"""Toy code implementing the TDVP for *finite* MPS."""
# Copyright (C) TeNPy Developers, Apache license
import numpy as np
import scipy.sparse.linalg
from scipy.sparse.linalg import expm
from .d_dmrg import SimpleHeff2
from . import a_mps
class SimpleTDVPEngine:
"""TDVP algorithm for finite systems, implemented as class holding the necessary data.
Note that this class is very similar to `d_dmrg.SimpleDMRGEngine`.
We could use a common base class; but to keep things maximally simple and readable,
we rather duplicate the code for the `__init__`, `update_LP`, and `update_RP` methods.
Also, here we generalize the sweep to temporarily change the MPS to a mixed canonical form
and directly save `A` tensors in it. This means that the SimpleMPS methods (which *assume*
that the tensors are all right-canonical) would give wrong results *during* the sweep; yet
we recover the all-right-canonical B form on each site at the end of the sweep.
Parameters
----------
psi, chi_max, eps:
See attributes below.
model :
The model with the Hamiltonian for time evolution as `model.H_mpo`.
Attributes
----------
psi : SimpleMPS
The current state to be evolved.
H_mpo : list of W tensors with legs ``wL wR i i*``
The Hamiltonian as an MPO.
chi_max, eps:
Truncation parameters, see :func:`a_mps.split_truncate_theta`.
Only used when we evolve two-site wave functions!
LPs, RPs : list of np.Array[ndim=3]
Left and right parts ("environments") of the effective Hamiltonian.
``LPs[i]`` is the contraction of all parts left of site `i` in the network ``<psi|H|psi>``,
and similar ``RPs[i]`` for all parts right of site `i`.
Each ``LPs[i]`` has legs ``vL wL* vL*``, ``RPs[i]`` has legs ``vR* wR* vR``
"""
def __init__(self, psi, model, chi_max, eps):
assert psi.L == model.L and psi.bc == model.bc # ensure compatibility
if psi.bc != 'finite':
raise ValueError("This TDVP implementation works only for finite MPS.")
self.H_mpo = model.H_mpo
self.psi = psi
self.LPs = [None] * psi.L
self.RPs = [None] * psi.L
self.chi_max = chi_max
self.eps = eps
# initialize left and right environment
D = self.H_mpo[0].shape[0]
chi = psi.Bs[0].shape[0]
LP = np.zeros([chi, D, chi], dtype=float) # vL wL* vL*
RP = np.zeros([chi, D, chi], dtype=float) # vR* wR* vR
LP[:, 0, :] = np.eye(chi)
RP[:, D - 1, :] = np.eye(chi)
self.LPs[0] = LP
self.RPs[-1] = RP
# initialize necessary RPs
for i in range(psi.L - 1, 0, -1):
self.update_RP(i, psi.Bs[i])
def sweep_one_site(self, dt):
"""Perform one one-site TDVP sweep to evolve |psi> -> exp(-i H_mpo dt) |psi>.
This does *not* grow the bond dimension of the MPS, but is strictly TDVP.
"""
psi = self.psi
L = self.psi.L
# sweep from left to right
theta = self.psi.get_theta1(0)
for i in range(L - 1):
theta = self.evolve_one_site(i, 0.5*dt, theta) # forward
Ai, theta = self.split_one_site_theta(i, theta, move_right=True)
# here theta is zero-site between site i and i+1
psi.Bs[i] = Ai # not in right canonical form, but expect this in right-to-left sweep
self.update_LP(i, Ai)
theta = self.evolve_zero_site(i, -0.5*dt, theta) # backward
j = i + 1
Bj = self.psi.Bs[j]
theta = np.tensordot(theta, Bj, axes=(1, 0)) # vL [vL'], [vL] j vR
# here theta is one-site on site j = i + 1
# right boundary
i = L - 1
theta = self.evolve_one_site(i, dt, theta) # forward
theta, Bi = self.split_one_site_theta(i, theta, move_right=False)
self.psi.Bs[i] = Bi
self.update_RP(i, Bi)
# sweep from right to left
for i in reversed(range(L - 1)):
theta = self.evolve_zero_site(i, -0.5*dt, theta) # backward
Ai = self.psi.Bs[i] # still in left-canonical A form from the above right-sweep!
theta = np.tensordot(Ai, theta, axes=(2, 0)) # vL i [vR], [vR'] vR
theta = self.evolve_one_site(i, 0.5*dt, theta) # forward
theta, Bi = self.split_one_site_theta(i, theta, move_right=False)
self.psi.Bs[i] = Bi
self.update_RP(i, Bi)
# The last `evolve_one_site` brought the tensor on site 0 in right-canonical B form,
# recovering the right-canonical form on each MPS tensor (as the SimpleMPS assumes).
# It splitted the very left, trivial leg off theta,
# which should only have an arbitrary phase for the left, trivial singular vector,
# and a singular value 1 (if the state is normalized).
assert theta.shape == (1, 1)
assert abs(abs(theta[0]) - 1.) < 1.e-10
# To keep track of the phase, we put it back into the tensor.
self.psi.Bs[0] *= theta[0, 0]
def sweep_two_site(self, dt):
"""Perform one two-site TDVP sweep to evolve |psi> -> exp(-i H_mpo dt) |psi>.
