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Alternative quantum circuit implementation for 2D electromagnetic wave simulation with quasi-PEC modeling

Nicolas Bui, Alain Reineix, Christophe Guiffaut

Published Jul 6, 2022• 0 citations
OpenAlex

Abstract

Numerical simulation may benefits from quantum computing (QC) potential to solve particular problems faster than its classical counterpart. This motivates recent works [1, 2, 3] to solve the wave equation with a quantum computer. Here, we consider 2D wave propagation in free space based on an alternative implementation of the quantum algorithm developed in [2, 3]. We also propose a way to model quasi perfect electrical conductor (PEC). An example of a plane wave reflection is given, with QC on a simulator.

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Numerical simulation may benefits from quantum computing (QC) potential to solve particular problems faster than its classical counterpart. This motivates recent works [1, 2, 3] to solve the wave equation with a quantum computer. Here, we consider 2D wave propagation in free space based on an alternative implementation of the quantum algorithm developed in [2, 3]. We also propose a way to model quasi perfect electrical conductor (PEC). An example of a plane wave reflection is given, with QC on a simulator.
SUMMARY Quantum computing has attracted considerable attention in recent years because it promises speedups that conventional supercomputers cannot offer, at least for some applications. Though existing quantum computers (QCs) are, in most cases, still too small to solve significant problems, their future impact on domain sciences is already being explored now. Within this context, we present a quantum computing concept for 1-D elastic wave propagation in heterogeneous media with two components: a theoretical formulation and an implementation on a real QC. The method rests on a finite-difference approximation, followed by a sparsity-preserving transformation of the discrete elastic wave equation to a Schrödinger equation, which can be simulated directly on a gate-based QC. An implementation on an error-free quantum simulator verifies our approach and forms the basis of numerical experiments with small problems on the real QC IBM Brisbane. The latter produce simulation results that qualitatively agree with the error-free version but are contaminated by quantum decoherence and noise effects. Complementing the discrete transformation to the Schrödinger equation by a continuous version allows the replacement of finite differences by other spatial discretization schemes, such as the spectral-element method. Anticipating the emergence of error-corrected quantum chips, we analyse the computational complexity of the best quantum simulation algorithms for future QCs. This analysis suggests that our quantum computing approach may lead to wavefield simulations that run exponentially faster than simulations on classical computers.
Quantum computing, or quantum computation, has garnered increasing attention within the computational community in recent years, owing to its remarkable efficacy in addressing large-scale problems. In specific scenarios, quantum algorithms exhibit exponential enhancements over their classical counterparts executed on classical computing platforms. Quantum computing introduces myriad opportunities for simulating intricate physical systems, including but not limited to electromagnetic (EM) structures. Despite the extensive research devoted to quantum computing in contemporary literature, the exploration of quantum simulation techniques for EM structures remains relatively limited. In this paper, we will describe a quantum computing method for solving EM problems through finite element equations. The performance of this method will be verified by an EM example.
Numerical models are essential for investigating and understanding the physics and sound propagation and scattering, given the ocean’s complexity and the limitations of real-world field experiments. High-performance computing has become a primary tool for researchers to simulate long-range sound propagation. Quantum computing is an emerging technique that offers potential speedups over conventional binary computers. However, classical algorithms cannot be directly implemented in quantum systems due to fundamental differences in computing principles. This necessitates the development of quantum ocean acoustic models that remain interpretable within the same context as classical algorithms. This paper presents a quantum formulation for a finite-difference time-domain (FDTD) solver and a parabolic equation (PE) method. The results demonstrate that these quantum algorithms implemented on a simulator reproduce results identical to their classical counterparts, providing a foundation for further developments in quantum-centric ocean acoustics.
Quantum computers are ideally set up to solve linear systems which are of a form similar to the Schrodinger/Dirac equation of quantum mechanics. In the framework of linear response theory, the propagation and scattering of electromagnetic waves in a dielectric medium are described by Maxwell equations. The qubit lattice algorithm consists of a series of alternating unitary streaming and entanglement operators acting on qubit amplitudes constructed from the electric and magnetic fields. It is not a direct discretization of Maxwell equations, but recovers the desired equations to second order in lattice grid spacing. The resulting algorithm is implemented on a present-day supercomputer and is the basis of studying scattering of electromagnetic waves by an elliptical dielectric. As opposed to the steady state description of Mie scattering in frequency domain, the temporal evolution provides insights into transient scattering. The QLA simulations, reveal that a spatially localized wave packet propagating past an elliptic dielectric, embedded in vacuum, leads to several reflections generated by wave fields trapped within the dielectric. The physics insight brought forth by these simulations is not apparent from frequency domain studies of scattering. A complimentary simulation on transient scattering of a wave packet by an elliptical vacuum bubble inserted in a uniform dielectric demonstrates a stark contrast with respect to scattering off an elliptical dielectric in vacuum. Essentially, there is only a single internal reflection in which the field amplitudes are significantly smaller than those for side and forward scattering. A simple model based on the Kirchhoff tangent plane approximation helps explain the differences between these two scattering examples.
Quantum algorithms offer new avenues for solving partial differential equations (PDEs). While the potential for end-to-end quantum advantage is at present not well understood, recent literature presents explicit circuit constructions for solving certain classes of linear PDEs in the frequency domain and thus offers concrete examples to study. In this work, we develop end-to-end implementations of these quantum circuits compiled to machine-level instructions and benchmark them in both numerical simulations and IBMQ hardware experiments. We focus on the advection, wave, and Poisson equations and study quantum circuits that propagate the dynamics in frequency space via the quantum Fourier transform using approximate methods based on a first-order approximation which offer compact representations with uncontrollable approximation error, and polynomial approximation methods based on quantum signal processing (QSP) leading to deeper circuits with tunable algorithmic error. In addition, we experimentally demonstrate that the QSP-augmented algorithm can provide accurate solutions under realistic hardware constraints. Finally, we extend our method to address non-homogeneous Dirichlet boundary conditions and verify it numerically for a Poisson equation with source term obtained from high-fidelity physics simulations of a capacitively coupled plasma.
Electromagnetic waves are an inherent part of all plasmas—laboratory fusion plasmas or astrophysical plasmas. The conventional methods for studying properties of electromagnetic waves rely on discretization of Maxwell equations suitable for implementing on classical, present day, computers. The traditional methodology is not efficient for quantum computing implementation—a future computational source offering a tantalizing possibility of enormous speed up and a significant reduction in computational cost. This paper addresses two topics relevant to implementing Maxwell equations on a quantum computer. The first is on formulating a quantum Schrödinger representation of Maxwell equations for wave propagation in a cold, inhomogeneous, and magnetized plasma. This representation admits unitary, energy preserving, evolution and conveniently lends itself to appropriate discretization for a quantum computer. Riding on the coattails of these results, the second topic is on developing a sequence of unitary operators which form the basis for a qubit lattice algorithm (QLA). The QLA, suitable for quantum computers, can be implemented and tested on existing classical computers for accuracy as well as scaling of computational time with the number of available processors. In order to illustrate the QLA for Maxwell equations, results are presented from a time evolving, full wave simulation of propagation and scattering of an electromagnetic wave packet by non-dispersive dielectric medium localized in space.