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Quantum Lattice Boltzmann (QLB)

Sauro Succi A

Published Apr 19, 2018• 0 citations
Publisher

Abstract

Abstract The Lattice Boltzmann concepts and applications described so far refer to classical, i.e., non-quantum physics. However, the LB formalism is not restricted to classical Newtonian mechanics and indeed an LB formulation of quantum mechanics, going by the name of quantum LB (QLB) has been in existence for more than two decades. Even though it would far-fetched to say that QLB represents a mainstream, in the recent years it has captured some revived interest, mostly on account of recent developments in quantum-computing research. This chapter provides an account of the QLB formulation: stay tuned, LBE goes quantum!

DOI
10.1093/oso/9780199592357.003.0032

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Abstract The Lattice Boltzmann concepts and applications described so far refer to classical, i.e., non-quantum physics. However, the LB formalism is not restricted to classical Newtonian mechanics and indeed an LB formulation of quantum mechanics, going by the name of quantum LB (QLB) has been in existence for more than two decades. Even though it would far-fetched to say that QLB represents a mainstream, in the recent years it has captured some revived interest, mostly on account of recent developments in quantum-computing research. This chapter provides an account of the QLB formulation: stay tuned, LBE goes quantum!
Abstract The lattice Boltzmann concepts and applications described so far refer to classical (non-quantum) physics. Whether this limitation is strictly scientific or purely contingental in character, it is difficult to say. However, it appears plausible to speculate that at least part of the LBE assets should be exportable to the quantum world. In principle, this does not even require much of a conceptual jump, since it has been known for a long time that quantum mechanics can be formulated in a fluid dynamic language. This analogy is precisely the stepping stone for this final chapter: stay with us, LBE goes quantum mechanical! Intriguing analogies between quantum mechanics and fluid mechanics have been pointed out since the earliest days of quantum theory [212]. The orthodox tenet is that these analogies are purely formal in character and do not bear upon the basic physics of quantum phenomena. A less orthodox, albeit not minor, stream of thought insists instead that quantum mechanics and notably its philosophical hard-core, Heisenberg’s uncertainty principle, are nothing but a mirror of our ignorance of the underlying (hidden) microscopic physical level. This leads to the puzzling theory of ‘hidden variables’ which can be traced back to Einstein and subsequently to D. Bohm and others [213]. It is not our intention here to enter this fascinating and still open subject [214]. We turn to a practical question instead: what can the analogy do for us in terms of numerical modeling of evolutionary quantum mechanical phenomena? The question is legitimate because, regardless of its philosophical implications, the fluid analogy certainly provides an intuitive and sound basis to develop numerical methods for time-dependent quantum mechanics. In particular, it appears reasonable to ask whether the advantages brought about by LBE in fluid dynamics can (by means of the fluid analogy) be exported to the context of quantum mechanics. Before we put forward our discrete kinetic theory version of the analogy, it is useful to provide a cursory survey of the main ideas behind the analogy itself. To this end, a short recap of basic notions of quantum mechanics is in order.
Abstract Chapter 32 expounded the basic theory of quantum LB for the case of relativistic and non-relativistic wavefunctions, namely single-particle quantum mechanics. This chapter goes on to cover extensions of the quantum LB formalism to the overly challenging arena of quantum many-body problems and quantum field theory, along with an appraisal of prospective quantum computing implementations. Solving the single particle Schrodinger, or Dirac, equation in three dimensions is a computationally demanding task. This task, however, pales in front of the ordeal of solving the Schrodinger equation for the quantum many-body problem, namely a collection of many quantum particles, typically nuclei and electrons in a given atom or molecule.
The multidimensional formulation of the quantum lattice Boltzmann (QLB) scheme is validated against analytical solutions of the time-dependent nonrelativistic Schrödinger equation in two and three spatial dimensions. As a result, we demonstrate the viability of the quantum lattice Boltzmann scheme for the numerical solution of the time-dependent Schrödinger equation in multiple spatial dimensions. In addition, the links between the QLB scheme and sequential splitting methods for partial differential equations are also clarified.
The derivation of the quantum lattice Boltzmann model is reviewed with special emphasis on recent developments of the model, namely, the extension to a multi-dimensional formulation and the application to the computation of the ground state of the Gross-Pitaevskii equation (GPE). Numerical results for the linear and nonlinear Schrödinger equation and for the ground state solution of the GPE are also presented and validated against analytical results or other classical schemes such as Crank-Nicholson.
Lattice Boltzmann (LB) methods have proven to be powerful tools for simulating classical physical systems such as waves, diffusion and fluid dynamics. For quantum mechanics in one dimension there are two main schemes: one proposed by Meyer, where the Schrödinger equation is interpreted as a diffusion equation with imaginary diffusion constant; the other, proposed by Succi, is based on the Dirac equation, which in the mesoscopic limit reproduces the non-relativistic quantum behaviour in a similar way to lattice Boltzmann models for fluids. In the present work we compare these models by testing the quantum time evolution of a free Gaussian wavepacket. It is found to be in close correspondence with the results of the two models. Time oscillations are found both for the mean and variance of the wavepacket. Also we found their dependences from factors not present in the predicted values. These elements could provide clues for developing a good adaptation of lattice Boltzmann method for the quantum world.