This can grow the bond dimension, but is *not* stricly TDVP.
"""
psi = self.psi
L = self.psi.L
# sweep from left to right
theta = self.psi.get_theta2(0)
for i in range(L - 2):
j = i + 1
k = i + 2
Ai, S, Bj = self.evolve_split_two_site(i, 0.5*dt, theta) # forward
psi.Bs[i] = Ai # not in right canonical form, but expect this in right-to-left sweep
self.update_LP(i, Ai)
theta = np.tensordot(np.diag(S), Bj, axes=(1, 0)) # vL [vL'], [vL] j vC
# here theta is one-site on site j = i + 1
theta = self.evolve_one_site(j, -0.5*dt, theta) # backward
Bk = self.psi.Bs[k]
theta = np.tensordot(theta, Bk, axes=(2, 0)) # vL j [vC], [vC] k vR
# here theta is two-site on sites j, k = i + 1, i + 2
# right boundary
i = L - 2
j = L - 1
Ai, S, Bj = self.evolve_split_two_site(i, dt, theta) # forward
theta = np.tensordot(Ai, np.diag(S), axes=(2, 0)) # vL i [vC], [vC'] vC
self.psi.Bs[j] = Bj
self.update_RP(j, Bj)
# sweep from right to left
for i in reversed(range(L - 2)):
j = i + 1
# here, theta is one-site on site j = i + 1
theta = self.evolve_one_site(j, -0.5*dt, theta) # backward
Ai = self.psi.Bs[i] # still in left-canonical A form from the above right-sweep!
theta = np.tensordot(Ai, theta, axes=(2, 0)) # vL i [vR], [vR'] vR
# here, theta is two-site on sites i, j = i, i + 1
Ai, S, Bj = self.evolve_split_two_site(i, 0.5*dt, theta) # forward
self.psi.Bs[j] = Bj
self.update_RP(j, Bj)
theta = np.tensordot(Ai, np.diag(S), axes=(2, 0)) # vL i vC, [vC'] vC
self.psi.Bs[0] = theta # this is right-canonical, because for a finite system
# the left-most virtual bond is trivial, so `theta` and `B` are the same on site 0.
# So we recovered the right-canonical form on each MPS tensor (as the SimpleMPS assumes).
def evolve_zero_site(self, i, dt, theta):
"""Evolve zero-site `theta` with SimpleHeff0 right of site `i`."""
Heff = SimpleHeff0(self.LPs[i + 1], self.RPs[i])
theta = np.reshape(theta, [Heff.shape[0]])
theta = self.expm_multiply(Heff, theta, dt)
# no truncation necessary!
return np.reshape(theta, Heff.theta_shape)
def evolve_one_site(self, i, dt, theta):
"""Evolve one-site `theta` with SimpleHeff1 on site i."""
# get effective Hamiltonian
Heff = SimpleHeff1(self.LPs[i], self.RPs[i], self.H_mpo[i])
theta = np.reshape(theta, [Heff.shape[0]])
theta = self.expm_multiply(Heff, theta, dt)
# no truncation necessary!
return np.reshape(theta, Heff.theta_shape)
def evolve_split_two_site(self, i, dt, theta):
"""Evolve two-site `theta` with SimpleHeff2 on sites i and i + 1."""
j = i + 1
# get effective Hamiltonian
Heff = SimpleHeff2(self.LPs[i], self.RPs[j], self.H_mpo[i], self.H_mpo[j])
theta = np.reshape(theta, [Heff.shape[0]]) # group legs
theta = self.expm_multiply(Heff, theta, dt)
theta = np.reshape(theta, Heff.theta_shape) # split legs
# truncation necessary!
Ai, S, Bj = a_mps.split_truncate_theta(theta, self.chi_max, self.eps)
self.psi.Ss[j] = S
return Ai, S, Bj
def split_one_site_theta(self, i, theta, move_right=True):
"""Split a one-site theta into `Ai, theta` (right move) or ``theta, Bi`` (left move)."""
chivL, d, chivR = theta.shape
if move_right:
# group i to the left
theta = np.reshape(theta, [chivL * d, chivR])
A, S, V = a_mps.svd(theta, full_matrices=False) # vL vC, vC, vC i vR
S /= np.linalg.norm(S)
self.psi.Ss[i + 1] = S
chivC = len(S) # no truncation necessary!
