as well as the identical expression of [|a⟩⟨a|] s and [|b⟩⟨b|] s in Eq. 37. We further use the focused initial condition that replaces the sampling of the mapping variables in the dτ integral of Eq. 36 with specific values of the mapping variables, such that 1 2 (q 2 a +p 2 a -Γ) = 1 for initially occupied state |a⟩ and 1 2 (q 2 b + p 2 b -Γ) = 0 for the initially unoccupied states |b⟩. The angle variables {θ b } (Eq. 26) are randomly sampled 67 in the range of [0, 2π). More computational details for the γ-SQC and spin-LSC are provided in section III B.