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Radiation effects and defects in solids

One- and two-dimensional quantum lattice algorithms for Maxwell equations in inhomogeneous scalar dielectric media. II: Simulations

George Vahala, Min Soe, Linda L. Vahala, Abhay K. Ram

Published Feb 1, 2021• 9 citations
Publisher

Abstract

Long-time quantum lattice algorithm (QLA) simulations are performed for the multiple reflection–transmission of an initial electromagnetic pulse propagating normally to a boundary layer region joining two media of different refractive index. For these one-dimensional (1D) simulations, there is excellent agreement between x-, y- and z-representations, as well as very good agreement with nearly all the standard plane wave boundary condition results for reflection and transmission off a dielectric discontinuity. In the QLA simulation, no boundary conditions are imposed at the continuous, but sharply increasing, dielectric boundary layers. Two-dimensional (2D) QLA scattering simulations in the x–z plane are performed for an electromagnetic pulse interacting with a conical dielectric obstacle for the 8–16 qubit model.

DOI
10.1080/10420150.2021.1891059

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George Vahala

A quantum lattice algorithm (QLA) is developed for Maxwell equations in scalar dielectric media using the Riemann-Silberstein representation on a Cartesian grid. For x-dependent and y-dependent dielectric inhomogeneities, the corresponding QLA requires a minimum of 8 qubits/spatial lattice site. This is because the corresponding Pauli spin matrices have off-diagonal components which permit the local collisional entanglement of these qubits. However, z-dependent inhomogeneities require a QLA with a minimum of 16 qubits/lattice site since the Pauli spin matrix σz is diagonal. For 2 dimensional inhomogeneities, one can readily couple the 8-8 qubit schemes for x-y variations. z-x and y-z variations can be treated by either a 16-8 qubit scheme or a 16-16 qubit representation.

