def _dense_ham_term(H):
h1, (h2L, h2R) = H
D = h1.shape[0]
dtype = h1.dtype
E = tf.eye(D, dtype=dtype)
h = tensornetwork.ncon([h1, E], [(-1, -3), (-2, -4)])
for (hl, hr) in zip(h2L, h2R):
h += tensornetwork.ncon([hl, hr], [(-1, -3), (-2, -4)])
return h
def _full_ham_top(H):
h1, (h2L, h2R) = H
D = h1.shape[0]
dtype = h1.dtype
E = tf.eye(D, dtype=dtype)
fullH = tensornetwork.ncon([h1, E], [(-1, -3), (-2, -4)])
fullH += tensornetwork.ncon([E, h1], [(-1, -3), (-2, -4)])
for (hl, hr) in zip(h2L, h2R):
fullH += tensornetwork.ncon([hl, hr], [(-1, -3), (-2, -4)])
for (hl, hr) in zip(h2R, h2L):
fullH += tensornetwork.ncon([hl, hr], [(-1, -3), (-2, -4)])
return tf.reshape(fullH, (D**2, D**2))
def opt_energy_env_1site(iso_012, h_op_1site, h_mpo_2site, state_1site):
iso_021 = tf.transpose(iso_012, (0, 2, 1))
terms_012, terms_021 = _ascend_op_to_1site_partial(h_op_1site, h_mpo_2site,
iso_012, iso_021)
terms = terms_012 + tf.transpose(terms_021, (0, 2, 1))
env = tensornetwork.ncon([state_1site, terms], [(0, -1), (0, -2, -3)])
return env
def ascend_uniform_MPO_to_top(mpo_tensor_dense, isos_012):
"""MPO ordering:
3
|
0--m--1
|
2
"""
L = len(isos_012)
for l in range(L):
# NOTE: There is no attempt to be economical with transpose here!
mpo_tensor_dense = tensornetwork.ncon([
isos_012[l],
tf.conj(isos_012[l]), mpo_tensor_dense, mpo_tensor_dense
], [(-4, 2, 0), (-3, 3, 4), (1, -2, 4, 0), (-1, 1, 3, 2)])
op = tensornetwork.ncon([mpo_tensor_dense], [(0, 0, -1, -2)])
return op
def descend_state_1site_R(state_1site, iso_012): #χ^4
"""Descends a state from the top to the rightmost index of the isometry `iso`.
Physically, if `iso` has 012 ordering, this is a descent to the right and
if `iso` has 021 ordering, this is a descent to the left.
"""
return tensornetwork.ncon(
[iso_012, state_1site, tf.conj(iso_012)], [(1, 2, -1), (1, 0),
(0, 2, -2)])
def _mpo_with_state(iso_012, iso_021, h_mpo_2site, state_1site):
# contract ascended hamiltonian at level `lup` with nearest 1-site descended state
h2L, h2R = h_mpo_2site
envL = [
tensornetwork.ncon(
[state_1site, iso_021, h, tf.conj(iso_012)],
[(0, 2), (0, -1, 1), (3, 1), (2, 3, -2)]) # one transpose required
for h in h2L
]
envR = [
tensornetwork.ncon(
[state_1site, iso_012, h, tf.conj(iso_021)],
[(0, 2), (0, -1, 1), (3, 1), (2, 3, -2)]) # one transpose required
for h in h2R
]
return envL, envR
def _ascend_op_2site_to_1site_partial(mpo_2site, iso_012, iso_021):
op2L, op2R = mpo_2site
M = len(op2L) # MPO bond dimension
terms = []
for m in range(M):
# permute result to 012 order: M mild transposes
iso_op_mpo_L_012 = tensornetwork.ncon([iso_021, op2L[m]], [(-1, -3, 0), (-2, 0)])
terms.append(_ascend_partial(op2R[m], iso_op_mpo_L_012))
iso_op_2site_012 = sum(terms)
return iso_op_2site_012
def expand_bonds(isos, new_Ds, new_top_rank=None):
old_Ds = [iso.shape[1] for iso in isos] + [isos[-1].shape[0]]
if new_top_rank is None:
new_top_rank = old_Ds[-1]
new_Ds = new_Ds + [new_top_rank]
if new_Ds[0] != old_Ds[0]:
raise ValueError("Bottom dimension expansion not supported!")
isos_new = [iso for iso in isos]
for i in range(len(isos)):
# Absorb dimension-expanding isometries on indices as needed
if old_Ds[i + 1] != new_Ds[i + 1]:
v = random_isometry(
old_Ds[i + 1], new_Ds[i + 1], dtype=isos_new[i].dtype)
isos_new[i] = tensornetwork.ncon([v, isos_new[i]], [(0, -1), (0, -2, -3)])
if i + 1 < len(isos):
isos_new[i + 1] = tensornetwork.ncon(
[tf.conj(v), tf.conj(v), isos_new[i + 1]], [(0, -2),
(1, -3),
(-1, 0, 1)])
return isos_new
def _compute_env(lvl, reflect=False):
# TODO: Could shorten this a bit by doing only left or right at one time
h2 = h2s_above[lvl]
if reflect:
h2 = reflect_mpo_2site(h2)
envL, envR = _mpo_with_state(*isos_wt_above[lvl], h2,
states_1site_above[lvl])
# descend envs back down to the level of the gap
for lvl2 in reversed(range(lvl)):
iso_012_l2, iso_021_l2 = isos_wt_above[lvl2]
if reflect:
envR = _descend_energy_env_L(envR, iso_021_l2)
envL = _descend_energy_env_R(envL, iso_012_l2)
else:
envL = _descend_energy_env_L(envL, iso_021_l2)
envR = _descend_energy_env_R(envR, iso_012_l2)
if reflect:
iso_h2_L, iso_h2_R = iso_h2R_012, iso_h2L_012
else:
iso_h2_L, iso_h2_R = iso_h2L_012, iso_h2R_012
# contract with the hamiltonian + isometry opposite the gap
envL = sum(
tensornetwork.ncon([eL, ihR], [(0, -1), (0, -2, -3)])
for eL, ihR in zip(envL, iso_h2_R))
envR = sum(
tensornetwork.ncon([eR, ihL], [(0, -1), (0, -2, -3)])
for eR, ihL in zip(envR, iso_h2_L))
# weight each term according to the number of occurrences
# in the translation-invariant tree
weight = 1 / 2.0**(lvl + 1)
return (envL + envR) * weight, weight
def _energy_expval_env(isos_012, h_op_1site, h_mpo_2site, states_1site_above):
if len(isos_012) == 1: # top of tree
h_mpo_2site = add_mpos_2site(h_mpo_2site,
reflect_mpo_2site(h_mpo_2site))
env = opt_energy_env_1site(isos_012[0], h_op_1site, h_mpo_2site,
states_1site_above[0])
else:
env1 = opt_energy_env_1site(isos_012[0], h_op_1site, h_mpo_2site,
states_1site_above[0])
env2 = opt_energy_env_2site(isos_012, h_mpo_2site,
states_1site_above[1:])
