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cmpo.py
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cmpo.py
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""" utility functions for cMPO method
"""
import numpy as np
import torch
import os, io, subprocess
import os.path
torch.set_num_threads(int(os.environ['OMP_NUM_THREADS']))
def eigensolver(M):
""" Eigensolver
manually symmetrize M before the eigen decomposition
"""
return torch.linalg.eigh(0.5*(M+M.t()))
class LogTrExpm(torch.autograd.Function):
@staticmethod
def forward(self, beta, mat):
dtype, device = mat.dtype, mat.device
#rho = torch.matrix_exp(beta*mat)
#tr_rho = torch.trace(rho)
w, v = eigensolver(mat)
y = torch.logsumexp(beta*w, dim=0)
scaled_rho = beta * v @ torch.diag(torch.exp(beta*w-y)) @ v.t()
self.save_for_backward(scaled_rho)
return y
@staticmethod
def backward(self, dy):
scaled_rho = self.saved_tensors[0]
dmat = dy * scaled_rho.t()
return None, dmat
class cmpo(object):
""" the object for cMPO
dim: the physical dimension of the cMPO
the structure of cMPO
-- --
| I + dtau Q -- sqrt(dtau) R -- |
| |
| | |
| sqrt(dtau) L P |
| | |
-- --
"""
def __init__(self, Q, L, R, P):
self.dim = Q.shape[0]
self.dtype = Q.dtype
self.device = Q.device
self.Q = Q # 2 leg: D x D
self.L = L # 3 leg: d x D x D
self.R = R # 3 leg: d x D x D
self.P = P # 4 leg: d x d x D x D
def detach(self):
""" return the detached cMPO object, clear autograd information
"""
return cmpo(self.Q.detach(), self.L.detach(), self.R.detach(), self.P.detach())
def project(self, U):
""" perform a unitary transformation in the imaginary-time direction
if U is a square matrix, this is a guage transformation
"""
Q = U.t() @ self.Q @ U
L = U.t() @ self.L @ U
R = U.t() @ self.R @ U
P = U.t() @ self.P @ U
return cmpo(Q, L, R, P)
def t(self):
""" give the transpose of the cMPO
"""
Q = self.Q
L = self.R
R = self.L
P = torch.einsum('abmn->bamn', self.P)
return cmpo(Q, L, R, P)
class cmps(object):
""" the object for cMPS
dim: the physical dimension of the cMPS
the structure of cMPS
-- --
| I + dtau Q |
| |
| | |
| sqrt(dtau) R |
| | |
-- --
"""
def __init__(self, Q, R):
self.dim = Q.shape[0]
self.dtype = Q.dtype
self.device = Q.device
self.Q = Q
self.R = R
def detach(self):
""" return the detached cMPS object, clear autograd information
"""
return cmps(self.Q.detach(), self.R.detach())
def project(self, U):
""" perform a unitary transformation in the imaginary-time direction
if U is a square matrix, this is a guage transformation
"""
Q = U.t() @ self.Q @ U
R = U.t() @ self.R @ U
return cmps(Q, R)
def diagQ(self):
""" transform the cMPS to the gauge where Q is a diagonalized matrix
"""
_, U = eigensolver(self.Q)
return self.project(U)
def multiply(W, mps):
""" multiply a matrix to the left of the cMPS
-- -- -- --
| 1 0 ... 0| | I + dtau Q |
| 0 | | |
| : | | | |
| : W | | sqrt(dtau) R |
| 0 | | | |
-- -- -- --
"""
dtype, device = mps.dtype, mps.device
R1 = torch.einsum('mn, nab->mab', W, mps.R)
return cmps(mps.Q, R1)
def act(mpo, mps):
""" act the cmps to the right of cmpo
-- -- -- --
| I + dtau Q -- sqrt(dtau) R -- | | I + dtau Q |
| | | |
| | | | | |
| sqrt(dtau) L P | | sqrt(dtau) R |
| | | | | |
-- -- -- --
"""
dtype, device = mps.dtype, mps.device
Do, Ds = mpo.dim, mps.dim
d = mps.R.shape[0]
Io = torch.eye(Do, dtype=dtype, device=device)
Is = torch.eye(Ds, dtype=dtype, device=device)
Q_rslt = torch.einsum('ab,cd->acbd', mpo.Q, Is).contiguous().view(Do*Ds, Do*Ds) \
+ torch.einsum('ab,cd->acbd', Io, mps.Q).contiguous().view(Do*Ds, Do*Ds) \
+ torch.einsum('mab,mcd->acbd', mpo.R, mps.R).contiguous().view(Do*Ds, Do*Ds)
R_rslt = torch.einsum('mab,mcd->macbd', mpo.L, Is.repeat(d,1,1)).contiguous().view(d, Do*Ds, Do*Ds) \
+ torch.einsum('mnab,ncd->macbd', mpo.P, mps.R).contiguous().view(d, Do*Ds, Do*Ds)
return cmps(Q_rslt, R_rslt)
def Lact(mps, mpo):
""" act the cmps to the left of cmpo
-- -- -- --
| I + dtau Q | | I + dtau Q -- sqrt(dtau) R -- |
| | | |
| | | | | |
| sqrt(dtau) R | | sqrt(dtau) L P |
| | | | | |
-- -- -- --
"""
dtype, device = mps.dtype, mps.device
Do, Ds = mpo.dim, mps.dim
d = mps.R.shape[0]
Tmps = act(mpo.t(), mps)
Q = torch.einsum('abcd->badc', Tmps.Q.view(Do, Ds, Do, Ds)).contiguous().view(Do*Ds, Do*Ds)
