Surface code simulation

Current report

• report.ps

Progress log

May 2007:
• practiced coding simple circuits in qubit formalism.
• started writing surface code in stabilizer formalism.
• implemented and debugged Pauli operation subroutine [applyPauli].
• implemented and debugged Pauli measurement subroutine [measurePauli].
• implemented and debugged Bell-pair encoding subroutine [encode].
• implemented and debugged code-splitting subroutine [splitcode].
June 2007:
• understood surface code recovery strategy.
• implemented and debugged error generating subroutine [evolve]:
  • independent, probabilistic phase-flip errors with probability p (added other channels in August).
• studied Edmonds' algorithm for maximum matching without weights.
• studied Horodecki's proof of positivity of partial transpose as necessary and sufficient criterion for separability in 2x2 or 2x3 systems.
• created M.Sc. report outline.
July 2007:
• implemented and debugged perfect syndrome measurement subroutine [measuresyndrome]:
  • collected only primal syndrome (processed dual syndrome in August).
• incorporated and checked Cook's software for minimum-weight perfect matching.
• achieved major (10x) speedup in [applyPauli] and [measurePauli]:
  • modified stabilizer data structure (5x);
  • made use of operator type information in [symprod] (2x);
  • affecting [splitcode] and [evolve].
August 2007:
• designed final logical state measurement for Pauli channels.
• understood Edmonds' algorithm for maximum matching without weights.
• understood code splitting measurements and dependence of measurement outcomes.
• rewrote (encode + splitcode) =: [encode] subroutine:
  • redefined logicalZZ in [logicalop];
  • achieved speedup.
• achieved additional (2x) speedup in [applyPauli] and [measurePauli]:
  • used information about the type of each generator (2x);
  • rearranged the order of generators so that all X-type generators and logicalXX are grouped together before the Z-type generators and logicalZZ.
• added other Pauli channels in [evolve]:
  • justX - independent, probabilistic bit-flip errors only with probability p;
  • indXZ - independent, probabilistic X and Z errors with probability p each;
  • depol - depolarizing channel with probability p/3 for each type of errors.
• processed dual syndrome in subroutine [measuresyndrome].
• implemented and debugged primal recovery subroutine in case of perfect measurements [recover]:
  • added and debugged 2d [shortestpath], [weight], [obstacle] and [whichside].
• implemented and debugged Monte Carlo loops.
• implemented and debugged logical Bell state distinction function [distinctBell]:
  • by looking at the phases of the final stabilizer elements logicalXX and logicalZZ;
  • compared with Bell state obtained from (recorded) generated errors and recovery operations (for debugging).
• implemented and debugged logical '2-qubit Pauli state' distinction function [distinctPauli]:
  • obtained from (recorded) generated errors and recovery operations (for debuggin).
• implemented and debugged imperfect syndrome measurements in [measuresyndrome]:
  • measurement errors are independent, probabilistic classical bit flips with probability q.
• implemented and debugged primal recovery subroutine in case of imperfect measurements with q=p [recover]:
  • implemented relative matching with respect to t=tf;
  • broke up the 2 code blocks for matching;
  • revised to 3d [weight] and 1d [whichside], corrected [shortestpath];
  • introduced meaningful variables lelh (= left-edge-of-left-hole) etc.
September 2007:
• finished M.Sc. report.
• converted from simulating 2 code blocks to simulating 1 code block:
  • introduced recursion in a few subroutines.
• implemented and debugged dual recovery (dealt with all possible rough boundary selections):
  • added and debugged subroutines [path2rough] and [weight2rough].
• implemented and debugged 2D QEC at final time for q=p.
• implemented and debugged final analysis subroutines [Pauliprob] and [Bellstats].
• compute Bellstats probabilities.
• remove redundancies/debugging.
• production runs.
Future:
• q.ne.p (rational edge weights).**
• think about correlated errors.**
• think about general unitary errors.**
• understand the proof of partial transpose and separability.
• fault-tolerant measurements of logical XX and ZZ at tf boundary; use entanglement witness.**
• understand Edmonds' blossom algorithm.

Subroutines

1st level subroutines:
• encode(l,minh,maxh,nlh,nSt,nStX,St)
• recursive evolve(channeltyp,p,l,minh,maxh,nlh,nSt,nStX,St,occurred)
• recursive measuresyndrome(channeltyp,pq,t,tf,l,minh,maxh,nlh,nSt,St,Syndrome)
• recursive recover(channeltyp,pq,t,tf,l,minh,maxh,nlh,nSt,nStX,Syndrome,St,corrected)
• checkstabilizer(l,minh,maxh,nlh,nSt,nStX,St)
• checkindex(l,m,minh,maxh,nlh,nSt,nStX)
2nd level subroutines:
• shortestpath(c1,c2,typ,l,minh,maxh,nlh,S)
• path2rough(etyp,c1,typ,l,minh,maxh,nlh,S)
• weight2rough(roughbdy,cc1,ttf,typ,l,minh,maxh,eelen,eetyp)
• applyPauli(typS,nlh,nSt,nStX,S,St)
• recursive logicalop(op,l,minh,maxh,nlh,S)
• physicalop(op,i,j,dir,l,minh,maxh,nlh,S)
Functions:
• character(1) :: whichPauli(l,nlh,nSt,nStX,St,occurred,corrected)
• integer :: weight(dimen,cc1,cc2,typ,l,minh,maxh)
• character(2) :: obstacle(c1,c2,typ,minh,maxh)
• integer :: index1d(i,j,dir,l,minh,maxh,nlh)
• integer :: opindex1d(i,j,typ,l,minh,maxh,nStX)
• logical(1) :: inbound(i,j,dir,l)
• logical(1) :: opinbound(i,j,typ,l)
• logical(1) :: inhole(i,j,dir,minh,maxh)
• logical(1) :: opinhole(i,j,typ,minh,maxh)
• integer :: symprod(typS2,nlh,S1,S2)