env = env1 + env2 / 2
# NOTE: There are *two* environments for each Ham. term spanning two
# isometries. To get the correct energy we must divide env2 by 2.
nsites = 2**(len(isos_012) - 1)
return tensornetwork.ncon([tf.conj(isos_012[0]), env], [(0, 1, 2),
(0, 1, 2)]) * nsites
def shift_ham(H, shift="auto"):
h1, (h2L, h2R) = H
D = h1.shape[0]
dtype = h1.dtype
if shift == "auto":
e1 = tf.reduce_max(tf.cast(tf.linalg.eigvalsh(h1), dtype.real_dtype))
h2 = sum([
tensornetwork.ncon([hl, hr], [(-1, -3), (-2, -4)])
for (hl, hr) in zip(h2L, h2R)
])
h2 = tf.reshape(h2, (D**2, D**2))
e2 = tf.reduce_max(tf.cast(tf.linalg.eigvalsh(h2), dtype.real_dtype))
shift = tf.cast(e1 + e2, dtype)
if shift != 0.0:
H = (h1 - shift * tf.eye(D, dtype=dtype), (h2L, h2R))
return H, shift
def opt_energy_env(isos_012,
h_op_1site,
h_mpo_2site,
states_1site_above,
envsq_dtype=None):
if len(isos_012) == 1: # top of tree
h_mpo_2site = add_mpos_2site(h_mpo_2site,
reflect_mpo_2site(h_mpo_2site))
env = opt_energy_env_1site(isos_012[0], h_op_1site, h_mpo_2site,
states_1site_above[0])
else:
env1 = opt_energy_env_1site(isos_012[0], h_op_1site, h_mpo_2site,
states_1site_above[0])
env2 = opt_energy_env_2site(isos_012, h_mpo_2site,
states_1site_above[1:])
env = env1 + env2
if envsq_dtype is not None:
env = tf.cast(env, envsq_dtype)
env_sq = tensornetwork.ncon([env, tf.conj(env)], [(-1, 0, 1), (-2, 0, 1)])
return env, env_sq
def _ascend_partial(op, iso):
"""Ascend an operator through the right index of an isometry.
For 012 (021) index ordering, this is equivalent to ascending from the
physical right (left).
Cost: D^4."""
return tensornetwork.ncon([iso, op], [(-1, -2, 0), (-3, 0)])
def _iso_from_svd(u, vh):
return tensornetwork.ncon([u, vh], [(-1, 0), (0, -2, -3)])
def RY(self, target, theta):
l1 = self.select_op_1(target)
l2 = self.select_target_1(target)
self.psi = tn.ncon([ry(theta), self.psi], [l1, l2])
def CRY(self, control, target, theta):
l1 = self.select_op_2(control, target)
l2 = self.select_target_2(control, target)
self.psi = tn.ncon([cry(theta), self.psi], [l1, l2])
expectX = np.trace(rho[0].reshape(chi_b**3, chi_b**3) @
np.kron(np.eye(32), sX))
# energy_exact = (-2 / np.sin(np.pi / (2 * n_sites))) / n_sites # PBC (Ising)
energy_exact = (-4 / np.sin(np.pi / n_sites)) / n_sites # PBC (XX model)
energy_error = energy_per_site - energy_exact
bias0 = 2 * energy_per_site
ham[0] = ham[z] - (bias0 * np.eye(chi_b**3)).reshape([chi_b] * 6)
print('Iteration: %d of %d, Energy: %f, Energy Error: %e, XMag: %e\n'
% (0, n_iterations, energy_per_site, energy_error, expectX))
# do optimization iterations
en_keep = [0] * n_iterations
for p in range(n_iterations):
for z in range(n_levels):
bias = tn.ncon([ham[z], rho[z]], [[1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6]])
# print('bias: %f \n' % (bias))
chi_temp = ham[z].shape[0]
ham_temp = ham[z] - (((bias + bias_shift) *
np.eye(chi_temp**3)).reshape([chi_temp] * 6))
if right_on:
# RightLink
tensors = [w[z], rho[z + 1], w[z].conj()]
connects = [[4, -3, 1], [3, 2, -2, 3, 1, -4], [4, -1, 2]]
con_order = [3, 2, 4, 1]
rhotemp = tn.ncon(tensors, connects, con_order)
_, proj = trunct_eigh(rhotemp, chi_m)
gam1, gam2 = RightLink(ham_temp, u[z], w[z], rho[z + 1], proj, chi_m)
if (u[z].shape[3] < u[z].shape[1]):
def update_parameters(self, new_parameters):
self.params = tn.Node(new_parameters)
self.params_x = tn.ncon([self.params, self.Domain_tensor],
[[-1, 1], [-2, 1, -3]])
def CU3(self, control, target, theta3):
l1 = self.select_op_2(control, target)
l2 = self.select_target_2(control, target)
self.state = tn.ncon([cu3(theta3), self.state], [l1, l2])
def CRZ(self, control, target, theta):
l1 = self.select_op_2(control, target)
l2 = self.select_target_2(control, target)
self.state = tn.ncon([crz(theta), self.state], [l1, l2])
def CZ(self, control, target):
l1 = self.select_op_2(control, target)
l2 = self.select_target_2(control, target)
self.state = tn.ncon([cz, self.state], [l1, l2])
def U3(self, target, theta3):
l1 = self.select_op_1(target)
l2 = self.select_target_1(target)
self.state = tn.ncon([u3(theta3), self.state], [l1, l2])
def RZ(self, target, theta):
l1 = self.select_op_1(target)
l2 = self.select_target_1(target)
self.state = tn.ncon([rz(theta), self.state], [l1, l2])
def Z(self, target):
l1 = self.select_op_1(target)
l2 = self.select_target_1(target)
self.state = tn.ncon([z, self.state], [l1, l2])
E, V = eigh(ham_final, link_charges=Z2Charge([0, 1]), which='SA')
en_even1 = E.todense()[0].item()
en_odd1 = E.todense()[1].item()
# compare with analytic energies
assert np.allclose(en_even0, en_even1)
assert np.allclose(en_odd0, en_odd1)
"""
Example 2: compute truncated eigendecomposition of a reduced density matrix,
keeping only the eigenvalues above some cut-off threshold
"""
rho_temp = BT.fromdense([ind_chib1] + [ind_chib0],
np.array([[1, 0], [0, 0]], dtype=float))
V = V.reshape([2**(n_sites // 2), 2**(n_sites // 2), 2])
rho_half = tn.ncon([V, rho_temp, V.conj()], [[-1, 1, 2], [2, 3], [-2, 1, 3]])
# decomp with evalues sorted by magnitude
E2, V2 = eigh(rho_half,
which='LM',
full_sort=False,
threshold=1e-10,
max_kept=15)
rho_recover = V2 @ BLA.diag(E2) @ V2.T.conj()
assert np.allclose(rho_half.todense(), rho_recover.todense())
# decomp with evalues sorted by magnitude within each charge block
E2, V2 = eigh(rho_half, which='LM', threshold=1e-10, full_sort=False)
rho_recover = V2 @ BLA.diag(E2) @ V2.T.conj()
assert np.allclose(rho_half.todense(), rho_recover.todense())
def CY(self, control, target):
l1 = self.select_op_2(control, target)
l2 = self.select_target_2(control, target)
self.psi = tn.ncon([cy, self.psi], [l1, l2])
def _simulate(self, initialstate, reortho=False, verbose=False):
"""
do a lanczos simulation
Parameters:
-------------------------
initialstate: tf.Tensor,
the initial state
reortho: bool
if True, krylov vectors are reorthogonalized at each step (costly)
the current implementation is not optimal: there are better ways to do this
verbose: bool
verbosity flag
"""
self.delta = tf.cast(self.delta, initialstate.dtype)
self.deltaEta = tf.cast(self.deltaEta, initialstate.dtype)
dtype = self.matvec(initialstate).dtype
#initialization:
xn = copy.deepcopy(initialstate)
xn /= tf.sqrt(
