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------------------------------------------------------------------------------- Experiments on Quantum Annealing Gabriel Aeppli University College of London The experimental basis, including materials and techniques, for studies of quantum annealing is described. We show how certain rare earth fluorides meet the criteria of readily tunable quantum fluctuations and disorder which such studies require, and then describe the experiments which establish the underlying quantum mechanical nature of the dynamics dominating the approach to equilibrium at low temperatures, as well as those which reveal the quite different outcomes of classical and quantum annealing protocols. ------------------------------------------------------------------------------ An Introduction to Transverse Ising Models and Quantum Annealing in Kinetically Constrained Tranverse Ising Systems. Bikas K. Chakrabarti Saha Institute of Nuclear Physics, Kolkata We introduce the transverse Ising model as a prototype for discussing quantum phase transition. Mean field theory, application to superconductivity (BCS) and real space renormalization group technique (in one dimension) are discussed. Next we introduce Suzuki-Trotter formalism to show the correspondence between $d$-dimensional quantum system with a $(d+1)$-dimensional classical system. We then discuss, in the context of transverse Ising spin glass models, the possible relationship between Quantum Annealing and Replica Symmetry Restoration in quantum spin glasses. Next, we will discuss the quantum annealing (combinations of annealing schedules) in kinetically constrained Ising model (e.g., East model) in presence of (annealed out) transverse field (work in collaboration with A. Das and R. B. Stinchcombe; 2004) ------------------------------------------------------------------------------ Quantum Annealing of a ± J Ising Spin Glass Using a Zero Temperature Monte Carlo Technique. Arnab Das Saha Institute of Nuclear Physics, Kolkata Here we investigate how quantum annealing helps in reaching the ground state of a NP-hard classical ± J Ising spin glass (in infinite dimension) using a zero temperatute quantum Monte Carlo. The main idea behind using that particular Monte Carlo algorithm is that in this algorithm, the tunneling probability is linear in energy (scaled by the tunneling field) unlike the finite temperature ones, where it is exponential. This allows a faster movement through the configuration space which can increases the efficiency of annealing, specially in certain cases, where there are many locally optimal states separated by divereging energy barriers. The ground states for the corrosponding transverse Ising systems upto 30 spins (450 bonds) are determined exactly using an exhaustive search algorithm and the data are compared with those obtained using a zero temperature quantum Monte Carlo. We investigate the relaxation behaviour of the system by keeping track of the average energy during quantum annealing. The study for higher system sizes has also been made. ------------------------------------------------------------------------------ Evolution of decoherence and quantum couplings in quantum systems with a noisy environment Andrew J. Fisher University College London I will review the theory of quantum systems coupled to noisy condensed-phase environments, enphasising the central role played by the spectral functions of the environmental fluctuations. I will show how the application of the fluctuation-dissipation theorem to these functions leads to important connections between the coherent couplings and incoherent dynamics induced by the environment, and give examples of this connection from condensed-matter and quantum-optical systems. I will describe how such couplings may be expected to evolve under scaling transformations. I will also decribe how to tailor responsne functions in such a way as to optimise the coherent evolution of the system, and describe some novel proposals to exploit local excitation to control the evolution of quantum states. ------------------------------------------------------------------------------ Finding Suzuki-Trotter decompositions of higher orders Naomichi Hatano Institute of Industrial Science, University of Tokyo Mathematical methods of finding higher-order Suzuki-Trotter decomposi- tions are reviewed. Higher-order decompositions are useful particularly in nu- merically exact computations such as transfer-matrix calculation and numerical integeration. The simplist decomposition exp (x(A+B)) = exp(xA)exp(xB) + O(x2) .... (1) is immediately improved as the second-order decompostion exp(x(A+B))= exp(Ax/2)exp(Bx/2)exp(Ax/2) + O(x3) .... (2) How can we make higher-order decompositions in the form exp(x(A+B)) = exp(p1xA)exp(p2xB)exp(p3xA)exp(p4xB) + O(xm)? .... (3) M. Suzuki has developed a general theory of finding such higher-order decom- positions as the fractal decomposition and quantum analysis. These techniques are summarized in this lecture. ------------------------------------------------------------------------------ Cluster algorithm of quantum Monte