectrum of the (alpha + beta) type protein HU from Bacillus stearothermophilus. The arrangement of the elucidated elements of secondary structure is very similar to that of the x-ray and nmr structures of HU. In addition, our analysis revealed a pattern of interstrand nuclear Overhauser effect in the beta-arm region (residues 53-76) of HU, which suggest irregularities, not reported in the x-ray structure, in both strands of the beta-arm at Ala57 and Pro72, respectively. At these residues, both strands of the beta-arm appear to flip inside out before continuing as a regular antiparallel beta-sheet.

Graph-theoretical assignment of secondary structure in multidimensional protein NMR spectra: Application to the lac repressor headpiece

A novel procedure is presented for the automatic identification of secondary structures in proteins from their corresponding NOE data. The method uses a branch of mathematics known as graph theory to identify prescribed NOE connectivity patterns characteristic of the regular secondary structures. Resonance assignment is achieved by connecting these patterns of secondary structure together, thereby matching the connected spin systems to specific segments of the protein sequence. The method known as SERENDIPITY refers to a set of routines developed in a modular fashion, where each program has one or several well-defined tasks. NOE templates for several secondary structure motifs have been developed and the method has been successfully applied to data obtained from NOESY-type spectra. The present report describes the application of the SERENDIPITY protocol to a 3D NOESY-HMQC spectrum of the 15N-labelled lac repressor headpiece protein. The application demonstrates that, under favourable conditions, fully automated identification of secondary structures and semi-automated assignment are feasible.

Non-square tensor algebra and applications in the NMR of multiply connected spin systems: II

Non-square tensor algebra and applications in the NMR of multiply connected spin systems: II

Non-square tensor algebra and applications in the NMR of multiply connected spin systems: I

Several new results in the theory of angular momentum are presented, which have their origin in NMR experiments on multiply connected spin-systems. In particular, Fano's ‘rectangular’ unit spherical tensors are used to extend Racah tensor algebra to deal with ‘non-square’ matrices. The properties of ‘non-square commutation relationships’ are examined in some detail, and illustrated using a simple three coupled spin-1/2 system. It is shown that Racah algebra is a subset of that which applies to the Fano spherical tensors.

Dynamics of spin systems with randomly asymmetric bonds: Langevin dynamics and a spherical model.

Neural networks contain, very often, asymmetric bonds. The interactions J;, and J,; between the ith and the jth neurons are not identical. In this paper we study the Langevin dynamics of fully connected spin systems whose interaction matrix contains a random antisymmetrie part. The symmetric part consists of independent random bonds whose mean is either zero or ferromagnetic. We also consider a more general class of systems such as the asymmetric Hopfield model and other neural-network models. Within the framework of mean-field theory, the spin fluctuations are viewed as local, thermally averaged, time-dependent magnetic moments. These moments are induced by excess (i.e., nonthermal) internal noise which, in the presence of asymmetry, is time dependent and does not vanish even in the high-temperature phase. The mean-field equations are solved using a simplified, spherical model, in which the spins are linear variables except for a global constraint on the total level of their fluctuations. Random asymmetry of arbitrary strength destroys spin-glass freezing. Ferromagnetic phases, as well as retrieval states in neural networks, are affected only slightly by weak random asymmetry, in agreement with the conclusions of Hertz et al. The dynamical behavior of a system with weak asymmetry is studied in some detail. In the spin-glass case at low temperatures, when the strength of the asymmetry decreases, the internal excess noise does not vanish but slows down with a characteristic correlation time ~ ~ k The parameter k denotes the relative strength of the antisymmetric components of the bonds. The system behaves as a frozen symmetric spin glass on time scales t &&~ and as a paramagnet on scales t &&~. The thermal fluctuations decay with a characteristic time ~T ~ k . The spherical model exhibits a completely frozen spin-glass state at zero temperature. As T~O, fluctuations exhibit a critical slowing down with time ~~ T for all values of k ~0. This T =0 spin-glass transition is probably an artifact of the spherical model and is not expected to exist in nonlinear systems. The relevance of the results to the performance of neural networks is discussed.