CHAPTER 5 : CONCLUSION
A.1 Main Program
B.1.1 Three Bit Gate Subroutine
!
! Subroutine to take as input three bits, construct the tensors for SVD: !
! M = | A0.B0.C0 A0.B0.C1 | = U.S.(V^T)
! | A0.B1.C0 A0.B1.C1 | ! | A1.B0.C0 A1.B0.C1 | ! | A1.B1.C0 A1.B1.C1 | ! ! C’ = (S^(1/2)).(V^T) ! ! M1 = U.(S^(1/2)) = | M1[[1]] | ! | M1[[2]] | ! | M1[[3]] | ! | M1[[4]] | ! ! M1’ = | M1[[1]] M1[[2]] | = W.X.(Y^T) ! | M1[[3]] M1[[4]] | ! ! A’ = W.(X^(1/2)) ! B’ = (X^(1/2)).(Y^T) ! !--- ! S. Pelton 04/15 !--- !
& A, B, C, Atil, Btil, Ctil, beta, gate) ! implicit none ! !--- ! Variable declarations !--- ! ! External variables:
integer, intent(in) :: al, r1, r2, br, r1til, r2til, gate integer, intent(out) :: r1new, r2new
real (kind=8), intent(in) :: beta
real (kind=8), intent(in), dimension(al,r1,0:1) :: A real (kind=8), intent(in), dimension(r1,r2,0:1) :: B real (kind=8), intent(in), dimension(r2,br,0:1) :: C
real (kind=8), intent(out), dimension(al,r1til,0:1) :: Atil real (kind=8), intent(out), dimension(r1til,r2til,0:1) :: Btil real (kind=8), intent(out), dimension(r2til,br,0:1) :: Ctil !
! Internal variables:
integer :: k, l, M, N, M1, N1, M2, N2
real (kind=8), dimension(0:1,0:1,0:1) :: prefactor real (kind=8), allocatable, dimension(:,:) :: U, V real (kind=8), allocatable, dimension(:) :: S
real (kind=8), allocatable, dimension(:,:) :: matz, matz1 ! !--- ! Main !--- ! !--- ! Construct the matrix M = U.S.(V^T).
!--- !
M = 4*al N = 2*br !
! memory allocation for auxiliary array allocate (matz(M,N))
prefactor = exp(beta) if (gate .eq. 1) then
prefactor(0,0,0) = exp(-beta) else if (gate .eq. 2) then
prefactor(0,0,1) = exp(-beta) else if (gate .eq. 3) then
prefactor(0,1,0) = exp(-beta) else if (gate .eq. 4) then
prefactor(0,1,1) = exp(-beta) else if (gate .eq. 5) then
prefactor(1,0,0) = exp(-beta) else if (gate .eq. 6) then
prefactor(1,0,1) = exp(-beta) else if (gate .eq. 7) then
prefactor(1,1,0) = exp(-beta) else if (gate .eq. 8) then
prefactor(1,1,1) = exp(-beta) else
prefactor = 1 end if
!
! Fill the matrix for the decomposition. matz(1:al,1:br) = prefactor(0,0,0)*matmul(A(:,:,0),(matmul(B(:,:,0),C(:,:,0)))) matz(1:al,(1+br):2*br) = prefactor(0,0,1)*matmul(A(:,:,0),(matmul(B(:,:,0),C(:,:,1)))) matz((1+al):2*al,1:br) = prefactor(0,1,0)*matmul(A(:,:,0),(matmul(B(:,:,1),C(:,:,0)))) matz((1+al):2*al,(1+br):2*br) = prefactor(0,1,1)*matmul(A(:,:,0),(matmul(B(:,:,1),C(:,:,1)))) matz((1+2*al):3*al,1:br) = prefactor(1,0,0)*matmul(A(:,:,1),(matmul(B(:,:,0),C(:,:,0)))) matz((1+2*al):3*al,(1+br):2*br) = prefactor(1,0,1)*matmul(A(:,:,1),(matmul(B(:,:,0),C(:,:,1)))) matz((1+3*al):4*al,1:br) = prefactor(1,1,0)*matmul(A(:,:,1),(matmul(B(:,:,1),C(:,:,0)))) matz((1+3*al):4*al,(1+br):2*br) = prefactor(1,1,1)*matmul(A(:,:,1),(matmul(B(:,:,1),C(:,:,1)))) ! ! Call SVD.
allocate (U(M,M), V(N,N), S(min(M,N))) !
call singvaldec (M, N, matz, S, U, V) !
! Write the results to the rightmost bit. r2new = 1 do k = 2, min(M,N) if (S(k)/S(1) .gt. 1.d-8) r2new = r2new + 1 end do r2new = min(r2new,100) ! Ctil = 0.d0 !
! Write out the new bit k to k0_til and k1_til. do k = 1, r2new do l = 1, br Ctil(k,l,0) = (S(k)**(1/2.d0))*V(k,l) Ctil(k,l,1) = (S(k)**(1/2.d0))*V(k,l+br) end do end do ! deallocate(matz) ! !--- ! Construct the second matrix M1 = U.(S^(1/2)) --> M1’
!--- !
! Build the auxilliary matrix. allocate (matz(M,min(M,N))) matz = 0.d0 do k = 1, M do l = 1, min(M,N) matz(k,l) = U(k,l)*((S(l))**(1/2.d0)) end do end do !
! Build the second matrix for decomposing from matz. M1 = al
M2 = 2*al N1 = r2til N2 = 2*r2til
matz1(1:M1,(1+N1):2*N1) = matz((1+M1):2*M1,1:N1) matz1((1+M1):2*M1,1:N1) = matz((1+2*M1):3*M1,1:N1)
matz1((1+M1):2*M1,(1+N1):2*N1) = matz((1+3*M1):4*M1,1:N1) !
! Deallocate and reallocate usable matrices. deallocate(matz,U,V,S)
allocate (U(M2,M2), V(N2,N2), S(min(M2,N2))) !
! Call SVD for the second construction. call singvaldec (M2, N2, matz1, S, U, V) !
! Write the results to the middle and rightmost bit. r1new = 1 do k = 2, min(M2,N2) if (S(k)/S(1) .gt. 1.d-8) r1new = r1new + 1 end do r1new = min(r1new, 100) ! Atil = 0.d0 Btil = 0.d0 !
! Write out the new bit i to i0_til and i1_til. do k = 1, al do l = 1, r1new Atil(k,l,0) = U(k,l)*sqrt(S(l)) Atil(k,l,1) = U(k+al,l)*sqrt(S(l)) end do end do !
! Write out the new bit j to j0_til and j1_til do k = 1, r1new do l = 1, r2new Btil(k,l,0) = sqrt(S(k))*V(k,l) Btil(k,l,1) = sqrt(S(k))*V(k,l+N1) end do end do ! deallocate(U, V, S, matz1) !
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