Figure 3

Quantum circuit for solving the one-dimensional heat equation with spectral accuracy on \(N=16\) grid points with \(O(\log ^2N)\) gates. Periodic boundary conditions are used. The circuit implements the products in Eqs. (16) and (18). The unitary U is defined in Eq. (19), and the ancilla is measured to be \(\left| 0\right\rangle\) after each application. The outer CNOT gates flip the second half of the spectrum to exploit the mirroring of the squared wavenumbers, as discussed in the text.