Transport Phenomena In Kinetic Theory Of Gases

[Margit Szermer]

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transport phenomena in kinetic theory of gases

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Transport phenomena still stand as one of the most challenging problems in computational physics. By exploiting the analogies between Dirac and lattice Boltzmann equations, we develop a quantum simulator based on pseudospin-boson quantum systems, which is suitable for encoding fluid dynamics transport phenomena within a lattice kinetic formalism. It is shown that both the streaming and collision processes of lattice Boltzmann dynamics can be implemented with controlled quantum operations, using a heralded quantum protocol to encode non-unitary scattering processes. The proposed simulator is amenable to realization in controlled quantum platforms, such as ion-trap quantum computers or circuit quantum electrodynamics processors.

Transport phenomena in fluid flows play a crucial role for many applications in science and engineering. Indeed, a large variety of natural and industrial processes depend critically on the transport of mass, momentum and energy of chemical species by means of fluid flows across material media of assorted nature1. The numerical simulation of such transport phenomena still presents a major challenge to modern computational fluid dynamics. Among the reasons for this complexity stand out the presence of strong heterogeneities and huge scale separation in the basic mechanisms, namely advection, diffusion and chemical reactions2,3. In the last two decades, a novel concept for the solution of transport phenomena in fluid flows has emerged in the form of a minimal lattice Boltzmann (LB) kinetic equation. This approach is based on the statistical viewpoint typical of kinetic theory4,5. LB is currently used across a broad range of problems in fluid dynamics, from fully developed turbulence in complex geometries to micro and nanofluidics6,7, all the way down to lattice gas automata8 and quark-gluon applications9.

Recent improvements in ion trap and superconducting circuit experiments make these platforms ideal for challenging quantum information and simulation tasks. Trapped-ion experiments have demonstrated quantum information and simulation capabilities10,11,12, including the quantum simulation of highly correlated fermionic systems13, fermionic-bosonic models14,15 and lattice gauge theories16. Superconducting circuit setups can host nowadays top-end quantum information protocols, such as quantum teleportation17 and topological phase transitions18. These quantum devices are approaching the complexity required to simulate both classical and quantum nontrivial problems, as proposed by Feynman some decades ago19. Efforts in designing quantum algorithms for the implementation of fluid dynamics make use of quantum computer networks20,21. In these works, the quantum degrees of freedom are used on the same ground as classical parameters and the exponential gain of quantum computers is not properly exploited. In contrast, systems described by pseudospins coupled to bosonic modes, such as the aforementioned ion-trap and superconducting circuit platforms, can enjoy quantum superposition and have advantages with respect to pure-qubit quantum computers in simulating fluids.

In this article, we propose a quantum simulation of lattice Boltzmann dynamics, using coupled pseudospin-boson quantum platforms. Based on previously established analogies between Dirac and LB equations, we define here a full quantum mapping of transport equations in fluid flows. The LB dynamics is simulated sequentially by performing particle streaming and collision steps. The non-unitary collision process can be implemented with an heralded protocol, by sequential collapses of an ancillary qubit. The proposed mapping is amenable to realization in trapped-ion and superconducting circuit platforms.

Here, is the ith component of the particle fluid density associated with the lattice site at the time t and with discrete velocity . The macroscopic fluid density at the site is retrieved as , while the fluid velocity is defined as the weighted sum of the discrete velocities, . The velocity components , with , satisfy mass-momentum-energy conservation laws and rotational symmetry. Typical lattices are D2Q9 or D3Q15 models, for the case of two dimensions with 9 speeds and three dimensions with 15 speeds, respectively22.

Collisional properties are here expressed in scattering-relaxation form, making use of the local equilibrium distribution . The LB approach to compute the dynamics associated with Eq. (1) uses sequential computational steps. One initially performs a displacement (free-streaming) of each distribution component towards the nearest-neighbor lattice site pointed at by the discrete velocity . From there, the equilibrium distribution function is computed and the outcome of the collisional process is retrieved. Further iterations of these calculations allow the propagation of the lattice dynamics in time. We address the question of whether all these steps can be performed in a quantum simulator with practical quantum computing protocols.