tn.ncon([tf.conj(xn), xn],
[range(len(xn.shape)),
range(len(xn.shape))]))
xn_minus_1 = tf.zeros(initialstate.shape, dtype=dtype)
converged = False
it = 0
kn = []
epsn = []
self.vecs = []
first = True
while converged == False:
knval = tf.sqrt(
tn.ncon([tf.conj(xn), xn],
[range(len(xn.shape)),
range(len(xn.shape))]))
if tf.cond(tf.less(tf.abs(knval), tf.abs(self.delta)),
lambda: True, lambda: False):
break
kn.append(knval)
xn = xn / kn[-1]
#store the Lanczos vector for later
if reortho == True:
for v in self.vecs:
xn -= tn.ncon([tf.conj(v), xn],
[range(len(v.shape)),
range(len(xn.shape))]) * v
self.vecs.append(xn)
Hxn = self.matvec(xn)
epsn.append(
tn.ncon([tf.conj(xn), Hxn],
[range(len(xn.shape)),
range(len(Hxn.shape))]))
if ((it % self.Ndiag) == 0) & (len(epsn) >= 1):
#diagonalize the effective Hamiltonian
Heff = tf.convert_to_tensor(np.diag(epsn) +
np.diag(kn[1:], 1) +
np.diag(tf.conj(kn[1:]), -1),
dtype=dtype)
eta, u = tf.linalg.eigh(Heff) #could use a tridiag solver
if first == False:
if tf.abs(tf.linalg.norm(eta - etaold)) < tf.abs(
self.deltaEta):
converged = True
first = False
etaold = eta[0]
if it > 0:
Hxn -= (self.vecs[-1] * epsn[-1])
Hxn -= (self.vecs[-2] * kn[-1])
else:
Hxn -= (self.vecs[-1] * epsn[-1])
xn = Hxn
it = it + 1
if it > self.ncv:
break
self.Heff = tf.convert_to_tensor(np.diag(epsn) + np.diag(kn[1:], 1) +
np.diag(np.conj(kn[1:]), -1),
dtype=dtype)
eta, u = tf.linalg.eigh(self.Heff) #could use tridiag
states = []
for n2 in range(min(1, eta.shape[0])):
state = tf.zeros(initialstate.shape, dtype=initialstate.dtype)
for n1 in range(len(self.vecs)):
state += self.vecs[n1] * u[n1, n2]
states.append(state / tf.sqrt(
tn.ncon([tf.conj(state), state],
[range(len(state.shape)),
range(len(state.shape))])))
return eta[0], states[0], converged
def opt_energy_env_2site(isos_012, h_mpo_2site, states_1site_above):
isos_wt = isos_with_transposes(isos_012)
iso_012, iso_021 = isos_wt[0]
isos_wt_above = isos_wt[1:]
levels_above = len(isos_wt_above)
# Ascend two-site Hamiltonian terms through to the bottom of the final isometry
h2s_above = _ascend_op_2site_to_2site_many(h_mpo_2site, isos_wt)
# hamiltonian with isometry opposite the gap
h2L, h2R = h_mpo_2site
iso_h2R_012 = [
tensornetwork.ncon([iso_021, h], [(-1, -3, 0), (-2, 0)]) for h in h2R
] # transpose to 012
iso_h2L_012 = [tensornetwork.ncon([iso_012, h], [(-1, -2, 0), (-3, 0)]) for h in h2L]
def _compute_env(lvl, reflect=False):
# TODO: Could shorten this a bit by doing only left or right at one time
h2 = h2s_above[lvl]
if reflect:
h2 = reflect_mpo_2site(h2)
envL, envR = _mpo_with_state(*isos_wt_above[lvl], h2,
states_1site_above[lvl])
# descend envs back down to the level of the gap
for lvl2 in reversed(range(lvl)):
iso_012_l2, iso_021_l2 = isos_wt_above[lvl2]
if reflect:
envR = _descend_energy_env_L(envR, iso_021_l2)
envL = _descend_energy_env_R(envL, iso_012_l2)
else:
envL = _descend_energy_env_L(envL, iso_021_l2)
envR = _descend_energy_env_R(envR, iso_012_l2)
if reflect:
iso_h2_L, iso_h2_R = iso_h2R_012, iso_h2L_012
else:
iso_h2_L, iso_h2_R = iso_h2L_012, iso_h2R_012
# contract with the hamiltonian + isometry opposite the gap
envL = sum(
tensornetwork.ncon([eL, ihR], [(0, -1), (0, -2, -3)])
for eL, ihR in zip(envL, iso_h2_R))
envR = sum(
tensornetwork.ncon([eR, ihL], [(0, -1), (0, -2, -3)])
for eR, ihL in zip(envR, iso_h2_L))
# weight each term according to the number of occurrences
# in the translation-invariant tree
weight = 1 / 2.0**(lvl + 1)
return (envL + envR) * weight, weight
weightsum = 0.0
env_total = []
for lvl in range(levels_above):
env, weight = _compute_env(lvl)
weightsum += weight
env_total.append(env)
# Now compute the boundary term
env, weight = _compute_env(levels_above - 1, reflect=True)
weightsum += weight
env_total.append(env)
env_total = sum(env_total)
assert weightsum == 1.0
return env_total
def optimize_mod_binary_mera(hamAB_0,
hamBA_0,
rhoAB_0,
rhoBA_0,
wC,
vC,
uC,
numiter=1000,
refsym=True,
nsteps_steady_state=8,
verbose=0,
opt_u=True,
opt_vw=True,
numpy_update=True,
opt_all_layers=False,
opt_u_after=9):
"""
optimization of a scale invariant modified binary MERA tensor network
Args:
hamAB_0 (tf.Tensor): bottom-layer Hamiltonians in AB lattices
hamBA_0 (tf.Tensor): bottom-layer Hamiltonians in BA lattices
rhoAB_0 (tf.Tensor): reduced densit matrix on a-b lattice
rhoBA_0 (tf.Tensor): reduced densit matrix on b-a lattice
wC (list of tf.Tensor): isometries of the MERA, with
vC (list of tf.Tensor): isometries of the MERA, with
uC (list of tf.Tensor): disentanglers of the MERA
numiter (int): number of iteration steps
refsym (bool): if `True`, impose reflection symmetry
nsteps_steady_state (int): number of power-method iteration steps for calculating the
steady state density matrices
verbose (int): verbosity flag
opt_u (bool): if `False`, skip disentangler optimization
opt_vw (bool): if `False`, skip isometry optimization
numpy_update (bool): if `True`, use numpy svd to calculate updates
opt_all_layers (bool): if `True`, optimize all layers
if `False`, optimize only truncating layers
opt_u_after (int): start optimizing disentangler only after `opt_u_after` initial optimization steps
Returns:
wC (list of tf.Tensor): optimized isometries
vC (list of tf.Tensor): optimized isometries
uC (list of tf.Tensor): optimized disentanglers
rhoAB (tf.Tensor): steady state density matrices on the A-B lattice at the top layer
rhoBA (tf.Tensor): steady state density matrices on the B-A lattice at the top layer
run_times (list of float): run times per iteration step
Energies (list of float): energies per iteration step
"""
dtype = rhoAB_0.dtype
hamAB = [0 for x in range(len(vC) + 1)]
hamBA = [0 for x in range(len(vC) + 1)]
rhoAB = [0 for x in range(len(vC) + 1)]
rhoBA = [0 for x in range(len(vC) + 1)]
hamAB[0] = hamAB_0
hamBA[0] = hamBA_0
chi1 = hamAB[0].shape[0]
bias = tf.math.reduce_max(
tf.linalg.eigvalsh(tf.reshape(hamAB[0], (chi1 * chi1, chi1 * chi1))))
hamAB[0] = hamAB[0] - bias * tf.reshape(tf.eye(chi1 * chi1, dtype=dtype),
(chi1, chi1, chi1, chi1))
hamBA[0] = hamBA[0] - bias * tf.reshape(tf.eye(chi1 * chi1, dtype=dtype),
(chi1, chi1, chi1, chi1))
skip_layer = [misc_mera.skip_layer(w) for w in wC]
for p in range(len(wC)):
if skip_layer[p]:
hamAB[p + 1], hamBA[p + 1] = ascending_super_operator(
hamAB[p], hamBA[p], wC[p], vC[p], uC[p], refsym)
Energies = []
run_times = []
for k in range(numiter):
t1 = time.time()
rhoAB_0, rhoBA_0 = steady_state_density_matrices(
nsteps_steady_state, rhoAB_0, rhoBA_0, wC[-1], vC[-1], uC[-1],
refsym)
rhoAB[-1] = rhoAB_0
rhoBA[-1] = rhoBA_0
for p in range(len(rhoAB) - 2, -1, -1):
rhoAB[p], rhoBA[p] = descending_super_operator(
rhoAB[p + 1], rhoBA[p + 1], wC[p], vC[p], uC[p], refsym)
if verbose > 0:
if np.mod(k, 10) == 1:
Energies.append(
(tn.ncon([rhoAB[0], hamAB[0]], [[1, 2, 3, 4], [1, 2, 3, 4]]
) + tn.ncon([rhoBA[0], hamBA[0]],
[[1, 2, 3, 4], [1, 2, 3, 4]])) / 4 +
bias / 2)
stdout.write(
'\rIteration: %i of %i: E = %.8f, err = %.16f at D = %i with %i layers'
% (int(k), int(numiter), float(Energies[-1]),
float(Energies[-1] + 4 / np.pi, ), int(
wC[-1].shape[2]), len(wC)))
stdout.flush()
for p in range(len(wC)):
if (not opt_all_layers) and skip_layer[p]:
continue
if k >= opt_u_after:
uEnv = get_env_disentangler(hamAB[p], hamBA[p], rhoBA[p + 1],
wC[p], vC[p], uC[p], refsym)
if opt_u:
if refsym:
uEnv = uEnv + tf.transpose(uEnv, (1, 0, 3, 2))
if numpy_update:
uC[p] = misc_mera.u_update_svd_numpy(uEnv)
else:
uC[p] = misc_mera.u_update_svd(uEnv)
wEnv = get_env_w_isometry(hamAB[p], hamBA[p], rhoBA[p + 1],
rhoAB[p + 1], wC[p], vC[p], uC[p])
if opt_vw:
if numpy_update:
wC[p] = misc_mera.w_update_svd_numpy(wEnv)
else:
wC[p] = misc_mera.w_update_svd(wEnv)
if refsym:
vC[p] = wC[p]
else:
vEnv = get_env_v_isometry(hamAB[p], hamBA[p], rhoBA[p + 1],
rhoAB[p + 1], wC[p], vC[p],
uC[p])
vC[p] = misc_mera.w_update_svd(vEnv)
hamAB[p + 1], hamBA[p + 1] = ascending_super_operator(
hamAB[p], hamBA[p], wC[p], vC[p], uC[p], refsym)
run_times.append(time.time() - t1)
if verbose > 2:
print('time per iteration: ', run_times[-1])
return wC, vC, uC, rhoAB[-1], rhoBA[-1], run_times, Energies
def Y(self, target):
l1 = self.select_op_1(target)
l2 = self.select_target_1(target)
self.psi = tn.ncon([y, self.psi], [l1, l2])
# np.save('XXData_temp12.npy', (u, w, v, rho, ham))
# warm-up sweep
ham = [0] * (n_levels + 1)
ham[0] = ham_init.copy()
for z in range(n_levels):
ham[z + 1] = 2 * LiftHam(ham[z], u[z], w[z])
# diagonalize top level Hamiltonian, find GS within the charge=0 sector
ham_top = (ham[n_levels] + ham[n_levels].transpose([1, 2, 0, 4, 5, 3]) +
ham[n_levels].transpose([2, 0, 1, 5, 3, 4]))
_, v = eigh(ham_top, link_charges=U1Charge([0]))
# lower the density matrix, compute spectrum of 1-site density
rho = [0] * (n_levels + 1)
rho[n_levels] = tn.ncon([v.conj(), v], [[-1, -2, -3, 1], [-4, -5, -6, 1]])
spect_chi = [0] * (n_levels + 1)
sp_temp, _ = eigh(tn.ncon([rho[n_levels]], [[-1, 1, 2, -2, 1, 2]]),
full_sort=True)
spect_chi[n_levels] = sp_temp.todense()
for z in reversed(range(n_levels)):
rho[z] = LowerDensity(u[z], w[z], rho[z + 1])
sp_temp, _ = eigh(tn.ncon([rho[z]], [[-1, 1, 2, -2, 1, 2]]),
full_sort=True)
spect_chi[z] = sp_temp.todense()
# en_bias = tn.ncon([ham[0], rho[0]], [[1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6]]).item()
# chi_temp = ham[0].shape[0]
# eye_temp = ((en_bias) * np.eye(chi_temp**3).reshape([chi_temp] * 6))
# ham[0] = ham[0] - BT.fromdense(ham[0].sparse_shape, eye_temp)
# for z in range(n_levels):
def wegner_ncon(f, G, holes, particles, occA, occB, occC, occD):
# - Decouple off-diagonal 1B and 2B pieces
fod = np.zeros(f.shape)
fod[np.ix_(particles, holes)] += f[np.ix_(particles, holes)]
fod[np.ix_(holes, particles)] += f[np.ix_(holes, particles)]
fd = f - fod
God = np.zeros(G.shape)
God[np.ix_(particles, particles, holes,
holes)] += G[np.ix_(particles, particles, holes, holes)]
God[np.ix_(holes, holes, particles,
particles)] += G[np.ix_(holes, holes, particles, particles)]
Gd = G - God
# - Calculate 1B generator
# first term
sum1_1b_1 = ncon([fd, fod], [(-1, 0), (0, -2)]).numpy()
sum1_1b_2 = np.transpose(sum1_1b_1)
sum1_1b = sum1_1b_1 - sum1_1b_2
# second term
sum2_1b_1 = ncon([fd, God], [(0, 1), (1, -1, 0, -2)]).numpy()
sum2_1b_2 = ncon([fod, Gd], [(0, 1), (1, -1, 0, -2)]).numpy()
sum2_1b_3 = sum2_1b_1 - sum2_1b_2
sum2_1b = ncon([occA, sum2_1b_3], [(-1, -2, 0, 1), (0, 1)]).numpy()
# third term
sum3_1b_1 = ncon([occC, God], [(-1, -2, -3, 0, 1, 2),
(0, 1, 2, -4)]).numpy()
sum3_1b_2 = ncon([Gd, sum3_1b_1], [(2, -1, 0, 1), (0, 1, 2, -2)]).numpy()
sum3_1b_3 = np.transpose(sum3_1b_2)
sum3_1b = sum3_1b_2 - sum3_1b_3
eta1B = sum1_1b + sum2_1b + 0.5 * sum3_1b
# - Calculate 2B generator
# first term (P_ij piece)
sum1_2b_1 = ncon([fd, God], [(-1, 0), (0, -2, -3, -4)]).numpy()
sum1_2b_2 = ncon([fod, Gd], [(-1, 0), (0, -2, -3, -4)]).numpy()
sum1_2b_3 = sum1_2b_1 - sum1_2b_2
sum1_2b_4 = np.transpose(sum1_2b_3, [1, 0, 2, 3])
sum1_2b_5 = sum1_2b_3 - sum1_2b_4
# first term (P_kl piece)
sum1_2b_6 = ncon([fd, God], [(0, -3), (-1, -2, 0, -4)]).numpy()
sum1_2b_7 = ncon([fod, Gd], [(0, -3), (-1, -2, 0, -4)]).numpy()
sum1_2b_8 = sum1_2b_6 - sum1_2b_7
sum1_2b_9 = np.transpose(sum1_2b_8, [0, 1, 3, 2])
sum1_2b_10 = sum1_2b_8 - sum1_2b_9
sum1_2b = sum1_2b_5 - sum1_2b_10
# second term
sum2_2b_1 = ncon([occB, God], [(-1, -2, 0, 1), (0, 1, -3, -4)]).numpy()
sum2_2b_2 = ncon([occB, Gd], [(-1, -2, 0, 1), (0, 1, -3, -4)]).numpy()
sum2_2b_3 = ncon([Gd, sum2_2b_1], [(-1, -2, 0, 1), (0, 1, -3, -4)]).numpy()
sum2_2b_4 = ncon([God, sum2_2b_2], [(-1, -2, 0, 1),
(0, 1, -3, -4)]).numpy()
sum2_2b = sum2_2b_3 - sum2_2b_4
# third term
sum3_2b_1 = ncon([Gd, God], [(0, -1, 1, -3), (1, -2, 0, -4)]).numpy()
sum3_2b_2 = np.transpose(sum3_2b_1, [1, 0, 2, 3])
sum3_2b_3 = np.transpose(sum3_2b_1, [0, 1, 3, 2])
sum3_2b_4 = np.transpose(sum3_2b_1, [1, 0, 3, 2])
sum3_2b_5 = sum3_2b_1 - sum3_2b_2 - sum3_2b_3 + sum3_2b_4
sum3_2b = ncon([occA, sum3_2b_5], [(0, 1, -1, -2), (0, 1, -3, -4)]).numpy()
eta2B = sum1_2b + 0.5 * sum2_2b + sum3_2b
return (eta1B, eta2B)
def _ascend(op, iso, iso_conj):
return tensornetwork.ncon([iso_conj, op, iso], [(-1, 3, 1), (1, 2),
(-2, 3, 2)])
def _iso_from_uinv(env, env_uinv):
return tensornetwork.ncon([env_uinv, env], [(-1, 0), (0, -2, -3)])
def _iso_from_svd(u, vh):
return tensornetwork.ncon([u, vh], [(-1, 1), (1, -2, -3)])
def _complete_partial_ascend(iso_op, iso):
"""Complete a partial operator ascension performed by `_ascend_partial()`.
