for i in range(N - 1):
(n1, n2, n3) = M[i].shape
def opt_fun(x): # Function to be called by Davidson algorithm
x_reshape = np.reshape(x, M[i].shape)
in_sum1 = lib.einsum('ijk,lmk->ijlm', F[i + 1], x_reshape)
in_sum2 = lib.einsum('njol,ijlm->noim', W[i], in_sum1)
fin_sum = lib.einsum('pnm,noim->opi', F[i], in_sum2)
return -np.reshape(fin_sum, -1)
def precond(dx, e, x0):
return dx
init_guess = np.reshape(M[i], -1)
u, v = lib.eig(opt_fun,
init_guess,
precond,
nroots=min(target_state + 1, n1 * n2 * n3 - 1))
# State Averaging
for j in range(len(v)):
M_tmp = np.reshape(v[j], (n1, n2, n3))
M_reshape_tmp = np.reshape(M_tmp, (n1 * n2, n3))
(U, s, V) = np.linalg.svd(M_reshape_tmp, full_matrices=False)
#print(np.diag(np.einsum('ij,jk->ik',np.einsum('i,ij->ij',s,V),np.conj(np.einsum('i,ij->ij',s,V).T))))
#print(s**2)
if j is 0:
s_avg = s**2
#rho_avg = np.einsum('i,ij->ij',s,V)
else:
s_avg += s**2
#rho_avg += np.einsum('i,ij->ij',s,V)
s_avg = np.sqrt(s_avg)
def main():
t0 = time.time()
######## Inputs ##############################
# SEP Model
N = 50
alpha = 0.35 # In at left
beta = 2. / 3. # Exit at right
s = -1. # Exponential weighting
gamma = 0. # Exit at left
delta = 0. # In at right
p = 1. # Jump right
q = 0. # Jump Left
# Optimization
tol = 1e-5
maxIter = 0
maxBondDim = 10
useCTF = True
##############################################
# Create MPS #################################
# PH - Make Isometries, Center Site
mpiprint('Generating MPS')
M = []
for i in range(int(N / 2)):
tmp = np.random.rand(2,
min(2**(i),maxBondDim),
min(2**(i+1),maxBondDim))\
+0.j
M.append(ctf.from_nparray(tmp))
for i in range(int(N / 2))[::-1]:
tmp = np.random.rand(2,
min(2**(i+1),maxBondDim),
min(2**i,maxBondDim))\
+0.j
M.append(ctf.from_nparray(tmp))
##############################################
# Create MPO #################################
mpiprint('Generating MPO')
# Simple Operators
Sp = np.array([[0., 1.], [0., 0.]])
Sm = np.array([[0., 0.], [1., 0.]])
n = np.array([[0., 0.], [0., 1.]])
v = np.array([[1., 0.], [0., 0.]])
I = np.array([[1., 0.], [0., 1.]])
z = np.array([[0., 0.], [0., 0.]])
# List to hold MPOs
W = []
# First Site
site_0 = ctf.astensor(
[[alpha * (np.exp(-s) * Sm - v),
np.exp(-s) * Sp, -n, I]])
W.append(site_0)
# Central Sites
for i in range(N - 2):
site_i = ctf.astensor([[I, z, z, z], [Sm, z, z, z], [v, z, z, z],
[z, np.exp(-s) * Sp, -n, I]])
W.append(site_i)
# Last Site
site_N = ctf.astensor([[I], [Sm], [v], [beta * (np.exp(-s) * Sp - n)]])
W.append(site_N)
##############################################
# Canonicalize MPS ###########################
for i in range(int(N) - 1, 0, -1):
M_reshape = ctf.transpose(M[i], axes=[1, 0, 2])
(n1, n2, n3) = M_reshape.shape
M_reshape = M_reshape.reshape(n1, n2 * n3)
(U, S, V) = ctf.svd(M_reshape)
M_reshape = V.reshape(n1, n2, n3)
M[i] = ctf.transpose(M_reshape, axes=[1, 0, 2])
M[i - 1] = ctf.einsum('klj,ji,i->kli', M[i - 1], U, S)
##############################################
# Canonicalize MPS ###########################
def pick_eigs(w, v, nroots, x0):
idx = np.argsort(np.real(w))
w = w[idx]