•Jan 1, 2022•0 citations
Long-time quantum lattice algorithm (QLA) simulations are performed for the multiple reflection–transmission of an initial electromagnetic pulse propagating normally to a boundary layer region joining two media of different refractive index. For these one-dimensional (1D) simulations, there is excellent agreement between x-, y- and z-representations, as well as very good agreement with nearly all the standard plane wave boundary condition results for reflection and transmission off a dielectric discontinuity. In the QLA simulation, no boundary conditions are imposed at the continuous, but sharply increasing, dielectric boundary layers. Two-dimensional (2D) QLA scattering simulations in the x–z plane are performed for an electromagnetic pulse interacting with a conical dielectric obstacle for the 8–16 qubit model.
Long time quantum lattice algorithm (QLA) simulations are performed for the mul- tiple reflection-transmission of an initial electromagnetic pulse propagating normally to a boundary layer region joining two media of different refractive index. For these one dimensional (1D) sim- ulations, there is excellent agreement between x-, y- and z- representations, as well as very good agreement with nearly all the standard plane wave boundary condition results for reflection and transmission off a dielectric discontinuity. In the QLA simulation, no boundary conditions are im- posed at the continuous, but sharply increasing, dielectic boundary layers. Two dimensional (2D) QLA scattering simulations in the x-z plane are performed for an electromagnetic pulse interacting with a conical dielectric obstacle for the 8-16 qubit model.
A quantum lattice algorithm (QLA) is developed for the solution of Maxwell equations in scalar dielectric media using the Riemann-Silberstein representation. For x-dependent and y-dependent inhomogeneities, the corresponding QLA requries 8 qubits/spatial lattice site. This is because the corresponding Pauli spin matrices have off-diagonal components which permit the collisional entanglement of two qubits. However, z-dependent inhomogeneities require a QLA with 16 qubits/lattice site since the Pauli spin matrix σzσ_zσz​ is diagonal. QLA simulations are performed for the time evolution of an initial electromagnetic pulse propagating normally to a boundary layer region joining two media of different refractive index. There is excellent agreement between all three representations, as well as very good agreement with nearly all the standard plane wave boundary condition results for reflection and transmission off a dielectric discontinuity. In the QLA simulation, no boundary conditions are imposed at the continuous, but sharply increasing, boundary layer.
There is considerable interest in the application of quantum information science to advance computations in plasma physics. A particular point of curiosity is whether it is possible to take advantage of quantum computers to speed up numerical simulations relative to conventional computers. Many of the topics in fusion plasma physics are classical in nature. In order to implement them on quantum computers it will require couching a classical problem in the language of quantum mechanics. Electromagnetic waves are routinely used in fusion experiments to heat a plasma or to generate currents in the plasma. The propagation of electromagnetic waves is described by Maxwell equations with an appropriate description of the plasma as a dielectric medium. Before advancing to the tensor dielectric of a magnetized plasma, this paper considers electromagnetic wave propagation in a one-dimensional inhomogeneous scalar dielectric. The classic theory of scattering of plane electromagnetic waves at a planar interface, separating two different dielectric media, leads to Fresnel equations for reflection and transmission coefficients. In contrast to plane waves, this paper is on the reflection and transmission of a spatially confined electromagnetic pulse. Following an analytical formulation for the scattering of a Gaussian pulse, it is deduced that the maximum transmission coefficient for a pulse is n2/n1\sqrt{n_2/n_1}n2​/n1​​ times that for a plane wave; the incident and transmitted pulses propagating in dielectric media with refractive indices n1n_1n1​ and n2n_2n2​, respectively. The analytical theory is complemented by numerical simulations using a quantum lattice algorithm for Maxwell equations. The algorithm, based on the Riemann-Silberstein-Weber representation of the electromagnetic fields and expressed in term of qubits, is an interleaved sequence of entangling operators at each lattice site and unitary streaming operators which transmit information from one site to an adjacent lattice site. Besides substantiating results from the theory for Gaussian pulses, numerical simulations show their validity for non-Gaussian pulses. Apart from their time-asymptotic forms, the simulations display an interplay between the incident, reflected, and transmitted pulses in the vicinity of the transition region between two dielectric media.
There is considerable interest in the application of quantum information science to advance computations in plasma physics. A particular point of curiosity is whether it is possible to take advantage of quantum computers to speed up numerical simulations relative to conventional computers. Many of the topics in fusion plasma physics are classical in nature. In order to implement them on quantum computers, it will require couching a classical problem in the language of quantum mechanics. Electromagnetic waves are routinely used in fusion experiments to heat a plasma or to generate currents in the plasma. The propagation of electromagnetic waves is described by Maxwell equations with an appropriate description of the plasma as a dielectric medium. Before advancing to the tensor dielectric of a magnetized plasma, this paper considers electromagnetic wave propagation in a one-dimensional inhomogeneous scalar dielectric. The classic theory of scattering of plane electromagnetic waves at a planar interface, separating two different dielectric media, leads to Fresnel equations for reflection and transmission coefficients. In contrast to plane waves, this paper is on the reflection and transmission of a spatially confined electromagnetic pulse. Following an analytical formulation for the scattering of a Gaussian pulse, it is deduced that the maximum transmission coefficient for a pulse is n2/n1 times that for a plane wave; the incident and transmitted pulses propagate in dielectric media with refractive indices n1 and n2, respectively. The analytical theory is complemented by numerical simulations using a quantum lattice algorithm for Maxwell equations. The algorithm, based on the Riemann–Silberstein–Weber representation of the electromagnetic fields and expressed in terms of qubits, is an interleaved sequence of entangling operators at each lattice site and unitary streaming operators, which transmit information from one site to an adjacent lattice site. Besides substantiating results from the theory for Gaussian pulses, numerical simulations show their validity for non-Gaussian pulses. Apart from their time-asymptotic forms, the simulations display an interplay between the incident, reflected, and transmitted pulses in the vicinity of the transition region between two dielectric media.
A quantum lattice algorithm (QLA) is developed for Maxwell equations in scalar dielectric media using the Riemann–Silberstein representation on a Cartesian grid. For x-dependent and y-dependent dielectric inhomogeneities, the corresponding QLA requires a minimum of 8 qubits/spatial lattice site. This is because the corresponding Pauli spin matrices have off-diagonal components which permit the local collisional entanglement of these qubits. However, z-dependent inhomogeneities require a QLA with a minimum of 16 qubits/lattice site since the Pauli spin matrix σz is diagonal. For two-dimensional inhomogeneities, one can readily couple the 8–8 qubit schemes for x−y variations. z−x and y−z variations can be treated by either a 16–8 qubit scheme or a 16–16 qubit representation.
A quantum lattice algorithm (QLA) is developed for Maxwell equations in scalar dielectric media using the Riemann-Silberstein representation on a Cartesian grid. For x-dependent and y-dependent dielectric inhomogeneities, the corresponding QLA requires a minimum of 8 qubits/spatial lattice site. This is because the corresponding Pauli spin matrices have off-diagonal components which permit the local collisional entanglement of these qubits. However, z-dependent inhomogeneities require a QLA with a minimum of 16 qubits/lattice site since the Pauli spin matrix σz is diagonal. For 2 dimensional inhomogeneities, one can readily couple the 8-8 qubit schemes for x-y variations. z-x and y-z variations can be treated by either a 16-8 qubit scheme or a 16-16 qubit representation.