| 1. Inputs: the decomposition (19) of H, parameterized quantum circuits U({\boldsymbol{\theta }}) with initial parameters {\boldsymbol{\theta }}, and error tolerance ε. |
| 2. For any 1\leqslant i\leqslant n, applying the Hadamard test in figure 4 to compute the value: |
| {f}_{i}({\boldsymbol{\theta }})=\displaystyle \frac{1}{2}\left(\mathrm{tr}({U}_{i}U({\boldsymbol{\theta }}))+\mathrm{tr}({U}^{\dagger }({\boldsymbol{\theta }}){U}_{i}^{\dagger })\right). (20) |
| 3. Compute the value of loss function: |
| {l}_{3}({\boldsymbol{\theta }})=\sum _{i=1}^{n}{k}_{i}{f}_{i}({\boldsymbol{\theta }}). (21) |
| 4. Perform optimization process to maximize {l}_{3}({\boldsymbol{\theta }}), update {\boldsymbol{\theta }}. |
| 5. Repeat steps 2–4, until the loss function satisfies | {\rm{\Delta }}{l}_{3}({\boldsymbol{\theta }})| \leqslant \epsilon . |