Quantum Unitary Matrix Representation of Lattice Boltzmann Model for Low Reynolds Fluid Flow Simulation

TL;DR

This paper introduces a quantum algorithm for simulating low Reynolds number fluid flows using a matrix representation of the Lattice Boltzmann method, employing unitary decompositions and quantum gates.

Contribution

It presents a novel quantum algorithm that encodes Lattice Boltzmann models as unitary matrices for efficient quantum simulation of fluid dynamics.

Findings

Tested on linear flow problems with up to 216 grid points

Gate counts closely match theoretical limits

High two-qubit gate count (~10^7) highlights circuit complexity

Abstract

We propose a quantum algorithm for the Lattice Boltzmann (LB) method to simulate fluid flows in the low Reynolds number regime. First, we encode the particle distribution functions (PDFs) as probability amplitudes of the quantum state and demonstrate the need to control the state of the ancilla qubit during the initial state preparation. Second, we express the LB algorithm as a matrix-vector product by neglecting the quadratic non-linearity in the equilibrium distribution function, wherein the vector represents the PDFs, and the matrix represents the collision and streaming operators. Third, we employ classical singular value decomposition (SVD) to decompose the non-unitary collision and streaming operators into a product of unitary matrices. Finally, we show the importance of having a Hadamard gate between the collision and the streaming operations. Our approach has been tested on linear/linearized flow problems such as the advection-diffusion of a Gaussian hill, Poiseuille flow, Couette flow, and lid-driven cavity problems. We provide counts for two-qubit controlled-NOT (CNOT) and single-qubit U gates for test cases involving 9 to 12 qubits, with grid sizes ranging from 24 to 216 points. While the gate count aligns closely with theoretical limits, the high number of two-qubit gates on the order of $10^7$ necessitates careful attention to circuit synthesis.