text: 2/3 (S1  S2 + S3  S4 + S1  S3 
  p 
  · 
  + S2  · 
  S4 + S1  · 
  S4 + S2 
  
  This exact construction contains high order physical 
  spin interactions, which is undesirable for practical im- 
  plementation. We described two possible approaches to 
  reduce this problem: generating the high order spin in- 
  teractions by perturbative expansion of the coupling to 
  optical phonon, or the magnetic coupling between clus- 
  ters. This perturbative construction will introduce trun- 
  cation error of perturbation series, which may be con- 
  trolled by small expansion parameters. Whether these 
  constructions can be experimentally engineered is how- 
  ever beyond the scope of this study. It is conceivable that 
  other perturbative expansion can also generate these high 
  order spin interactions, but this possibility will be left for 
  future works. 
  
  S3). 
  ·  ·  · 
  
  The functions f E 
  but are reproduced here, 
  1,2 for the E modes have been given before 
  
  f T2 
  1 = (S2 
  f T2 
  2 = (S1 
  f T2 
  3 = (S1 
  
  ·  −  · 
  S3  S2  S4), 
  ·  −  · 
  S2  S3  S4)  Acknowledgments 