| （英） |
We conducted numerical experiments on the linear combination of Hamiltonian simulation (LCHS) method, which represents linear non-unitary time evolution as a linear combination of Hamiltonian simulations, to obtain practical design guidelines.
First, for homogeneous problems, we measured the LCHS approximation error, integral discretization error, and QSP polynomial approximation error using vector norms, and searched for an error allocation that minimizes the QSP degree d while satisfying the target accuracy. Since the query complexity of LCHS is governed by the QSP degree required to implement its constituent unitaries, reducing d directly lowers the query cost. The numerical results showed that accounting for empirical errors yielded a modest reduction in d. However, substantial reductions were difficult to achieve because the approximation error of approximate LCHS dominated the total error and showed little reduction when evaluated empirically.
Second, for inhomogeneous problems with a constant source term, we compared Lap-LCHS, based on the variation-of-constants formula, with an approach that applies LCHS after converting the inhomogeneous system into an augmented homogeneous system. The numerical results showed that the time T_amp^* obtained by comparing the linear-combination-of-unitaries (LCU) normalization factors, closely matched the crossover time at which the cost ratio ρ(T)=Q_aug/Q_Lap equals one. This agreement demonstrated the usefulness of T_amp^* as a practical criterion for selecting between the two methods. |