Quantum Simulator for Transport Phenomena in Fluid Flows

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.


Quantum Simulator for Transport Phenomena in Fluid Flows
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 nature 1 . 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 reactions 2,3 . 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 theory 4,5 . LB is currently used across a broad range of problems in fluid dynamics, from fully developed turbulence in complex geometries to micro and nanofluidics 6,7 , all the way down to lattice gas automata 8 and quark-gluon applications 9 .
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 capabilities [10][11][12] , including the quantum simulation of highly correlated fermionic systems 13 , fermionic-bosonic models 14,15 and lattice gauge theories 16 . Superconducting circuit setups can host nowadays top-end quantum information protocols, such as quantum teleportation 17 and topological phase transitions 18 . These quantum devices are approaching the complexity required to simulate both classical and quantum nontrivial problems, as proposed by Feynman some decades ago 19 . Efforts in designing quantum algorithms for the implementation of fluid dynamics make use of quantum computer networks 20,21 . 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 i i , with = , , ...
, satisfy mass-momentumenergy 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, respectively 22 .
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 ( ) 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.
The formal analogy between the Dirac and LB equations was first highlighted in 4,23 , where the velocity distribution of the particle is treated in a similar fashion as a relativistic spinor. This analogy is further exploited in the Majorana representation of the Dirac equation, by using real spinors 24 . The Dirac (Majorana) equation reads (ħ = 1 here and in the following) where we have defined the Dirac (Majorana) streaming matrices α ij b , mass term β ij , and the imaginary prefactor i proper of quantum mechanical evolution.
Notice that the streaming matrices of the LB equation are diagonal, while the α ij , which generate a Clifford algebra, cannot be simultaneously diagonalized. Additionally, the mass matrix β ij 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 ( , )  f x t i can be encoded in a set of quantum states Ψ i 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 is a real distribution and ( ) x 1 2 the eigenstate of the quadrature of the first (second) bosonic mode. Several quantum distributions Ψ i 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 In order to keep a real-valued representation of Ψ , 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 proposed 25 and afterwards realized in a trapped-ion experiment 26 . In general, streaming interactions involving matrices in the Dirac or Majorana representation α ∇ where K b stands for the pseudospin-boson coupling and α b act upon the pseudospin degrees of freedom. Thus, the three streaming matrices α ij b are written in the Dirac representation as α σ σ = − ⊗  . In this way, a purely imaginary streaming step iβ∇ b can be built, which mimics the diagonal streaming of the LB equation. The total wavefunction after the streaming steps can be retrieved with a sequential implementation, following the operator splitting method 23 . For example, in a 2-dimensional lattice, one has The last collision step C, which scrambles particle distributions in different directions, is discussed below.
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 ∂ Ψ = Ω Ψ i t i ij j , providing a non-unitary evolution operator that describes lattice collisions = (− ΩΔ ) C i t exp . The collision operator can be decomposed in a weighted sum of two commuting unitary operators, , assuming without loss of generality that γ > 0. Given a specific diagonalizable collision operator C and weight γ, one can then find its decomposition in terms of unitaries. In order to find a decomposition in terms of unitaries, C must first be diagonalized as = † C VDV . This reduces the problem of finding U α and U β down to an eigenvalue equation, δ i = α i + γβ i , with δ i , α i and β i being the ith eigenvalues of the collision and unitary operators respectively. Notice that, due to the properties of the scattering matrix, δ ∈ + R i . Taking into account the normalization conditions, one has the system of equations By defining δ M and δ m 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 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 U α and U β gates 27 . 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 . As P f is an increasing function of γ, choosing γ δ δ = − + , − + min{ 1 1 } m M 0 maximizes the probability of success. This directly connects the simulation time of the scattering process C with the best choice for γ. To propagate the dynamics of a given collision process C, one can split the step time Δ t into N time intervals Δ t/N and perform the heralded protocol at each step, such that At each step, one has a collision operator , with an optimal γ 0 . In this way, as the step size gets smaller, the success probabilities for each step increase, while the total success probability accumulates single success rates from the individual steps. In Fig. 2a, we plot the success probability ( ) = − ( ) P N P N 1 s f of the simulation of the single step, as a function of N, for random symmetric purely imaginary matrices. As expected, the success probability per step increases as the size for the single time step gets smaller. The success of the whole protocol P s N is constant and does not depend on N. In Fig. 2b is shown that the optimal protocol is performed at γ = γ 0 .

Discussion
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 is the scalar field transported by a fluid with space-dependent velocity . Here, velocities are numbered 1 ÷ 4 counterclockwise starting from the + x direction.
The latter defines the quantum scattering matrix as composed of three contributions, namely is proportional to the position quadrature of the bosonic mode associated with the y direction. The three contributions to the scattering matrix represent classical linear wave propagation and damping, mass conservation and macroscopic advection, respectively. They can be implemented with the quantum simulation protocol previously introduced. The bounds to γ can be obtained, e.g., for the first contribution to the scattering matrix − A ij , by computing the spectrum of = − Δ C e A t for different time steps Δ t, for D = 0.05, in units of 1/ω 4 . The result is shown in Fig. 3.
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  7) and shadowed in the picture. traps, in the internal level and motion modes of the ions 28 , while in a superconducting architectures, one can use the first levels of charge-like qubits, e.g. transmon qubits, and microwave resonators 29 . One may consider opening similar avenues in other quantum technologies as is the case of quantum photonics 30 and Bose-Einstein condensates 31 .
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 architectures 32 or in ion-trap platforms 33 . 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