Carleman-lattice-Boltzmann quantum circuit with matrix access oracles

TL;DR

This paper introduces a quantum algorithm for simulating fluid flows using Carleman linearization of the Lattice Boltzmann method, achieving reduced gate complexity but facing challenges with success probability in multi-step evolution.

Contribution

It develops a quantum circuit leveraging matrix oracles for the Carleman-lattice-Boltzmann method, significantly reducing complexity and exploring its accuracy for fluid dynamics simulation.

Findings

Achieves relative error of about 10^{-3} with two Carleman iterates.

Reduces gate complexity from exponential to quadratic.

Faces low success probability due to high ancilla qubit requirements.

Abstract

We apply Carleman linearization of the Lattice Boltzmann (CLB) representation of fluid flows to quantum emulate the dynamics of a 2D Kolmogorov-like flow. We assess the accuracy of the result and find a relative error of the order of $10^{-3}$ with just two Carleman iterates, for a range of the Reynolds number up to a few hundreds. We first define a gate-based quantum circuit for the implementation of the CLB method and then exploit the sparse nature of the CLB matrix to build a quantum circuit based on block-encoding techniques which makes use of matrix oracles. It is shown that the gate complexity of the algorithm is thereby dramatically reduced, from exponential to quadratic. However, due to the need of employing up to seven ancilla qubits, the probability of success of the corresponding circuit for a single time step is too low to enable multi-step time evolution. Several possible directions to circumvent this problem are briefly outlined.