Carlo simulations Naomichi Hatano Institute of Industrial Science, University of Tokyo A new development of quantum Monte Carlo algorithm is reviewed. In the last decade, the application of the cluster algorithm to the quantum Monte Carlo simulation has led to much higher efficiency. The cluster algorithm has two major advantages. First, the Trotter limit is taken before carrying out the simulation. Second, the relaxation of the simulation dynamics is accellerated much. Some details of the algorithm as well as exampls of results are reviewed. This lecture is mainly based on a recent review article by N. Kawashima and Harada. ------------------------------------------------------------------------------ Quantum Spin Glasses, Quantum Annealing and Probabilistic Information Processing Jun-ichi Inoue Hokkaido University We present several applications of quantum spin glasses (random field Ising model, Sherrington-Kirkpatrick model, Ising spin glasses with $p$-body interaction in a transverse field) to probabilistic information processing, especially to the problems of image restoration and error-correcting codes. As a related optimization method, quantum annealing is also introduced to these research fields and its performance is investigated by using the quantum Markov chain Monte Carlo method. After a short review of the previous work [J. Inoue, Physical Review E 63, 046114 (2001)], we evaluate the performance of both the Maximum A Priori (MAP for short) and the Maximizer of Posterior Marginal (MPM for short) image restorations which are purely induced by quantum fluctuation (without any thermal fluctuation). The Nishimori condition, on which the best possible performance of the quantum MPM estimation is achieved, is derived as a condition on the effective amplitude of the transverse field. We show the lowest values of the bit-error rate for both the thermal and the quantum MPM estimations are exactly the same. We next discuss an extension of the Sourlas codes by means of the Ising spin glasses with $p$-body interactions in a transverse field. We investigate the tolerance of error-less (or quite low-error) ferromagnetic state to quantum uncertainties in the prior distribution. We find that there exist some critical amplitudes of the transverse field, and at the critical point the system changes from the low-error state to the poor error-correction state as the second order ($p=2$) and as the first order ($p \geq 3$) phase transitions. The relation between the amplitude of the transverse field and the Shannon's information bound is also discussed in the limit of $p \to \infty$ for a given effective amplitude of the transverse field. We show that the Shannon's bound is not violated in this limit by the quantum fluctuation in the prior. In last part of this article, we apply quantum annealing, which is an optimization method based on quantum fluctuations, to the problem of image restoration. We compare the results of the thermal MAP and the quantum MAP estimations by using the simulated (thermal) and the quantum annealings, respectively. We find that a fine restoration of image is achieved by the quantum annealing and its performance measured by the bit-error rate is slightly superior to that of the simulated annealing. ------------------------------------------------------------------------------ Ergodicity, Replica Symmetry, Fluctuations and Their Related Phenomena of Glass Transition in Classical and Quantum Spin Glass Systems Jong-Jean Kim Korea Advanced Institute of Science and Technology Pedagogical problems of ergodicity, replica symmetry, ultrametricity, and fluctuation-dissipation will be recapitulated as important themes of glass transition physics to introduce recent developments in the studies of quantum phase transitions and quantum spin glasses. The simplest model system, that is, the Sherrington-Kirkpatrick(SK) model with a transverse field will be discussed on the basis of the simplest calculations by using the imaginary-time replica formalism under the static approximation to illustrate differences between classical and quantum spin glass systems. ------------------------------------------------------------------------------ Exploring complex landscapes with classical Monte Carlo (I): "equilibrium" super-cooled liquids Victor Martin-Mayor Departamento de Fisica Teorica I Universidad Complutense de Madrid super cooled liquids are "complex" systems where there is no obvious quenched disorder. Nevertheless, their behaviour is in some senses quite similar to a particular types of spin-glasses. We will first review some of their puzzling properties, then we shall review some of our recent results[1,2,3,4] in the study of their equilibrium properties. This will include the use of an optimized Monte Carlo algorithm that completely avoids the "caging" effect and which is able of "thermalizing" the supercooled liquid well below the Mode Coupling temperature, the use of euclidean random-matrices to characterize the local properties of the potential-energy landscape, and the study of Fluctuation-Dissipation relations. We shall comment on the gathered evidence for a