Beating the Efficiency of Both Quantum and Classical Simulations with a Semiclassical Method

Cesare Mollica, Jiri Vanicek
American Physical Society, 2011
Article

Résumé

While rigorous quantum dynamical simulations of many-body systems are extremely difficult (or impossible) due to exponential scaling with dimensionality, the corresponding classical simulations ignore quantum effects. Semiclassical methods are generally more efficient but less accurate than quantum methods and more accurate but less efficient than classical methods. We find a remarkable exception to this rule by showing that a semiclassical method can be both more accurate and faster than a classical simulation. Specifically, we prove that for the semiclassical dephasing representation the number of trajectories needed to simulate quantum fidelity is independent of dimensionality and also that this semiclassical method is even faster than the most efficient corresponding classical algorithm. Analytical results are confirmed with simulations of fidelity in up to 100 dimensions with 21700-dimensional Hilbert space.

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https://infoscience.epfl.ch/record/170214?ln=fr

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Publications associées (3)

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This thesis is focused on classical and semiclassical approximations of a specific quantum time correlation function, the “quantum fidelity.” Namely, we rigorously study the efficiency of a continuous class of algorithms for the evaluation of its classical limit and both the cost and the error of a particular semiclassical approximation of it. In the physics domain, quantum fidelity can be used to study the cross section of inelastic neutron scattering and phenomena like decoherence and irreversibility in NMR experiments. In chemical physics, quantum fidelity can be employed to compute the spectrum of a quantum state, to evaluate the accuracy of quantum dynamics on an approximate potential energy surface, and to verify the importance of nonadiabatic electronic transitions in molecules. Unfortunately, as for any numerical simulation of a quantum dynamical object, many-dimensional calculations of quantum fidelity are extremely expensive computationally. Classical methods, not including any quantum effect, are generally believed to be efficient approximations for the numerical evaluation of large systems’ properties when quantum effects are not important. We prove that, previous to this research, the computational costs of available algorithms for the simulation of classical fidelity increased exponentially with the number of degrees of freedom and we single out, within a continuous family of algorithms, the only one for which the number of trajectories needed for convergence is independent of the system’s dimensionality. Recently a semiclassical method, called “dephasing representation” (DR), has been proposed to approximate quantum fidelity accurately and efficiently. Although semiclassical methods, approximately including all types of quantum effects, are generally more accurate but much less efficient than classical methods, we prove that for the semiclassical DR there exists at least one algorithm for which the number of trajectories needed for convergence is independent of dimensionality and also that this semiclassical algorithm is even faster than the most efficient classical fidelity algorithm previously singled out. Encouraged by the surprising efficiency of the specific semiclassical method, we investigate the error of the DR compared to the quantum benchmark.

EPFL2012

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Dynamic triangulations for efficient 3D simulation of granular materials

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Granular materials are omnipresent in many fields ranging from civil engineering to food, mining and pharmaceutical industries. Often considered a fourth state of matter, they exhibit specific phenomena such as segregation, arching effects, pattern formation, etc. Due to its potential capability of realistically rendering these behaviors, the Distinct Element Method (DEM) is a very enticing simulation technique. Indeed it makes it possible to analyze and observe phenomena that are barely if at all accessible experimentally. DEM works by tracking every particle in the system individually, maintaining for each a trajectory influenced by external factors such as gravitation or contacts with boundary objects and by the interactions with other grains. The mathematical problem of identifying pairs of grains that interact and locating precisely where the contact occurs is highly dependent on the shape of the grains. We focus in this thesis on 3D spherical grains and use dynamic weighted Delaunay triangulations to track the collisions. The triangulation is built on the centers of the grains and evolves to follow their motion. We prove that all potentially colliding pairs of spheres are adjacent in the triangulation. As there are 6n to 8n edges for n spheres in most practical cases, the complexity of the collision detection becomes linear instead of quadratic in the number of particles, with a small overhead in maintaining the triangulation with efficient local operations. For the physical problem of realistically rendering the collision in a numerical contact model suitable for computer simulation, we have used widely accepted theories such as the viscoelastic model of Cundall, but have also tested some recent, more sophisticated developments in the field. The collision detection and contact models have been implemented in a modular DEM simulation code with advanced features in data structures storing the triangulation, in numerical robustness of the geometric computations, and in parallel processing on shared memory computers. Optimal packing of powders is important in many industrial processes, yet no theoretical result exists when dealing with grains of different sizes. We have performed simulations of such cases and could compare our results with experimental data. Preliminary results have been obtained regarding the relation between the size and proportion of grains and the density of the packing. Other simulations have also been performed, such as the granular flow through an hourglass. As no efficient simulation method is currently known for non-spherical 3D grains, we propose an intermediate approach of gluing spheres together into arbitrary shaped clusters and show some examples based on this approach.

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Radicals play an important role in many areas of chemistry, such as atmospheric, aquatic, polymer, and biological, to name a few. Radicals are often highly reactive species which are short-lived and therefore harder to study by experimental techniques. In principle, computational quantum chemistry enables the study of arbitrary chemical species, including elusive radicals. However, the approximate quantum chemical methods which are used for production simulations need to be validated for the type of problem in question. First, it is necessary to validate that highly accurate reference methods commonly applied to closed-shell systems are equally applicable to radicals. I show that the “gold standard” for closed-shell systems (coupled cluster with singles, doubles, and perturbative triples excitations (CCSD(T))) can produce molecular geometries and vibrational frequencies suitable for (sub-)kJ/mol thermochemistry for radical species as well. However, additional diagnosis is necessary, especially for vibrational frequency calculations. Coupled cluster methods are then used to compute benchmark binding energies between small radicals and water or hydrogen fluoride as a proxy to the aqueous condensed phase. I find that many common density functional theory (DFT) methods cannot reproduce these binding energies to within chemical accuracy (1 kcal/mol). However, some functionals, for example MPW1K and BHandHLYP, perform better than others. If more accurate ab initio methods are computationally too demanding for the system of interest, these DFT methods should be used for (micro-)solvated radicals. Complexes of radicals with polar solvent are found to be more strongly bound than expected. This can not be explained by dispersive and electrostatic interactions alone. Using natural bond orbitals, I show that additional donor-acceptor interactions may be responsible for the strong intermolecular binding observed. One of the most fundamental processes involving aqueous organic radicals is the one-electron oxidation of closed-shell organics. I study the vertical ionization of neutral organics. That is, the ionization is faster than the reorganization of the solvent (and solute), and the conformation is assumed to be unchanged during the ionization process. I find that the difference between the gas-phase and aqueous-phase vertical ionization energy, ∆VIE = VIEaq − VIEgas, ranges from ≈ 0 to -1.0 eV for small aromatic compounds. The differences in ∆VIE between different compounds seem to be caused by the solvent structures of the ≈ 70 water molecules closest to the solute. In natural surface waters, oxidized organics arise through the reaction of photochemically created radicals with organic molecules already present in the water column. One example for such a reaction is the oxidation of sulfonamide antibiotics. I show that upon oxidation of sulfadiazine and related compounds, an aniline radical cation is formed. The reactivity of this oxidized species is strikingly different from its reduced counterpart: the barrier for a nucleophilic attack on the aniline ring is drastically lowered, and a fast rearrangement reaction can take place. I demonstrate that current quantum chemical protocols are suitable for mechanistic studies of solvated radicals, and I successfully apply them to such problems. To achieve chemical accuracy for quantum computational protocols, further developments are necessary. To validate novel approaches, the benchmark data presented in this thesis may be used.

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