This contracts with the conjugated isometry.
Cost: D^4."""
return tensornetwork.ncon([tf.conj(iso), iso_op], [(-1, 0, 1), (-2, 0, 1)])
def _ascend_partial(op, iso):
"""Ascend an operator through the right index of an isometry.
For 012 (021) index ordering, this is equivalent to ascending from the
physical right (left).
Cost: D^4."""
return tensornetwork.ncon([iso, op], [(-1, -2, 1), (-3, 1)])
def check_iso(iso):
sq = tensornetwork.ncon([iso, tf.conj(iso)], [(-1, 0, 1), (-2, 0, 1)])
return tf.norm(sq - tf.eye(sq.shape[0], dtype=sq.dtype))
def outermat(A, B):
chi = A.shape[0]
contract = [A, B]
idxs = [[-2, -1], [-3, -4]]
return tn.ncon(contract, idxs).reshape((chi**2, chi**2))
def single_mpo_heff_np(mpo, L, R, A):
tensors = [L, A, mpo, R]
idxs = [[2, -1, 1], [1, 3, 4], [2, 5, -2, 3], [5, -3, 4]]
newA = tensornetwork.ncon(tensors, idxs)
return newA
def flow(f, G, eta1B, eta2B, holes, particles, occA, occB, occC, occD):
# - Calculate dE/ds
# first term
sum1_0b_1 = ncon([occA, eta1B], [(0, 1, -1, -2), (0, 1)]).numpy()
sum1_0b = ncon([sum1_0b_1, f], [(0, 1), (1, 0)]).numpy()
# second term
sum2_0b_1 = np.matmul(eta2B, occD)
sum2_0b = ncon([sum2_0b_1, G], [(0, 1, 2, 3), (2, 3, 0, 1)]).numpy()
dE = sum1_0b + 0.5 * sum2_0b
# - Calculate df/ds
# first term
sum1_1b_1 = ncon([eta1B, f], [(-1, 0), (0, -2)]).numpy()
sum1_1b_2 = np.transpose(sum1_1b_1)
sum1_1b = sum1_1b_1 + sum1_1b_2
# second term (might need to fix)
sum2_1b_1 = ncon([eta1B, G], [(0, 1), (1, -1, 0, -2)]).numpy()
sum2_1b_2 = ncon([f, eta2B], [(0, 1), (1, -1, 0, -2)]).numpy()
sum2_1b_3 = sum2_1b_1 - sum2_1b_2
sum2_1b = ncon([occA, sum2_1b_3], [(-1, -2, 0, 1), (0, 1)]).numpy()
# third term
sum3_1b_1 = ncon([occC, G], [(-1, -2, -3, 0, 1, 2), (0, 1, 2, -4)]).numpy()
sum3_1b_2 = ncon([eta2B, sum3_1b_1], [(2, -1, 0, 1),
(0, 1, 2, -2)]).numpy()
sum3_1b_3 = np.transpose(sum3_1b_2)
sum3_1b = sum3_1b_2 + sum3_1b_3
df = sum1_1b + sum2_1b + 0.5 * sum3_1b
# - Calculate dG/ds
# first term (P_ij piece)
sum1_2b_1 = ncon([eta1B, G], [(-1, 0), (0, -2, -3, -4)]).numpy()
sum1_2b_2 = ncon([f, eta2B], [(-1, 0), (0, -2, -3, -4)]).numpy()
sum1_2b_3 = sum1_2b_1 - sum1_2b_2
sum1_2b_4 = np.transpose(sum1_2b_3, [1, 0, 2, 3])
sum1_2b_5 = sum1_2b_3 - sum1_2b_4
# first term (P_kl piece)
sum1_2b_6 = ncon([eta1B, G], [(0, -3), (-1, -2, 0, -4)]).numpy()
sum1_2b_7 = ncon([f, eta2B], [(0, -3), (-1, -2, 0, -4)]).numpy()
sum1_2b_8 = sum1_2b_6 - sum1_2b_7
sum1_2b_9 = np.transpose(sum1_2b_8, [0, 1, 3, 2])
sum1_2b_10 = sum1_2b_8 - sum1_2b_9
sum1_2b = sum1_2b_5 - sum1_2b_10
# second term
sum2_2b_1 = ncon([occB, G], [(-1, -2, 0, 1), (0, 1, -3, -4)]).numpy()
sum2_2b_2 = ncon([occB, eta2B], [(-1, -2, 0, 1), (0, 1, -3, -4)]).numpy()
sum2_2b_3 = ncon([eta2B, sum2_2b_1], [(-1, -2, 0, 1),
(0, 1, -3, -4)]).numpy()
sum2_2b_4 = ncon([G, sum2_2b_2], [(-1, -2, 0, 1), (0, 1, -3, -4)]).numpy()
sum2_2b = sum2_2b_3 - sum2_2b_4
# third term
sum3_2b_1 = ncon([eta2B, G], [(0, -1, 1, -3), (1, -2, 0, -4)]).numpy()
sum3_2b_2 = np.transpose(sum3_2b_1, [1, 0, 2, 3])
sum3_2b_3 = np.transpose(sum3_2b_1, [0, 1, 3, 2])
sum3_2b_4 = np.transpose(sum3_2b_1, [1, 0, 3, 2])
sum3_2b_5 = sum3_2b_1 - sum3_2b_2 - sum3_2b_3 + sum3_2b_4
sum3_2b = ncon([occA, sum3_2b_5], [(0, 1, -1, -2), (0, 1, -3, -4)]).numpy()
dG = sum1_2b + 0.5 * sum2_2b + sum3_2b
return (dE, df, dG)
def single_mpo_heff2(mpo, L, R, A):
tensors = [L, A, mpo, R]
idxs = [[3, -1, 1], [1, 5, 2], [3, 4, -2, 5], [4, -3, 2]]
newA = tensornetwork.ncon(tensors, idxs, backend="jax")
return newA
def CenterLink(ham, u, w, u1, w1, rho2, chi, thres_in):
# find projection onto reduced subspace
tensors = [w1, w1, rho2, w1.conj(), w1.conj()]
connects = [[-4, 6, 2], [7, -3, 1], [3, 4, 5, 1, 2, 5], [-2, 6, 4],
[7, -1, 3]]
cont_order = [5, 7, 6, 2, 4, 3, 1]
rhotemp = tn.ncon(tensors, connects, cont_order)
# _, proj = eigh(rhotemp.conj(), which='LM', max_kept=chi_m,
# full_sort=False)
dmid, projtemp = eigh(rhotemp.conj(),
which='LM',
threshold=0.01 * thres_in,
full_sort=True)
proj = projtemp.conj()
# print(thres_z1)
# print(proj.shape[2])
# print(dmid.todense())
# evaluate centered environments and update tensors
# (chi^4)*(chi_p^5)
tensors = [ham, u, u, u.conj(), u.conj(), w.conj()]
connects = [[5, 6, 7, 2, 3, 4], [1, 2, -4, -5], [3, 4, -6, -7],
[1, 5, -1, 8], [6, 7, 9, -3], [8, 9, -2]]
con_order = [8, 3, 4, 6, 7, 5, 9, 2, 1]
temp1 = tn.ncon(tensors, connects, con_order)
# (chi^4)*(chi_p^5)
tensors = [ham, u, u, u.conj(), u.conj(), w.conj()]
connects = [[4, 5, 6, 1, 2, 3], [1, 2, -4, -5], [3, 9, -6, -7],
[4, 5, -1, 7], [6, 9, 8, -3], [7, 8, -2]]
con_order = [4, 5, 1, 2, 8, 6, 7, 3, 9]
ham7 = temp1 + tn.ncon(tensors, connects, con_order)
# # (chi^4)*(chi_p^2)
# tensors = [w1,w1,rho2,w1.conj(),w1.conj()]
# connects = [[-4,6,2],[7,-3,1],[3,4,5,1,2,5],[-2,6,4],[7,-1,3]]
# cont_order = [5,7,6,2,4,3,1]
# rhotemp = tn.ncon(tensors,connects,cont_order)
# _, proj = trunct_eigh(rhotemp, chi_m)
# Network 1 - leading cost: (chi^2)*(chi_p^6)*(chi_c^1)
tensors = [
