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An exact energy conservation property of the quantum lattice Boltzmann algorithm. (English) Zbl 1255.81133

Summary: The quantum lattice Boltzmann algorithm offers a unitary and readily parallelisable discretisation of the Dirac equation that is free of the fermion-doubling problem. The expectation of the discrete time-advance operator is an exact invariant of the algorithm. Its imaginary part determines the expectation of the Hamiltonian operator, the energy of the solution, with an accuracy that is consistent with the overall accuracy of the algorithm. In the one-dimensional case, this accuracy may be increased from first to second order using a variable transformation. The three-dimensional quantum lattice Boltzmann algorithm uses operator splitting to approximate evolution under the three-dimensional Dirac equation by a sequence of solutions of one-dimensional Dirac equations. The three-dimensional algorithm thus inherits the energy conservation property of the one-dimensional algorithm, although the implementation shown remains only first-order accurate due to the splitting error.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
39A12 Discrete version of topics in analysis
81-08 Computational methods for problems pertaining to quantum theory
81T80 Simulation and numerical modelling (quantum field theory) (MSC2010)
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs

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References:

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