topological transition at the Mode Coupling temperature and for a true second-order transition at lower temperatures. [1] T.S. Grigera, V. Martin-Mayor, G. Parisi, P. Verrocchio Phys. Rev. Lett. 87, 085502 (2001) [2] T. S. Grigera, V. Martin-Mayor, G. Parisi, P. Verrocchio Nature 422, 289 - 292 (2003) [3] T. S. Grigera, V. Martin-Mayor, G. Parisi, P. Verrocchio, Phys. Rev. B 70, 014202 (2004) [4] L.A. Fernandez, V. Martin-Mayor, P. Verrocchio, in preparation. ------------------------------------------------------------------------------ Exploring complex landscapes with classical Monte Carlo (II): out-of-equilibrium spin-glasses. Victor Martin-Mayor Departamento de Fisica Teorica I Universidad Complutense de Madrid The recently discovered "rejuvenation" and "memory" properties of spin-glasses seriously question the validity of the annealing approach for obtaining low-temperature properties of spin-glasses. Furthermore, it is by no means obvious that the standard theoretical models for spin-glass behaviour are true spin-glasses[1]. We shall briefly recall the relevant experimental facts, then present recent results [2,3] obtained using the dedicated computer SUE designed and built in the Universidad de Zaragoza (Spain). The longer time scales that can be achieved using SUE allow to shed some light on the crucial role of a growing coherence-length to analyze these phenomena. [1] Andrea Maiorano, Enzo Marinari, Federico Ricci-Tersenghi, cond-mat/0409577 [2] S. Jimenez, V. Martin-Mayor, G. Parisi, A. Tarancon J. Phys. A: Math. and Gen. 36, 10755 - 10771 (2003) [3] S. Jimenez, V. Martin-Mayor, S. Perez-Gaviro, cond-mat/0406365 and in preparation. ------------------------------------------------------------------------------ 1. Disordered systems, ground states and combinatorial optimization Heiko Rieger Saarlandes University The physical properties of solid materials which contain a substantial degree of quenched disorder have been an experimental and a theoretical challenge for many decades. Analytic studies of models for these systems are usually based on perturbation theories valid for weak disorder, on phenomenological scaling pictures or on mean-field approximations. Therefore the demand for efficient numerical techniques that allow the investigation of the model Hamiltonians of disordered systems has always been high. Three facts make life difficult here: 1) The regime, where disorder effects are most clearly seen, are at low temperatures -- and are even best visible at zero temperature; 2) the presence of disorder slows the dynamics of theses systems down, they become glassy, such that for instance conventional Monte-Carlo or molecular dynamics simulations encounter enormous equilibration problems; 3) any numerical computation of disordered systems has to incorporate an extensive disorder average. In recent years more and more model systems with quenched disorder were found that can be investigated numerically 1) at zero temperature, 2) without equilibration problems, 3) extremely fast, in polynomial time (for a review on these developments see [1-3]. This is indeed progress, which became possible by the application of exact combinatorial optimization algorithms developed by mathematicians and computer scientists over the last few decades. This gift is not for free: first a mapping of the problem of finding the exact ground state of the model Hamiltonian under consideration onto a standard combinatorial optimization problem has to be found. If one is lucky, this problem falls into the class of P-problems, for which polynomial algorithms exist. If not, the intellectual challenge for the theoretical physicist remains to reformulate the model Hamiltonian in such a way that its universality class is not changed but a mapping on a P-problem becomes feasible. In this lecture I will review some of the recent progress that has been made in this direction. In particular I will discuss, with an emphasis on the computational point of view, interfaces in random bond Ising ferromagnets, the random field Ising model, spin glasses, the collective behavior of a flux line ensemble in a disordered environment, the disorder induced roughening transition in a periodic elastic medium and the critical properties of the random bond Potts model in the large q-limit. [1] H. Rieger: Ground state properties of frustrated systems; Lecture Notes in Physics 501 (ed. J. Kertesz and I. Kondor), p. 122--158 (Springer Verlag, Berlin-Heidelberg-NewYork, 1998). [2] M. Alava, P. Duxbury, C. Moukarzel and H. Rieger, Combinatorial optimization and disordered systems, Phase Transition and Critical Phenomena, Vol. 18 (ed. C. Domb and J. L. Lebowitz), p.141-317, (Academic Press, Cambridge, 2000). [3] A. Hartmann and H. Rieger, Optimization Algorithms in Physics (Wiley VCH, Berlin, 2002); New optimization Algorithms in Physics (Wiley VCH, Berlin, 2004). *********************************************************************** 2. Strongly disordered quantum magnets and quantum spin glasses Heiko Rieger Saarlandes University The interplay of quenched disorder and quantum fluctuations has a profound effect on the low