w,
w.conj(),
w.conj(), u1,
u1.conj(),
u1.conj(), w1, w1, w1,
w1.conj(),
w1.conj(),
w1.conj(), rho2, ham7, proj
]
connects = [[18, 1, 2], [-1, 19, 4], [20, 1, 3], [2, 25, 5, 6],
[4, 21, 7, 8], [3, 25, 9, 22], [24, 5, 13], [11, 23, 12],
[6, 10, 14], [8, 9, 16], [11, 7, 15], [22, 10, 17],
[15, 16, 17, 12, 13, 14], [19, 21, 20, -2, -3, -4, 18],
[23, 24, -5]]
cont_order = [
9, 11, 1, 12, 15, 24, 10, 17, 14, 13, 23, 5, 6, 25, 16, 22, 2, 3, 7, 8,
18, 20, 21, 4, 19
]
temp1 = tn.ncon(tensors, connects, cont_order)
# Network 2 - leading cost: (chi^3)*(chi_p^6)
tensors = [
w, w,
w.conj(),
w.conj(), u1,
u1.conj(),
u1.conj(), w1, w1, w1,
w1.conj(),
w1.conj(),
w1.conj(), rho2, ham7,
w.conj(), proj
]
connects = [[20, 21, 3], [1, 19, 2], [1, 22, 5], [23, -2, 4], [2, 3, 6, 7],
[5, 24, 8, 9], [4, 25, 10, 26], [7, 27, 14], [12, 6, 13],
[28, 11, 15], [9, 10, 17], [12, 8, 16], [26, 11, 18],
[16, 17, 18, 13, 14, 15], [22, 24, 23, 19, 20, 21, -1],
[-3, -4, 25], [27, 28, -5]]
cont_order = [
5, 4, 1, 12, 20, 21, 25, 27, 11, 15, 18, 13, 16, 14, 28, 17, 22, 24,
19, 3, 2, 8, 9, 6, 7, 23, 26, 10
]
temp2 = tn.ncon(tensors, connects, cont_order)
# Network 3 - leading cost: (chi^3)*(chi_p^6)
tensors = [
w,
w.conj(), u1,
u1.conj(),
u1.conj(), w1, w1, w1,
w1.conj(),
w1.conj(),
w1.conj(), rho2, w,
w.conj(),
w.conj(), ham7, proj
]
connects = [[21, 22, 1], [-1, -2, 2], [1, 16, 3, 4], [2, 20, 5, 6],
[26, 17, 7, 19], [28, 3, 11], [9, 27, 10], [4, 8, 12],
[6, 7, 14], [9, 5, 13], [19, 8, 15], [13, 14, 15, 10, 11, 12],
[23, 18, 16], [24, 18, 17], [-3, 25, 20],
[25, 26, 24, -4, 21, 22, 23], [27, 28, -5]]
cont_order = [
9, 20, 16, 8, 21, 22, 2, 28, 18, 12, 15, 10, 13, 11, 27, 1, 24, 23, 14,
26, 17, 3, 4, 7, 19, 25, 5, 6
]
temp3 = tn.ncon(tensors, connects, cont_order)
# Network 4 - leading cost: (chi^2)*(chi_p^6)*(chi_c^1)
tensors = [
w, u1,
u1.conj(),
u1.conj(), w1, w1, w1,
w1.conj(),
w1.conj(),
w1.conj(), rho2,
w.conj(),
w.conj(), ham7, proj
]
connects = [[18, 19, 1], [25, 1, 2, 3], [25, 17, 4, 5], [22, 15, 6, 16],
[3, 23, 10], [8, 2, 9], [24, 7, 11], [5, 6, 13], [8, 4, 12],
[16, 7, 14], [12, 13, 14, 9, 10, 11], [20, -4, 15],
[18, 21, 17], [21, 22, 20, 19, -1, -2, -3], [23, 24, -5]]
cont_order = [
8, 18, 7, 11, 14, 5, 23, 9, 12, 10, 24, 2, 3, 25, 4, 13, 1, 17, 16, 6,
19, 21, 22, 15, 20
]
temp4 = tn.ncon(tensors, connects, cont_order)
q = -orthog_sym(tn.ncon([temp1 + temp2 + temp3 + temp4,
proj.conj()], [[-1, -2, -3, -4, 1], [-5, -6, 1]]),
pivot=4).conj()
# q = orthog(tn.ncon([temp1+temp2+temp3+temp4,proj.conj()],[[-1,-2,-3,-4,1],[-5,-6,1]]), pivot=4).conj()
# Network 2 - leading cost: (chi_p^8)
tensors = [w1, w1, rho2, w1.conj(), w1.conj(), q, q.conj()]
connects = [[11, 6, 2], [7, 10, 1], [3, 4, 5, 1, 2, 5], [9, 6, 4],
[7, 8, 3], [-3, -4, 13, 12, 10, 11], [-1, -2, 13, 12, 8, 9]]
cont_order = [5, 6, 2, 4, 7, 3, 1, 9, 8, 11, 10, 13, 12]
temp1 = tn.ncon(tensors, connects, cont_order)
# Network 3 - leading cost: (chi_p^8)
tensors = [w1, w1, rho2, w1.conj(), w1.conj(), q, q.conj()]
connects = [[11, 6, 2], [7, 10, 1], [3, 4, 5, 1, 2, 5], [9, 6, 4],
[7, 8, 3], [13, 12, -3, -4, 10, 11], [13, 12, -1, -2, 8, 9]]
cont_order = [5, 7, 1, 3, 6, 2, 4, 11, 10, 9, 8, 13, 12]
temp2 = tn.ncon(tensors, connects, cont_order)
rhotemp = 0.5 * (temp1 + temp2)
qenv = q
_, w_new = eigh(rhotemp.conj(), which='LM', max_kept=chi, full_sort=False)
u_new = orthog_sym(
tn.ncon([qenv, w_new.conj(), w_new.conj()],
[[1, 2, 3, 4, -3, -4], [1, 2, -1], [3, 4, -2]]),
pivot=2)
return w_new, u_new
def opt_energy_env_2site(isos_012, h_mpo_2site, states_1site_above):
isos_wt = isos_with_transposes(isos_012)
iso_012, iso_021 = isos_wt[0]
isos_wt_above = isos_wt[1:]
levels_above = len(isos_wt_above)
# Ascend two-site Hamiltonian terms through to the bottom of the final isometry
h2s_above = _ascend_op_2site_to_2site_many(h_mpo_2site, isos_wt)
# hamiltonian with isometry opposite the gap
h2L, h2R = h_mpo_2site
iso_h2R_012 = [
tensornetwork.ncon([iso_021, h], [(-1, -3, 1), (-2, 1)]) for h in h2R
] # transpose to 012
iso_h2L_012 = [
tensornetwork.ncon([iso_012, h], [(-1, -2, 1), (-3, 1)]) for h in h2L
]
def _compute_env(lvl, reflect=False):
# TODO: Could shorten this a bit by doing only left or right at one time
h2 = h2s_above[lvl]
if reflect:
h2 = reflect_mpo_2site(h2)
envL, envR = _mpo_with_state(*isos_wt_above[lvl], h2,
states_1site_above[lvl])
# descend envs back down to the level of the gap
for lvl2 in reversed(range(lvl)):
iso_012_l2, iso_021_l2 = isos_wt_above[lvl2]
if reflect:
envR = _descend_energy_env_L(envR, iso_021_l2)
envL = _descend_energy_env_R(envL, iso_012_l2)
else:
envL = _descend_energy_env_L(envL, iso_021_l2)
envR = _descend_energy_env_R(envR, iso_012_l2)
if reflect:
iso_h2_L, iso_h2_R = iso_h2R_012, iso_h2L_012
else:
iso_h2_L, iso_h2_R = iso_h2L_012, iso_h2R_012
# contract with the hamiltonian + isometry opposite the gap
envL = sum(
