Algorithm 2 DFSP diffusion step with splitting

INPUT:nk, τD, TransitionMatrixn, StateListn, ErrorTolerancen

OUTPUT: Output State

1: initialize once: PDVLookupTable, nmax←∞, L←0

2: ifnk>=nmaxthen

3:  return Output State←DFSP_Diffusion(⌊nk∕2⌋,L+1) +DFSP_Diffusion(⌈nk∕2⌉,L+1)

4: else

5:  ifPDVLookupTable contains nkthen

6:   PDV←PDVLookupTable[nk]

7: else

8:   PDV←exp(TransitionMatrixnk×D∕ℓ2×τD)×[1,0,0,⋯,0]T

9:   ifPDV[end]>ErrorTolerance∕2Lthen

10:    nmax←nk

11:    return Output State←DFSP_Diffusion(⌊nk∕2⌋) +DFSP_Diffusion(⌈nk∕2⌉)

12:   end if

13:   PDVLookupTable[nk]←PDV

14:  end if

15:  Generate a random number X∊U(0,1)

16:  Find the smallest integer μ such that ${\sum }_{j=1}^{\mu }\mathtt{PDV}\left[j\right]>X$

17:  return Output State←StateListnk[μ]

18: end if