temperature properties of magnetic systems. At zero temperature quantum phase transitions, which are driven by quantum rather than thermal fluctuations, can occur and display novel universal behavior. They are accompanied by disorder induced Griffiths-McCoy singularities in a whole region around them, displaying for instance divergent susceptibilities even away from the critical point. In some systems the effect of the disorder is so strong that a new class of fixed points emerges - the infinite randomness fixed point (IRFP), which is characterized by unconventional scaling laws at the critical point, in which the characteristic time scale depend exponentially rather than algebraically upon the length scale. Among them are e.g. random Ising magnets in a transverse field or random Heisenberg or XY spin chains. In this lecture I will review this peculiar behavior of strongly disordered magnets. As a paradigmatic example serves the random transverse Ising chain, with which theoretical concepts like the strong disorder renormalization group (SDRG, also called the Dasgupta-Ma-Hu RG) and the IRFP scaling scenario can be introduced. The analytical predictions are compared with exact diagonalization studies. The corresponding transverse Ising systems in higher dimensions need a numerical implementation of the SDRG, which have to be compared with results of quantum Monte-Carlo (QMC) simulations, the results of which I will also discuss. Finally I will work on the Ising spin glass in a transverse field. Then random bond Heisenberg and XY chains are considered, in which the IRFP is manifested by a random singlet phase. Also here the RG predictions are compared with QMC simulations and exact diagonalization calculations. In higher dimensions Heisenberg systems do not show a IRFP, and we will critically examine the predictions of the SDRG by comparing them with stochastic series expansion (SSE) computations. Finally the effect of frustration, in particular in the case of Heisenberg spin glasses, is discussed. ------------------------------------------------------------------------------- Quantum versus Classical Annealing: deterministic and stochastic approaches to complex (and simple) optimization problems. Giuseppe E. Santoro, Demian Battaglia and Lorenzo Stella SISSA We present some of the recent work done at SISSA in order to understand better the potential of Quantum Annealing algorithms, in comparison with standard thermal Simulated Annealing, in various problems of interest in the field of optimization. The optimization problems which have been tackled so far are the Random Ising Model [1], instances of the Travelling Salesman Problem [2], and problems in SATISFIABILITY, notably the random 3-SAT problem [3]. In all those cases, the technique used was a Path Integral Monte Carlo-based quantum annealing algorithm (PIMC-QA), compared to a standard Metropolis Simulated Annealing (SA). Remarkably, while PIMC-QA definitely works better than SA in the Ising and TSP case [1,2], the outcome is reversed in the SAT case [3]. More recent work on the use of Green Function Monte Carlo QA algorithms will be also reported, both for TSP and 3-SAT. In parallel, in order to understand some of the key features behind a quantum versus a classical annealing dynamics, we are also studying much simpler problems, like double well potentials, where the landscape which one is trying to optimize is well under control. These studies employ not only stochastic (Monte Carlo) approaches, but also deterministic ones, comparing, for instance, Fokker-Planck versus real-time and imaginary-time Schroedinger evolution annealings. [1] ``Theory of Quantum Annealing of an Ising Spin Glass'' Giuseppe E. Santoro, Roman Martonak, Erio Tosatti, Roberto Car, SCIENCE 295, p. 2427 (2002). [2] ``Quantum annealing of the Travelling Salesman Problem'', Roman Martonak, Giuseppe E. Santoro, and Erio Tosatti, to appear on PRE. [3] ``Understanding a Quantum Annealing algorithm: the random 3-SAT Problem as a case study'', Demian Battaglia, Giuseppe E. Santoro, and Erio Tosatti, in preparation. ------------------------------------------------------------------------------- Residual energies after slow quantum annealing Sei Suzuki and Masato Okada University of Tokyo An investigation on the residual energy after quantum annealing is presented. The residual energy is defined by the energy of an approximate ground state measured from that of the true ground state. Study on the residual energy is significant for revealing the efficiency of the method used to solve a problem. It has been discussed that the efficiency of the quantum annealing possibly surpasses that of the classical thermal annealing. However its evidence is insufficient. In our work, we carry out the quantum annealing using numerical method for a few models. We show by the result that the residual energy obeys a power law with respect to the annealing time in the slow limit of 'annealing'. We also show that this feature and evaluated exponent is derived from the theory of quantum adiabatic theorem. ------------------------------------------------------------------------------ |