Quantum algorithm for the collisionless Boltzmann equation

A novel quantum algorithm implementing a discrete-velocity method for the collisionless Boltzmann equation is introduced. The algorithm is designed for application on a quantum computer with a number of quantum bits feasible in the near future (e.g.40 - 50 ). Following the quantum-circuit model of quantum computation, the present works shows the quantum-circuit implementations for the convection or transport part of the kinetic model, inspired by work on quantum algorithms for the Dirac equation. The present work represents the advection step as a quantum walk process, implemented as a series of multiple-input controlled-NOT gates. A detailed discussion on the background to this new method is presented, including how a rarefied-flow problem can be encoded as the quantum state of a qubit register in a quantum computer. A complexity analysis is presented showing potential benefits of the proposed algorithm. Based on the concept of quantum parallelism, the extension to multiple species is demonstrated to not increase the number of required gate operations. A key aspect of the developed algorithm is the implementation of boundary conditions. This work describes how the specular-reflection boundary conditions can be effectively imposed with a quantum circuit implementation. The developed method is then applied to the supersonic flow around a blunt body as well as the free-molecular flow escaping from a rectangular container. As validation, the flow along the stagnation streamline of the blunt-body flow is compared with exact solutions for a piston-driven flow, showing excellent agreement. Finally, directions for future work are discussed in this work