Simulation of the diffusion equation on a type-II quantum computer

Abstract

A lattice-gas algorithm for the one-dimensional diffusion equation is realized using radio frequency pulses in a one-dimensional spin system. The model is a large array of quantum two-qubit nodes interconnected by the nearest-neighbor classical communication channels. We present a quantum protocol for implementation of the quantum collision operator and a method for initialization and reinitialization of quantum states. Numerical simulations of the quantum-classical dynamics are in good agreement with the analytic solution for the diffusion equation.

Lattice Boltzmann study on Kelvin-Helmholtz instability: Roles of velocity and density gradients

“…As a mesoscopic approach and a bridge between the molecular dynamics method at the microscopic level and the conventional numerical method at the macroscopic level, the lattice Boltzmann (LB) method [1] has been successfully applied to various fields during the past two decades, ranging from the multiphase system [2][3][4], magnetohydrodynamics [5][6][7], reaction-diffusion system [8][9][10], compressible fluid dynamics [11][12][13][14][15][16][17][18], simulations of linear and nonlinear partial differential equations [19], etc. Its popularity is mainly owed to its kinetic nature [20], which makes the physics at mesoscopic scale can be incorporated easily.…”

Twisting of filamentary vortex solitons demarcated by fast Poincaré recursion

“…the nonlinear interaction term in the Gross Pitaevskii equation, the effective equation of motion of a low-temperature BEC superfluid. With this representation, previously we have predicted solutions to a number of nonlinear classical and quantum systems [4,13,14,15,16]. An advantage of our approach over the standard computational physics GP solvers is that the simple unitary collidestream-rotate operations give rise to an algorithm that approaches pseudo-spectral accuracy [17] in much the same way as the simple collide-stream steps of a lattice Botlzmann algorithm approach pseudo-spectral accuracy for fluid dynamics simulations [18,19].…”

“…From the order of the error term in (14), the Taylor expansion predicts that the quantum algorithm is second order convergent in space.…”

Implementation Schemes for the Factorized Quantum Lattice-Gas Algorithm for the One Dimensional Diffusion Equation using Persistent-Current Qubits

“…This matrix simply "half-way" swaps the middle two (first and second excited) computational states of the coupled system. In NMRQC, the coupled eigenstates are exactly those computational states, but there are no direct matrix elements connecting these states 16 . When the PC Qubits are coupled, the first and second excited states of the four-level system, denoted as |1 and |2 respectively, are in general not the same as the computational basis states the √ swap intends to affect.…”