The presence of waves in laboratory and space plasma is due to collective processes which are inherently classical in nature. The linear propagation of electromagnetic waves follows from Maxwell equations in which the constitutive relations are obtained using either a kinetic or a fluid description of the plasma. Even though these descriptions require a large number of charged particles inside a Debye sphere, the interparticle distance is long compared to the de Broglie wavelength of any electron or ion in the plasma. As such, quantum effects are negligible and classical treatments have been successful in providing insight into plasma behavior in fusion, terrestrial, and space plasma. However, this does not preclude the exploitation of quantum computers for simulating classical plasma. It requires us to formulate appropriate theoretical models that appear to be “quantum-like” while maintaining the integrity of the classical description of electromagnetic wave propagation in plasma.
The presence of waves in laboratory and space plasma is due to collective processes which are inherently classical in nature. The linear propagation of electromagnetic waves follows from Maxwell equations in which the constitutive relations are obtained using either a kinetic or a fluid description of the plasma. Even though these descriptions require a large number of charged particles inside a Debye sphere, the interparticle distance is long compared to the de Broglie wavelength of any electron or ion in the plasma. As such, quantum effects are negligible and classical treatments have been successful in providing insight into plasma behavior in fusion, terrestrial, and space plasma. However, this does not preclude the exploitation of quantum computers for simulating classical plasma. It requires us to formulate appropriate theoretical models that appear to be “quantum-like” while maintaining the integrity of the classical description of electromagnetic wave propagation in plasma.
Quantum computers, through quantum entanglement and parallelization, offer an intriguing prospect of exponential speedup relative to classical computers for performing numerical simulations. Our interest is in the application of quantum information science to plasma physics in order to develop algorithms which can implemented on quantum computers. Concurrently, we want to test these algorithms on present conventional (classical) computers as large scale, error-correcting quantum computers with long coherence times are not yet forthcoming. Even though the constituents of laboratory and space plasmas are electrons and ions, the physics of these plasmas is dominated by collective processes. Quantum effects that are prominent at sub-atomic scales are averaged out in the statistical description of a plasma. Consequently, we need to render classical plasma physics in the language of quantum mechanics in preparation for quantum computers.
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.
The prospect that quantum computers could be exponentially faster than conventional computers has motivated our research on the application of quantum information science to plasma physics. Quantum computers leverage quantum parallelization, entanglement, and superposition to speed up computations. However, our usual approach to computations on topics in classical plasma physics is not directly suited to take advantage of quantum computers. In this presentation, we will put forth some ideas on formulating algorithms for electromagnetic wave propagation in plasmas which could be implemented on quantum computers.
Can quantum computers effectively simulate the propagation and scattering of electromagnetic waves in a classical plasma? This chapter introduces some of the basic concepts in mathematics and physics essential to answering that question. The numerical simulations of Maxwell equations for wave propagation in dielectrics are constrained by technological limitations of the present-day computers. In contrast, there has been ample fanfare around quantum computers and their potential to far exceed the performance of traditional computers. Whether the enhanced capabilities of a quantum computer can be put to use for simulating topics in classical physics is a source of intrigue and curiosity.
Quantum computing is gaining increased attention as a potential way to speed up simulations of physical systems, and it is also of interest to apply it to simulations of classical plasmas. However, quantum information science is traditionally aimed at modeling linear Hamiltonian systems of a particular form that is found in quantum mechanics, so extending the existing results to plasma applications remains a challenge. Here, we report a preliminary exploration of the long-term opportunities and likely obstacles in this area. First, we show that many plasma-wave problems are naturally representable in a quantumlike form and thus are naturally fit for quantum computers. Second, we consider more general plasma problems that include non-Hermitian dynamics (instabilities, irreversible dissipation) and nonlinearities. We show that by extending the configuration space, such systems can also be represented in a quantumlike form and thus can be simulated with quantum computers too, albeit that requires more computational resources compared to the first case. Third, we outline potential applications of hybrid quantum–classical computers, which include analysis of global eigenmodes and also an alternative approach to nonlinear simulations.
At the conceptual level, the quantization procedure replaces the pure particle description of classical mechanics by a particle-wave treatment satisfying the requirements of the fundamental wave–particle dualism. Technically, one can always start from a generalized wave equation and use it as a basis to discuss wave-front propagation in the form of rays following classical particle trajectories, see Sections 2.2.1 and 3.1. Consequently, any wave theory inherently contains that aspect of the wave–particle dualism. Especially, Maxwell's equations of electrodynamic theory include both wave and particle aspects of electromagnetic fields. Hence, it could be that there are no additional “quantum,” i.e., wave–particle dualistic aspects. However, as we discuss in this chapter, Maxwell's equations can be expressed in the framework of Hamilton's formulation of particle mechanics. This then suggests that in addition to the approximate ray-like behavior, electromagnetic fields do have a supplementary level of particle-like properties. Due to the fundamental axiom of wave–particle dualism, the additional particle-like aspects of electromagnetic fields must then also have a wave counterpart that can be axiomatically introduced via the canonical quantization scheme described in Section 3.2.3. Since the spatial dependency in Maxwell's equation contains wave–particle dualism already at the classical level, it is clear that the new level of quantization cannot be a simple real-space feature but has to involve some other space describing the structure of the electromagnetic fields. In fact, this quantization yields several very concrete implications such as quantization of the light energy, elementary fluctuations in the light-field amplitude and intensity, and more.