Figure 4

Quantum circuit for simulating a Trotter step of the advection–diffusion equation in a two-dimensional laminar shear flow using \(n=3\) qubits per spatial dimension, where the top three qubits correspond to the x direction and the next three qubits correspond to the y direction. Ancilla qubit measurements of \(\left| 0\right\rangle\) are required for successful execution. The boundaries are periodic in x and homogeneous Neumann in y, leading to \(\beta _1=Dt(2\pi /L)^2\) and \(\beta _2=Dt(\pi /L)^2\). Example advection block implementations are shown in Fig. 2. While the QCT is a unitary transformation, existing efficient circuits require one additional ancilla.