Notice that the streaming matrices of the LB equation are diagonal, while the αij, which generate a Clifford algebra, cannot be simultaneously diagonalized. Additionally, the mass matrix βij is Hermitian, while standard collision matrices come in real symmetric form in the LB equation. Therefore, a complete codification of the LB scheme in quantum language requires the implementation of diagonal streaming matrices and of purely imaginary symmetric scattering matrices.

The components of the fluid density distribution function can be encoded in a set of quantum states defined on a proper Fock space. For example, in two dimensions, the distribution of the fluid density over the two coordinates can be described by a real quantum wavefunction that encodes the state of two bosonic modes, as depicted in Fig. 1. In the x-quadrature representation, it reads , where fi(x1, x2) is a real distribution and the eigenstate of the quadrature of the first (second) bosonic mode. Several quantum distributions can be used by entangling the bosonic state to a multi-level system, such as a set of pseudospins, therefore the state of the complete system is given by , with ηi being real-valued coefficients. In order to keep a real-valued representation of , to be identified with a fluid density distribution function, one has to act only with purely imaginary interaction matrices.

The quantum simulation of the Dirac equation was originally proposed25 and afterwards realized in a trapped-ion experiment26. In general, streaming interactions involving matrices in the Dirac or Majorana representation can be implemented by using a pair of pseudospins coupled to one or more bosonic modes. In terms of creation and annihilation operators for the bosonic mode in the b direction, one can then consider and write Eq. (2) on the pseudospin-bosonic Hilbert space of ,

Standard collision operators in LB theory are represented by real symmetric matrices associated with non-unitary evolution operators. On the other hand, typical controlled quantum mechanics experiments produce unitary dynamics. Nevertheless, one can probabilistically encode non-unitary dynamics in a quantum device with a heralded protocol, by performing controlled operations conditioned on the state of an ancillary qubit and then using the state of the latter as a flag for the success of the protocol. We consider a purely imaginary symmetric scattering matrix Ω, whose quantum evolution equation reads , providing a non-unitary evolution operator that describes lattice collisions .

By defining δM and δm as the maximal and minimal eigenvalues of the spectrum of C, the system of inequalities in Eq. (7) can be reduced to one of the two inequalities or , respectively when or . If longer evolution times t are considered, the spectral range of C changes accordingly. The weighted γ-sum derived here can be implemented with quantum computing algorithms, using ancillary qubits and controlled Uα and Uβ gates27. By measuring the ancilla state, one can determine whether the desired operation has been performed or not. The success of the protocol depends on the weighted sum of unitary operators, with a failure probability .

The scheme proposed can be adapted to a variety of transport fluid problems. As an example, we consider the implementation of an advection-diffusion process in two spatial dimensions. The dynamics of the transported species, e.g. pollutants or bacteria, is described by the equation

Natural quantum platforms for prospective implementation of the proposed scheme could be ions trapped in linear Paul traps or superconducting circuit setups, in which the sequential streaming and collision steps in Eq. (4) can be realized. The pseudospin-bosonic state can be encoded, in the case of ion traps, in the internal level and motion modes of the ions28, while in a superconducting architectures, one can use the first levels of charge-like qubits, e.g. transmon qubits and microwave resonators29. One may consider opening similar avenues in other quantum technologies as is the case of quantum photonics30 and Bose-Einstein condensates31.

A practical implementation of the protocol proposed can make use of many-body interactions, involving couplings with bosonic modes. These type of gates have been considered in superconducting architectures32 or in ion-trap platforms33. For a four-speed lattice, the diagonal streaming processes can be realized with a combination of a qubit-boson interaction and two entangling gates among the qubits. For example, the corresponding evolution operator for the streaming in the X direction can be written as

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