v = v[:, idx]
return w, v, idx
##############################################
# Create Environment #########################
mpiprint('Generating Environment')
# Allocate empty environment
F = []
tmp = np.array([[[1.]]]) + 0.j
F.append(ctf.from_nparray(tmp))
for i in range(int(N / 2)):
tmp = np.zeros((min(2**(i + 1),
maxBondDim), 4, min(2**(i + 1), maxBondDim))) + 0.j
F.append(ctf.from_nparray(tmp))
for i in range(int(N / 2) - 1, 0, -1):
tmp = np.zeros(
(min(2**(i), maxBondDim), 4, min(2**i, maxBondDim))) + 0.j
F.append(ctf.from_nparray(tmp))
tmp = np.array([[[1.]]]) + 0.j
F.append(ctf.from_nparray(tmp))
# Calculate initial environment
for i in range(int(N) - 1, 0, -1):
tmp = ctf.einsum('eaf,cdf->eacd', M[i], F[i + 1])
tmp = ctf.einsum('ydbe,eacd->ybac', W[i], tmp)
F[i] = ctf.einsum('bxc,ybac->xya', ctf.conj(M[i]), tmp)
##############################################
# Optimization Sweeps ########################
converged = False
iterCnt = 0
E_prev = 0
while not converged:
# Right Sweep ----------------------------
tr = time.time()
mpiprint('Start Right Sweep {}'.format(iterCnt))
for i in range(N - 1):
(n1, n2, n3) = M[i].shape
# Make Hx Function
def Hfun(x):
x_reshape = ctf.array(x)
x_reshape = ctf.reshape(x_reshape, (n1, n2, n3))
tmp = ctf.einsum('ijk,lmk->ijlm', F[i + 1], x_reshape)
tmp = ctf.einsum('njol,ijlm->noim', W[i], tmp)
res = ctf.einsum('pnm,noim->opi', F[i], tmp)
return -ctf.reshape(res, -1).to_nparray()
def precond(dx, e, x0):
return dx
# Set up initial guess
guess = ctf.reshape(M[i], -1).to_nparray()
# Run eigenproblem
u, v = eig(Hfun, guess, precond, pick=pick_eigs)
E = -u
v = ctf.array(v)
M[i] = ctf.reshape(v, (n1, n2, n3))
# Print Results
mpiprint('\tEnergy at site {} = {}'.format(i, E))
# Right Normalize
M_reshape = ctf.reshape(M[i], (n1 * n2, n3))
(U, S, V) = ctf.svd(M_reshape)
M[i] = ctf.reshape(U, (n1, n2, n3))
M[i + 1] = ctf.einsum('i,ij,kjl->kil', S, V, M[i + 1])
# Update F
tmp = ctf.einsum('jlp,ijk->lpik', F[i], ctf.conj(M[i]))
tmp = ctf.einsum('lmin,lpik->mpnk', W[i], tmp)
F[i + 1] = ctf.einsum('npq,mpnk->kmq', M[i], tmp)
mpiprint('Complete Right Sweep {}, {} sec'.format(
iterCnt,
time.time() - tr))
# Left Sweep ------------------------------
tl = time.time()
mpiprint('Start Left Sweep {}'.format(iterCnt))
for i in range(N - 1, 0, -1):
(n1, n2, n3) = M[i].shape
# Make Hx Function
def Hfun(x):
x_reshape = ctf.array(x)
x_reshape = ctf.reshape(x_reshape, (n1, n2, n3))
tmp = ctf.einsum('ijk,lmk->ijlm', F[i + 1], x_reshape)
tmp = ctf.einsum('njol,ijlm->noim', W[i], tmp)
res = ctf.einsum('pnm,noim->opi', F[i], tmp)
return -ctf.reshape(res, -1).to_nparray()
def precond(dx, e, x0):
return dx
# Set up initial guess
guess = ctf.reshape(M[i], -1).to_nparray()
# Run eigenproblem
u, v = eig(Hfun, guess, precond, pick=pick_eigs)
E = -u
v = ctf.array(v)
M[i] = ctf.reshape(v, (n1, n2, n3))
# Print Results
mpiprint('\tEnergy at site {}= {}'.format(i, E))
# Right Normalize
M_reshape = ctf.transpose(M[i], (1, 0, 2))
M_reshape = ctf.reshape(M_reshape, (n2, n1 * n3))
(U, S, V) = ctf.svd(M_reshape)
M_reshape = ctf.reshape(V, (n2, n1, n3))
