File size: 10,126 Bytes
9425aed | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 | ! BOB Quantum Civilization Engine - Time Evolution
! Module: bob_evolution
! Purpose: Schrödinger evolution, Trotterization, Krylov, RK4, Magnus
! Standard: Fortran 2018
module bob_evolution
use bob_kinds
use bob_errors
use bob_state
use bob_hamiltonian
implicit none
private
public :: bob_evolve_exact
public :: bob_evolve_trotter
public :: bob_evolve_krylov
public :: bob_evolve_rk4
public :: bob_evolve_magnus
public :: bob_time_evolution_operator
public :: bob_time_integrator
public :: INTEGRATOR_EXACT, INTEGRATOR_TROTTER, INTEGRATOR_KRYLOV
public :: INTEGRATOR_RK4, INTEGRATOR_MAGNUS
integer(i4), parameter :: INTEGRATOR_EXACT = 1
integer(i4), parameter :: INTEGRATOR_TROTTER = 2
integer(i4), parameter :: INTEGRATOR_KRYLOV = 3
integer(i4), parameter :: INTEGRATOR_RK4 = 4
integer(i4), parameter :: INTEGRATOR_MAGNUS = 5
type, public :: bob_time_integrator
integer(i4) :: method = INTEGRATOR_RK4
real(wp) :: dt = 0.01_wp
integer(i4) :: trotter_order = 2
integer(i4) :: krylov_dim = 20
character(len=:), allocatable :: name
contains
procedure, public :: init => int_init
procedure, public :: step => int_step
end type bob_time_integrator
contains
!> Exact evolution: |ψ(t)⟩ = exp(-iHdt)|ψ(0)⟩
subroutine bob_evolve_exact(state, H, dt)
type(bob_quantum_state), intent(inout) :: state
complex(cwp), intent(in) :: H(:,:)
real(wp), intent(in) :: dt
complex(cwp), allocatable :: U(:,:), psi_new(:)
integer(i8) :: dim
if (.not. state%is_valid) then
call bob_set_error(BOB_ERROR_INVALID_STATE, "Invalid state", "bob_evolve_exact"); return
end if
dim = state%dim
if (size(H,1)/=dim .or. size(H,2)/=dim) then
call bob_set_error(BOB_ERROR_DIMENSION_MISMATCH, "H dim mismatch", "bob_evolve_exact"); return
end if
U = bob_time_evolution_operator(H, dt)
allocate(psi_new(dim)); psi_new = matmul(U, state%amplitudes)
state%amplitudes = psi_new; state%is_normalized = .true.
call bob_clear_error()
end subroutine bob_evolve_exact
!> Time evolution operator exp(-iHdt) via Padé (6,6) + scaling/squaring
function bob_time_evolution_operator(H, dt) result(U)
complex(cwp), intent(in) :: H(:,:)
real(wp), intent(in) :: dt
complex(cwp), allocatable :: U(:,:), A(:,:), I_mat(:,:)
complex(cwp), allocatable :: U_num(:,:), U_den(:,:), A2(:,:), A4(:,:), A6(:,:)
integer :: dim, n_squarings, i
real(wp) :: norm_H
dim = size(H,1)
allocate(U(dim,dim), A(dim,dim), I_mat(dim,dim))
allocate(U_num(dim,dim), U_den(dim,dim), A2(dim,dim), A4(dim,dim), A6(dim,dim))
I_mat = CZERO; do i=1,dim; I_mat(i,i)=CONE; end do
norm_H = maxval(abs(H))
if (norm_H > ZERO) then
A = -CI * H * dt / norm_H
n_squarings = max(0, ceiling(log(norm_H * abs(dt))/log(2.0_wp)))
else
A = -CI * H * dt; n_squarings = 0
end if
A2 = matmul(A, A)
A4 = matmul(A2, A2)
A6 = matmul(A4, A2)
U_num = I_mat + A/2.0_wp + A2/12.0_wp + matmul(A,A2)/240.0_wp + A4/10080.0_wp + &
matmul(A,A4)/725760.0_wp + A6/7257600.0_wp
U_den = I_mat - A/2.0_wp + A2/12.0_wp - matmul(A,A2)/240.0_wp + A4/10080.0_wp - &
matmul(A,A4)/725760.0_wp + A6/7257600.0_wp
call invert_matrix(U_den, U)
U = matmul(U, U_num)
do i = 1, n_squarings
U = matmul(U, U)
end do
end function bob_time_evolution_operator
subroutine invert_matrix(A, Ainv)
complex(cwp), intent(in) :: A(:,:)
complex(cwp), intent(out) :: Ainv(:,:)
integer :: n, i, k, pivot
complex(cwp), allocatable :: aug(:,:)
n = size(A,1)
allocate(aug(n, 2*n)); aug = CZERO
aug(:,1:n) = A
do i=1,n; aug(i,n+i)=CONE; end do
do i=1,n
pivot = i
do k=i+1,n; if (abs(aug(k,i)) > abs(aug(pivot,i))) pivot=k; end do
if (abs(aug(pivot,i)) < TOL_NORM) then
call bob_set_error(BOB_ERROR_CONVERGENCE, "Singular matrix", "invert_matrix")
Ainv = CZERO; return
end if
if (pivot /= i) aug([i,pivot],:) = aug([pivot,i],:)
aug(i,:) = aug(i,:) / aug(i,i)
do k=1,n
if (k /= i) aug(k,:) = aug(k,:) - aug(k,i) * aug(i,:)
end do
end do
Ainv = aug(:,n+1:2*n)
end subroutine invert_matrix
!> Krylov subspace (Lanczos) evolution
subroutine bob_evolve_krylov(state, H, dt, k)
type(bob_quantum_state), intent(inout) :: state
complex(cwp), intent(in) :: H(:,:)
real(wp), intent(in) :: dt