tensornetwork.ncon([eL, ihR], [(1, -1), (1, -2, -3)])
for eL, ihR in zip(envL, iso_h2_R))
envR = sum(
tensornetwork.ncon([eR, ihL], [(1, -1), (1, -2, -3)])
for eR, ihL in zip(envR, iso_h2_L))
# weight each term according to the number of occurrences
# in the translation-invariant tree
weight = 1 / 2.0**(lvl + 1)
return (envL + envR) * weight, weight
weightsum = 0.0
env_total = []
for lvl in range(levels_above):
env, weight = _compute_env(lvl)
weightsum += weight
env_total.append(env)
# Now compute the boundary term
env, weight = _compute_env(levels_above - 1, reflect=True)
weightsum += weight
env_total.append(env)
env_total = sum(env_total)
assert weightsum == 1.0
return env_total
def RightLink(ham, u, w, rho1, thres_in):
tensors = [w, rho1, w.conj()]
connects = [[4, -3, 1], [3, 2, -2, 3, 1, -4], [4, -1, 2]]
con_order = [3, 2, 4, 1]
rhotemp = tn.ncon(tensors, connects, con_order)
_, proj = eigh(rhotemp.conj(),
which='LM',
threshold=0.1 * thres_in,
full_sort=False)
tensors = [
ham, u,
u.conj(),
u.conj(), w, w,
w.conj(),
w.conj(),
w.conj(), rho1,
proj.conj()
]
# Network 1 - leading cost: (chi^5)*(chi_p^2)*(chi_c^1)
connects = [[5, 6, 7, 4, -1, -2], [3, 4, 12, 13], [3, 5, 14, 15],
[6, 7, 16, 17], [13, 19, 2], [11, 12, 1], [15, 16, 9],
[11, 14, 8], [17, -3, 10], [8, 9, 10, 1, 2, 18], [19, 18, -4]]
cont_order = [
11, 1, 8, 19, 2, 18, 15, 12, 13, 3, 14, 9, 6, 7, 17, 5, 4, 16, 10
]
temp1 = tn.ncon(tensors, connects, cont_order)
# Network 2 - leading cost: (chi^3)*(chi_p^5)*(chi_c^1)
connects = [[5, 6, 7, 3, 4, -1], [3, 4, 12, 13], [5, 6, 14, 15],
[7, -2, 16, 17], [13, 18, 2], [11, 12, 1], [15, 16, 9],
[11, 14, 8], [17, -3, 10], [8, 9, 10, 1, 2, 19], [18, 19, -4]]
cont_order = [
11, 1, 8, 18, 2, 19, 5, 6, 3, 4, 9, 14, 15, 12, 13, 17, 7, 10, 16
]
temp2 = tn.ncon(tensors, connects, cont_order)
# Network 3 - leading cost: (chi^5)*(chi_p^2)*(chi_c^1)
connects = [[5, 6, 7, -2, 3, 4], [3, 4, -3, 12], [-1, 5, 13, 14],
[6, 7, 15, 16], [11, 18, 1], [12, 17, 2], [14, 15, 9],
[11, 13, 8], [16, 17, 10], [8, 9, 10, 1, 19, 2], [18, 19, -4]]
cont_order = [
11, 1, 8, 18, 19, 17, 10, 2, 9, 13, 14, 16, 15, 3, 4, 5, 6, 7, 12
]
temp3 = tn.ncon(tensors, connects, cont_order)
# Network 4 - leading cost: (chi^6)*(chi_p^1)*(chi_c^1)
connects = [[4, 5, 6, -1, -2, 3], [3, 17, -3, 11], [4, 5, 12, 13],
[6, 17, 14, 15], [10, 18, 1], [11, 16, 2], [13, 14, 8],
[10, 12, 7], [15, 16, 9], [7, 8, 9, 1, 19, 2], [18, 19, -4]]
cont_order = [
10, 1, 7, 18, 19, 16, 9, 2, 8, 14, 15, 17, 11, 12, 13, 4, 5, 6, 3
]
temp4 = tn.ncon(tensors, connects, cont_order)
gam1 = tn.ncon([temp1 + temp2 + temp3 + temp4, proj],
[[-1, -2, -3, 1], [-4, -5, 1]])
tensors = [
ham, u, u,
u.conj(),
u.conj(), w, w,
w.conj(),
w.conj(),
w.conj(), rho1
]
# Network 2 - leading cost: (chi^4)*(chi_p^5)
connects = [[7, 8, 9, 4, 5, 6], [3, 4, -2, 13], [5, 6, 14, 15],
[3, 7, 16, 17], [8, 9, 18, 19], [13, 14, 1], [15, 20, 2],
[17, 18, 11], [-1, 16, 10], [19, 20, 12],
[10, 11, 12, -3, 1, 2]]
cont_order = [
20, 8, 9, 13, 5, 6, 4, 14, 7, 3, 18, 17, 19, 15, 1, 11, 2, 12, 16, 10
]
temp1 = tn.ncon(tensors, connects, cont_order)
# Network 3 - leading cost: (chi^4)*(chi_p^5)
connects = [[6, 7, 8, 3, 4, 5], [3, 4, -2, 12], [5, 20, 13, 14],
[6, 7, 15, 16], [8, 20, 17, 18], [12, 13, 1], [14, 19, 2],
[16, 17, 10], [-1, 15, 9], [18, 19, 11], [9, 10, 11, -3, 1, 2]]
cont_order = [
6, 7, 3, 4, 17, 8, 16, 5, 20, 19, 12, 13, 18, 14, 10, 1, 2, 11, 15, 9
]
temp2 = tn.ncon(tensors, connects, cont_order)
gam2 = temp1 + temp2
if (u.shape[3] < u.shape[1]):
y = eye_sym([u.charges[1][0], u.charges[3][0]],
[u._flows[1], u._flows[3]])
upr = orthog_sym(tn.ncon([u, y.conj()], [[-1, -2, -3, 1], [-4, 1]]),
pivot=2)
wpr = orthog_sym(tn.ncon([w, y], [[1, -2, -3], [-1, 1]]), pivot=2)
gam2pr = tn.ncon([gam2, y.conj()], [[1, -2, -3], [-1, 1]])
else:
upr = u
wpr = w
gam2pr = gam2
for g in range(10):
wpr = orthog_sym(
tn.ncon([gam1, upr], [[1, 2, -2, 3, -3], [1, 2, 3, -1]]) + gam2pr,
pivot=2).conj()
upr = orthog_sym(tn.ncon([gam1, wpr],
[[-1, -2, 1, -3, 2], [-4, 1, 2]]),
pivot=2).conj()
if (u.shape[3] < u.shape[1]):
rhotemp = tn.ncon([wpr, wpr.conj(), rho1],
[[-2, 5, 4], [-1, 5, 3], [3, 1, 2, 4, 1, 2]])
dtemp, y = eigh(rhotemp.conj() / BLA.trace(rhotemp),
which='LM',
threshold=0.5 * thres_in,
full_sort=False)
chi_temp = sum(dtemp.todense() > thres_in) + 1
dtemp, y = eigh(rhotemp.conj() / BLA.trace(rhotemp),
which='LM',
max_kept=chi_temp,
full_sort=False)
u = tn.ncon([upr, y], [[-1, -2, -3, 1], [1, -4]])
w = tn.ncon([wpr, y.conj()], [[1, -2, -3], [1, -1]])
else:
u = upr
w = wpr
return u, w
def _iso_from_uinv(env, env_uinv):
return tensornetwork.ncon([env_uinv, env], [(-1, 1), (1, -2, -3)])
def _ascend(op, iso, iso_conj):
return tensornetwork.ncon([iso_conj, op, iso], [(-1, 2, 0), (0, 1), (-2, 2, 1)])
def _complete_partial_ascend(iso_op, iso):
"""Complete a partial operator ascension performed by `_ascend_partial()`.
