Encoding Scheme for the Treatment of Non-linear Problems via the Lattice Boltzmann Method

Specifically, it is resorted to the classical representation of values as bitstrings, in which non-linear functions can be approximated quite easily via a mapping of input bitstrings to output bitstrings (cf. outline of the proposed algorithm concept in our preliminary work [5]). The quantum algorithm devised for this encoding scheme processes a superposition of bitstrings and can be regarded as an extension of the algorithm proposed for the lattice gas automaton by M. Schalkers and M. Möller [2] to the full LBM.

Correspondingly, the qubits are prepared in terms of a 'bottom level register' and a 'top level register': The bottom level register uses bitstring encoding to store the information needed for the computation of a specific number of time steps for at least one grid point. The quantum circuit that implements the mapping of the bitstrings according to a general equilibrium distribution function for realizing also non-linear collision operators hence involves only the qubits of this register. The top level register is then prepared in a superposition of basis states, where the bitstring of a basis state labels a grid point. The quantum state maintains this encoding format in the processing, so that the result of one run of this quantum algorithm would be one part of the resulting linear combination, which corresponds to the information about the flow quantities at one grid point after the set number of time steps.

Concerning the quantum circuits for the involved steps in the proposed algorithm concept, a minimal example for the approximation of a non-linear function via bitstring encoding, which was tested with IBM's QC simulation framework Qiskit, is presented in [5] and an implementation of the streaming step for periodic boundary conditions via SWAP-gates is discussed in [2].

The contribution will of course also address the crucial aspect of how efficiently the state vector of the required structure can be prepared and it is intended to demonstrate the functionality of the circuit implementations of the individual required procedures for the algorithm in the contribution via their simulation for minimal situations.

Since the circuit for the collision step of the LBM acts only on the bottom level register, the collision step is performed for this encoding scheme in parallel for all grid points, where the circuit depth of this step is independent of the number of grid points, which is an advantage w.r.t. to previous quantum algorithms for the LBM that are based on amplitude encoding. At least for the simple case of periodic boundary conditions, this applies also to the streaming step.

Since, in contrast to amplitude encoding, one measurement of the qubits yields already the information about the flow quantities at one grid point in this encoding format, it is considered to be promising specifically for computational aeroacoustics (CAA), for which a major objective is the recording of sound. To be able to record sound, the spatial resolution requirement is accompanied by the need to resolve the temporal dynamics of the fluid system as well, which means that the flow quantities have to be in fact read out permanently from the computation system after a few time steps.

For current QC hardware, the ratios of the coherence time to the execution times of native operations render it very challenging to run such non-variational quantum algorithms for industrial problem sizes on real hardware. In the contribution, an elaborated complexity analysis will be given, discussing the requirements for industrial scales.