M[i] = ctf.transpose(M_reshape, (1, 0, 2))
M[i - 1] = ctf.einsum('klj,ji,i->kli', M[i - 1], U, S)
# Update F
tmp = ctf.einsum('eaf,cdf->eacd', M[i], F[i + 1])
tmp = ctf.einsum('ydbe,eacd->ybac', W[i], tmp)
F[i] = ctf.einsum('bxc,ybac->xya', ctf.conj(M[i]), tmp)
mpiprint('Left Sweep {}, {} sec'.format(iterCnt, time.time() - tl))
# Convergence Test -----------------------
if np.abs(E - E_prev) < tol:
mpiprint('#' * 75 + '\nConverged at E = {}'.format(E) + '\n' +
'#' * 75)
converged = True
elif iterCnt > maxIter:
mpiprint('Convergence not acheived')
converged = True
else:
iterCnt += 1
E_prev = E
mpiprint('Total Time = {}'.format(time.time() - t0))
(n1, n2, n3) = Mr[i].shape
def opt_fun(x):
x_reshape = np.reshape(x, (n1, n2, n3))
Hx = lib.einsum('ijk,lmk->ijlm', F[i + 1], x_reshape)
Hx = lib.einsum('njol,ijlm->noim', W[i], Hx)
Hx = lib.einsum('pnm,noim->opi', F[i], Hx)
return np.reshape(Hx, -1)
def precond(dx, e, x0):
return dx
init_guess = np.reshape(Mr[i], -1)
E, vl, vr = lib.eig(opt_fun,
init_guess,
precond,
pick=pick_eigs,
left=True)
print('\tEnergy at site {}= {}'.format(i, E))
Mr[i] = np.reshape(vr, (n1, n2, n3))
Ml[i] = np.reshape(vl, (n1, n2, n3))
# Correct Normalization
norm_factor = np.einsum('j,ijk,k->', Fs[i - 1], Mr[i], Fs[i + 1])
Mr[i] /= norm_factor
Ml[i] /= np.einsum('ijk,ijk->', Mr[i], np.conj(Ml[i]))
# Put into Canonical Form
Mr_reshape = np.reshape(Mr[i], (n1 * n2, n3))
Ml_reshape = np.reshape(Ml[i], (n1 * n2, n3))
(ur, sr, vr) = np.linalg.svd(Mr_reshape, full_matrices=False)
(ul, sl, vl) = np.linalg.svd(Ml_reshape, full_matrices=False)
# Gauge Transform of Left State
#!/usr/bin/env python
import numpy
from pyscf import lib
n = 100
a = numpy.random.rand(n, n)
a = a + a.T
def matvec(x):
return a.dot(x)
precond = a.diagonal()
x_init = numpy.zeros(n)
x_init[0] = 1
#e, c = lib.eigh(matvec, x_init, precond, nroots=4, max_cycle=1000, verbose=5)
e, c = lib.eig(matvec, x_init, precond, nroots=4, max_cycle=1000, verbose=5)
print('Eigenvalues', e)
print('Right Sweep {}'.format(iterCnt))
for i in range(N - 1):
(n1, n2, n3) = M[i].shape
def opt_fun(x): # Function to be called by Davidson algorithm
x_reshape = np.reshape(x, M[i].shape)
in_sum1 = np.einsum('ijk,lmk->ijlm', F[i + 1], x_reshape)
in_sum2 = np.einsum('njol,ijlm->noim', W[i], in_sum1)
fin_sum = np.einsum('pnm,noim->opi', F[i], in_sum2)
return np.reshape(fin_sum, -1)
def precond(dx, e, x0): # A second dummy algorithm
return dx
init_guess = np.reshape(M[i], -1)
u, v = lib.eig(opt_fun, init_guess, precond, nroots=9)
# select max eigenvalue
sort_inds = np.argsort(np.real(u))[::-1]
E = u[sort_inds[target_state]]
v = v[sort_inds[target_state]]
print('\tEnergy at site {} = {}'.format(i, E))
M[i] = np.reshape(v, (n1, n2, n3))
# Right Normalize
M_reshape = np.reshape(M[i], (n1 * n2, n3))
(U, s, V) = np.linalg.svd(M_reshape, full_matrices=False)
M[i] = np.reshape(U, (n1, n2, n3))
M[i + 1] = np.einsum('i,ij,kjl->kil', s, V, M[i + 1])
# Update F
F[i + 1] = np.einsum('jlp,ijk,lmin,npq->kmq', F[i], np.conj(M[i]),
W[i], M[i])
# Left Sweep -----------------------------