R = torch.einsum('mabcd->mbadc', Tmps.R.view(d, Do, Ds, Do, Ds)).contiguous().view(d, Do*Ds, Do*Ds)
return cmps(Q, R)
def density_matrix(mps1, mps2):
""" construct the K matrix corresponding to <mps1|mps2>
-- -- -- --
| | | I + dtau Q2 |
| | | | |
| I + dtau Q1 sqrt(dtau) R1 | | sqrt(dtau) R2 | = I + dtau K
| | | | |
-- -- -- --
"""
dtype, device= mps1.dtype, mps1.device
D1, D2 = mps1.dim, mps2.dim
I1 = torch.eye(mps1.dim, dtype=dtype, device=device)
I2 = torch.eye(mps2.dim, dtype=dtype, device=device)
M = torch.einsum('ab,cd->acbd', mps1.Q, I2).contiguous().view(D1*D2, D1*D2) \
+ torch.einsum('ab,cd->acbd', I1, mps2.Q).contiguous().view(D1*D2, D1*D2) \
+ torch.einsum('mab,mcd->acbd', mps1.R, mps2.R).contiguous().view(D1*D2, D1*D2)
return M
def ln_ovlp(mps1, mps2, beta):
""" calculate log(<mps1|mps2>)
"""
M = density_matrix(mps1, mps2)
return LogTrExpm.apply(beta, M)
def Fidelity(psi, mps, beta):
""" calculate log [ <psi|mps> / sqrt(<psi|psi>) ]
"""
up = ln_ovlp(psi, mps, beta)
dn = ln_ovlp(psi, psi, beta)
return up - 0.5*dn
def energy_cut(mps, chi):
"""initialize the isometry
keep the chi largest eigenvalues in the Q matrix of the cMPS
"""
w, v = eigensolver(mps.Q)
P = v[:, -chi:]
return P
def interpolate_cut(cut1, cut2, theta):
""" interpolate two isometries
theta = pi/2: mix = cut1
theta = 0 : mix = cut2
"""
mix = np.sin(theta) * cut1 + np.cos(theta) * cut2
U, _, V = torch.svd(mix)
return [email protected]()
def adaptive_mera_update(mps, beta, chi, tol=1e-12, maxiter=50):
""" update the isometry using iterative SVD update with line search
mps: the original cMPS
beta: inverse temperature
chi: target bond dimension
return the compressed cMPS
"""
P = energy_cut(mps, chi)
last = 9.9e9
step = 0
while step < maxiter:
mps_new = mps.project(P.requires_grad_())
loss = ln_ovlp(mps_new, mps, beta) - 0.5 * ln_ovlp(mps_new, mps_new.detach(), beta)
diff = abs(loss.item() - last)
#print('adaptive', step, loss.item() - 0.5*ln_ovlp(mps, mps, beta).item())
if (diff < tol): break
print(step, end='\r')
grad = torch.autograd.grad(loss, P)[0]
last = loss.item()
Fidel0 = loss.item()
Fidel_test = 1e99
step += 1
U, _, V = torch.svd(grad)
#https://mathoverflow.net/questions/262560/natural-ways-of-interpolating-unitary-matrices
#https://groups.google.com/forum/#!topic/manopttoolbox/2zhx67doXaU
#interpolate between unitary matrices
theta = np.pi
proceed = False
while proceed == False:
theta = theta / 2
if theta < np.pi / 1.9**12:
theta = 0
P_test = P.data
else:
P_test = interpolate_cut([email protected](), P.data, theta)
#mix = np.sin(theta) * [email protected]() + np.cos(theta) * P.data
##then retraction back to unitary
#U, _, V = torch.svd(mix)
#P_test = [email protected]()
mps_test = mps.project(P_test)
Fidel1_test = Fidelity(mps_test, mps, beta)
if Fidel1_test > Fidel0 or np.isclose(theta, 0):
P = P_test
proceed=True
return mps_new
def variational_compr(mps, beta, chi, chkp_loc, init=None, tol=1e-12):
""" variationally optimize the compressed cMPS
mps: the original cMPS
beta: the inverse temperature
chi: target bond dimension
chkp_loc: the location to save check point datafile
tol: tolerance
return the compressed cMPS
"""
if init is None:
psi = adaptive_mera_update(mps, beta, chi, tol=tol)
psi = psi.diagQ()
else:
psi = init
Q = torch.nn.Parameter(torch.diag(psi.Q))
R = torch.nn.Parameter(psi.R)
psi_data = data_cmps(Q, R)
optimizer = torch.optim.LBFGS([Q, R], max_iter=20, tolerance_grad=0, tolerance_change=0, line_search_fn="strong_wolfe")
def closure():
optimizer.zero_grad()
psi = cmps(torch.diag(Q), R)
loss = - Fidelity(psi, mps, beta)
loss.backward()
return loss
is_converged = False
loss0 = 9.99e99
while not is_converged:
loss = optimizer.step(closure)
print('--> ' + '{:.12f}'.format(loss.item()), end='\r')
is_converged = np.isclose(loss.item(), loss0, rtol=tol, atol=tol)
loss0 = loss.item()
# "normalize"
with torch.no_grad():
Q -= torch.max(Q)
psi = cmps(torch.diag(Q), R)
# checkpoint
datasave(psi_data, chkp_loc)
return psi.detach()
# utility functions for save and load datafile
class data_cmps(torch.nn.Module):
def __init__(self, Q, R):
super(data_cmps, self).__init__()
self.Q = Q
self.R = R
def datasave(model, path):
torch.save(model.state_dict(), path)
def dataload(model, path):
model.load_state_dict(torch.load(path))
model.eval()