论文标题
运输方程的有效和故障安全量子算法
Efficient and fail-safe quantum algorithm for the transport equation
论文作者
论文摘要
我们提出了一种可扩展的算法，用于在两个和三个空间尺寸中求解传输方程，以在耐断层通用量子计算机上的可变网格大小和离散速度。作为我们的量子传输方法（QTM）概念的证明，我们描述了Qiskit中的全电路起点到端实现，并为2D流提供了数值结果。我们的QTM基于一种新型的流媒体方法，该方法与最先进的量子流方法相比，导致减少了CNOT门的数量。作为第二个亮点，我们提出了一种新颖的对象编码方法，该方法降低了编码壁所需的CNOT门的复杂性，该墙壁现在变得独立于壁的大小。最后，我们提出了一个新颖的量子编码，对粒子的离散速度进行了编码，该速度可以在反映粒子速度的成本上进行线性加速，现在它变得独立于编码的速度量。我们的主要贡献是对传输方程反射步骤的量子算法的故障安全实现的详细说明，该算法可以很容易地在物理量子计算机上实现。这种故障安全的实现允许各种初始条件和粒子速度，并导致障碍物的墙壁，边缘和角落在物理上正确的行为。结合这些结果，我们提出了一种用于传输方程式的新颖和故障安全的启动量子算法，可用于多种流量配置。我们最终表明，我们的方法是编码网格所需的量子数量的二次，以及在单个空间维度中编码离散速度所需的量子数量，这使我们的方法优于文献中已知的最新方法。
We present a scalable algorithm for solving the transport equation in two and three spatial dimensions for variable grid sizes and discrete velocities on a fault-tolerant universal quantum computer. As a proof of concept of our quantum transport method (QTM), we describe a full-circuit start-to-end implementation in Qiskit and present numerical results for 2D flows. Our QTM is based on a novel streaming approach which leads to a reduction in the amount of CNOT gates required in comparison to state-of-the-art quantum streaming methods. As a second highlight we present a novel object encoding method, that reduces the complexity of the amount of CNOT gates required to encode walls, which now becomes independent of the size of the wall. Finally we present a novel quantum encoding of the particles' discrete velocities that enables a linear speed-up in the costs of reflecting the velocity of a particle, which now becomes independent of the amount of velocities encoded. Our main contribution is a detailed description of a fail-safe implementation of a quantum algorithm for the reflection step of the transport equation that can be readily implemented on a physical quantum computer. This fail-safe implementation allows for a variety of initial conditions and particle velocities and leads to physically correct behavior around the walls, edges and corners of obstacles. Combining these results we present a novel and fail-safe start-to-end quantum algorithm for the transport equation that can be used for a multitude of flow configurations. We finally show that our approach is quadratic in the amount of qubits necessary to encode the grid and the amount of qubits necessary to encode the discrete velocities in a single spatial dimension, which makes our approach superior to state-of-the-art approaches known in the literature.