integer, intent(in), optional :: k
integer :: krylov_dim, dim, i, m
complex(cwp), allocatable :: V(:,:), T(:,:), beta(:), psi_krylov(:), w(:)
real(wp) :: norm
dim = state%dim
krylov_dim = 20; if (present(k)) krylov_dim = min(k, dim)
krylov_dim = min(krylov_dim, dim)
allocate(V(dim, krylov_dim), T(krylov_dim, krylov_dim), beta(krylov_dim))
allocate(psi_krylov(krylov_dim), w(dim))
V = CZERO; T = CZERO; beta = ZERO
V(:,1) = state%amplitudes
norm = sqrt(real(dot_product(conjg(V(:,1)), V(:,1))))
V(:,1) = V(:,1) / norm
m = krylov_dim
do i = 1, krylov_dim
w = matmul(H, V(:,i))
T(i,i) = dot_product(conjg(V(:,i)), w)
w = w - T(i,i) * V(:,i)
if (i > 1) w = w - beta(i-1) * V(:,i-1)
beta(i) = sqrt(real(dot_product(conjg(w), w)))
if (beta(i) < TOL_NORM .or. i == krylov_dim) then
m = i; exit
end if
V(:,i+1) = w / beta(i)
T(i,i+1) = beta(i); T(i+1,i) = beta(i)
end do
psi_krylov = CZERO; psi_krylov(1) = CONE
psi_krylov(1:m) = matmul(bob_time_evolution_operator(T(1:m,1:m), dt), psi_krylov(1:m))
state%amplitudes = matmul(V(:,1:m), psi_krylov(1:m))
state%is_normalized = .true.
call bob_clear_error()
end subroutine bob_evolve_krylov
!> 2nd-order Trotter-Suzuki decomposition
subroutine bob_evolve_trotter(state, H, dt, order)
type(bob_quantum_state), intent(inout) :: state
complex(cwp), intent(in) :: H(:,:)
real(wp), intent(in) :: dt
integer, intent(in), optional :: order
integer :: ord
complex(cwp), allocatable :: U(:,:), psi_new(:)
ord = 2; if (present(order)) ord = order
if (ord == 1) then
U = bob_time_evolution_operator(H, dt)
else
U = matmul(bob_time_evolution_operator(H, dt/2), bob_time_evolution_operator(H, dt/2))
end if
allocate(psi_new(state%dim))
psi_new = matmul(U, state%amplitudes)
state%amplitudes = psi_new; state%is_normalized = .true.
call bob_clear_error()
end subroutine bob_evolve_trotter
!> Magnus expansion (2nd order)
subroutine bob_evolve_magnus(state, H1, H2, dt)
type(bob_quantum_state), intent(inout) :: state
complex(cwp), intent(in) :: H1(:,:), H2(:,:)
real(wp), intent(in) :: dt
complex(cwp), allocatable :: Omega(:,:), U(:,:)
integer :: dim
dim = state%dim
allocate(Omega(dim,dim), U(dim,dim))
Omega = -CI * dt * (H1 + H2) / 2.0_wp
U = bob_time_evolution_operator(Omega/(-CI*dt), dt)
state%amplitudes = matmul(U, state%amplitudes)
call state%normalize()
call bob_clear_error()
end subroutine bob_evolve_magnus
subroutine bob_evolve_rk4(state, H, dt)
type(bob_quantum_state), intent(inout) :: state
complex(cwp), intent(in) :: H(:,:)
real(wp), intent(in) :: dt
complex(cwp), allocatable :: k1(:), k2(:), k3(:), k4(:), psi_temp(:)
integer(i8) :: dim
dim = state%dim
allocate(k1(dim), k2(dim), k3(dim), k4(dim), psi_temp(dim))
k1 = -CI * matmul(H, state%amplitudes)
psi_temp = state%amplitudes + dt/2 * k1; k2 = -CI * matmul(H, psi_temp)
psi_temp = state%amplitudes + dt/2 * k2; k3 = -CI * matmul(H, psi_temp)
psi_temp = state%amplitudes + dt * k3; k4 = -CI * matmul(H, psi_temp)
state%amplitudes = state%amplitudes + dt/6 * (k1 + 2*k2 + 2*k3 + k4)
call state%normalize()
call bob_clear_error()
end subroutine bob_evolve_rk4
subroutine int_init(this, method, dt, name)
class(bob_time_integrator), intent(inout) :: this
integer(i4), intent(in) :: method
real(wp), intent(in) :: dt
character(*), intent(in) :: name
this%method = method; this%dt = dt; this%name = name
end subroutine int_init
subroutine int_step(this, state, H)
class(bob_time_integrator), intent(inout) :: this
type(bob_quantum_state), intent(inout) :: state
type(bob_hamiltonian_operator), intent(inout) :: H
select case (this%method)
case (INTEGRATOR_EXACT)
call bob_evolve_exact(state, H%matrix, this%dt)
case (INTEGRATOR_TROTTER)
call bob_evolve_trotter(state, H%matrix, this%dt, this%trotter_order)
case (INTEGRATOR_KRYLOV)
call bob_evolve_krylov(state, H%matrix, this%dt, this%krylov_dim)
case (INTEGRATOR_RK4)
call bob_evolve_rk4(state, H%matrix, this%dt)
case (INTEGRATOR_MAGNUS)
call bob_evolve_exact(state, H%matrix, this%dt)
case default
call bob_set_error(BOB_ERROR_INVALID_ARGUMENT, "Unknown integrator", "int_step")
end select
end subroutine int_step
end module bob_evolution
|