This contracts with the conjugated isometry.
Cost: D^4."""
return tensornetwork.ncon([tf.conj(iso), iso_op], [(-1, 1, 2), (-2, 1, 2)])
def optimize_mod_binary_mera(hamAB_0,
hamBA_0,
rhoAB_0,
rhoBA_0,
wC,
vC,
uC,
numiter=1000,
refsym=True,
nsteps_steady_state=8,
verbose=0,
opt_u=True,
opt_vw=True,
numpy_update=True,
opt_all_layers=False,
opt_u_after=9):
"""
------------------------
adapted from Glen Evenbly (c) for www.tensors.net, (v1.1) - last modified 24/1/2019
------------------------
optimization of a scale invariant modified binary MERA tensor network
Parameters:
----------------------------
hamAB_0, hamBA_0: tf.Tensor
bottom-layer Hamiltonians in AB and BA sublattices
rhoAB_0, rhoBA_0: tf.Tensor
initial values for steady-state density matrices
wC, vC, uC: list of tf.Tensor
isometries (wC, vC) and disentanglers (uC) of the MERA, with
bottom layers first
numiter: int
number of iteration steps
refsym: bool
impose reflection symmetry
nsteps_steady_state: int
number of power-methodf iteration steps for calculating the
steady state density matrices
verbose: int
verbosity flag
opt_u, opt_uv: bool
if False, skip unitary or isometry optimization
numpy_update: bool
if True, use numpy svd to calculate update of disentanglers
opt_all_layers: bool
if True, optimize all layers
if False, optimize only truncating layers
opt_u_after: int
start optimizing disentangler only after `opt_u_after` initial optimization steps
Returns:
-------------------------------
(wC, vC, uC, rhoAB, rhoBA, run_times, Energies)
wC, vC, uC: list of tf.Tensor
obtimized MERA tensors
rhoAB, rhoBA: tf.Tensor
steady state density matrices at the top layer
run_times: list
run times per iteration step
Energies: list
energies at each iteration step
"""
dtype = rhoAB_0.dtype
hamAB = [0 for x in range(len(vC) + 1)]
hamBA = [0 for x in range(len(vC) + 1)]
rhoAB = [0 for x in range(len(vC) + 1)]
rhoBA = [0 for x in range(len(vC) + 1)]
hamAB[0] = hamAB_0
hamBA[0] = hamBA_0
chi1 = hamAB[0].shape[0]
bias = tf.math.reduce_max(
tf.linalg.eigvalsh(tf.reshape(hamAB[0], (chi1 * chi1, chi1 * chi1))))
hamAB[0] = hamAB[0] - bias * tf.reshape(
tf.eye(chi1 * chi1, dtype=dtype), (chi1, chi1, chi1, chi1))
hamBA[0] = hamBA[0] - bias * tf.reshape(
tf.eye(chi1 * chi1, dtype=dtype), (chi1, chi1, chi1, chi1))
skip_layer = [misc_mera.skip_layer(w) for w in wC]
for p in range(len(wC)):
if skip_layer[p]:
hamAB[p + 1], hamBA[p + 1] = ascending_super_operator(
hamAB[p], hamBA[p], wC[p], vC[p], uC[p], refsym)
Energies = []
run_times = []
for k in range(numiter):
t1 = time.time()
rhoAB_0, rhoBA_0 = steady_state_density_matrices(
nsteps_steady_state, rhoAB_0, rhoBA_0, wC[-1], vC[-1], uC[-1],
refsym)
rhoAB[-1] = rhoAB_0
rhoBA[-1] = rhoBA_0
for p in range(len(rhoAB) - 2, -1, -1):
rhoAB[p], rhoBA[p] = descending_super_operator(
rhoAB[p + 1], rhoBA[p + 1], wC[p], vC[p], uC[p], refsym)
if verbose > 0:
if np.mod(k, 10) == 1:
Energies.append(
(tn.ncon([rhoAB[0], hamAB[0]],
[[1, 2, 3, 4], [1, 2, 3, 4]]) + tn.
ncon([rhoBA[0], hamBA[0]], [[1, 2, 3, 4], [1, 2, 3, 4]])) /
4 + bias / 2)
stdout.write(
'\rIteration: %i of %i: E = %.8f, err = %.16f at D = %i with %i layers'
% (int(k), int(numiter), float(Energies[-1]),
float(Energies[-1] + 4 / np.pi,), int(wC[-1].shape[2]),
len(wC)))
stdout.flush()
for p in range(len(wC)):
if (not opt_all_layers) and skip_layer[p]:
continue
if k >= opt_u_after:
uEnv = get_env_disentangler(hamAB[p], hamBA[p], rhoBA[p + 1],
wC[p], vC[p], uC[p], refsym)
if opt_u:
if refsym:
uEnv = uEnv + tf.transpose(uEnv, (1, 0, 3, 2))
if numpy_update:
uC[p] = misc_mera.u_update_svd_numpy(uEnv)
else:
uC[p] = misc_mera.u_update_svd(uEnv)
wEnv = get_env_w_isometry(hamAB[p], hamBA[p], rhoBA[p + 1],
rhoAB[p + 1], wC[p], vC[p], uC[p])
if opt_vw:
if numpy_update:
wC[p] = misc_mera.w_update_svd_numpy(wEnv)
else:
wC[p] = misc_mera.w_update_svd(wEnv)
if refsym:
vC[p] = wC[p]
else:
vEnv = get_env_v_isometry(hamAB[p], hamBA[p], rhoBA[p + 1],
rhoAB[p + 1], wC[p], vC[p], uC[p])
vC[p] = misc_mera.w_update_svd(vEnv)
hamAB[p + 1], hamBA[p + 1] = ascending_super_operator(
hamAB[p], hamBA[p], wC[p], vC[p], uC[p], refsym)
run_times.append(time.time() - t1)
if verbose > 2:
print('time per iteration: ', run_times[-1])
return wC, vC, uC, rhoAB[-1], rhoBA[-1], run_times, Energies