A = np.reshape(A, [chivL, d, chivC])
theta = np.tensordot(np.diag(S), V, axes=(1, 0)) # vC [vC'], [vC] vR
return A, theta
else:
# group i to the right
theta = np.reshape(theta, [chivL, d * chivR])
U, S, B = a_mps.svd(theta, full_matrices=False) # vL i vC, vC, vC vR
S /= np.linalg.norm(S)
self.psi.Ss[i] = S
chivC = len(S) # no truncation necessary!
B = np.reshape(B, [chivC, d, chivR])
theta = np.tensordot(U, np.diag(S), axes=(1, 0)) # vL [vC], [vC'] vC
return theta, B
def update_RP(self, i, B):
"""Calculate RP environment right of site `i-1`.
Uses RP right of `i` and the given, right-canonical `B` on site `i`."""
j = (i - 1) % self.psi.L
RP = self.RPs[i] # vR* wR* vR
# B has legs vL i vR
Bc = B.conj() # vL* i* vR*
W = self.H_mpo[i] # wL wR i i*
RP = np.tensordot(B, RP, axes=(2, 0)) # vL i [vR], [vR*] wR* vR
RP = np.tensordot(RP, W, axes=([1, 2], [3, 1])) # vL [i] [wR*] vR, wL [wR] i [i*]
RP = np.tensordot(RP, Bc, axes=([1, 3], [2, 1])) # vL [vR] wL [i], vL* [i*] [vR*]
self.RPs[j] = RP # vL wL vL* (== vR* wR* vR on site i-1)
def update_LP(self, i, A):
"""Calculate LP environment left of site `i+1`.
Uses the LP left of site `i` and the given, left-canonical `A` on site `i`."""
j = (i + 1) % self.psi.L
LP = self.LPs[i] # vL wL vL*
# A has legs vL i vR
Ac = A.conj() # vL* i* vR*
W = self.H_mpo[i] # wL wR i i*
LP = np.tensordot(LP, A, axes=(2, 0)) # vL wL* [vL*], [vL] i vR
LP = np.tensordot(W, LP, axes=([0, 3], [1, 2])) # [wL] wR i [i*], vL [wL*] [i] vR
LP = np.tensordot(Ac, LP, axes=([0, 1], [2, 1])) # [vL*] [i*] vR*, wR [i] [vL] vR
self.LPs[j] = LP # vR* wR vR (== vL wL* vL* on site i+1)
def expm_multiply(self, H, psi0, dt):
from scipy.sparse.linalg import expm_multiply
from packaging import version
if version.parse(scipy.__version__) >= version.parse('1.9.0'):
traceH = H.trace() # new argument introduced in scipy 1.9.0
return expm_multiply((-1.j*dt) * H, psi0, traceA =1.j*dt*traceH)
return expm_multiply((-1.j*dt) * H, psi0)
# # alternatively, use custom lanczos implementation
# from .lanczos import lanczos_expm_multiply
# return lanczos_expm_multiply(H, psi0, dt)
class SimpleHeff1(scipy.sparse.linalg.LinearOperator):
"""Class for the effective Hamiltonian on 1 site.
Basically the same as d_dmrg.SimpleHeff2, but acts on a single site::
.--vL* vR*--.
| i* |
| | |
(LP)----(W1)----(RP)
| | |
| i |
.--vL vR--.
"""
def __init__(self, LP, RP, W1, prefactor=1.):
self.LP = LP # vL wL* vL*
self.RP = RP # vR* wR* vR
self.W1 = W1 # wL wR i i*
chi1, chi2 = LP.shape[0], RP.shape[2]
d1 = W1.shape[2]
self.theta_shape = (chi1, d1, chi2) # vL i vR
self.shape = (chi1 * d1 * chi2, chi1 * d1 * chi2)
self.dtype = W1.dtype
def _matvec(self, theta):
"""Calculate |theta'> = H_eff |theta>."""
x = np.reshape(theta, self.theta_shape) # vL i vR
x = np.tensordot(self.LP, x, axes=(2, 0)) # vL wL* [vL*], [vL] i vR
x = np.tensordot(x, self.W1, axes=([1, 2], [0, 3])) # vL [wL*] [i] vR, [wL] wR i [i*]
x = np.tensordot(x, self.RP, axes=([1, 2], [0, 1])) # vL [vR] [wR] i, [vR*] [wR*] vR
x = np.reshape(x, self.shape[0])
return x
def _adjoint(self):
"""Define self as hermitian."""
return self
def trace(self):
"""The trace of the operator.
Only needed for expm_multiply in scipy version > 1.9.0 to avoid warnings,
but cheap to calculate anyways.
"""
return np.inner(np.trace(self.LP, axis1=0, axis2=2), # [vL] wL* [vL*]
np.dot(np.trace(self.W1, axis1=2, axis2=3), # wL wR [i] [i*]
np.trace(self.RP, axis1=0, axis2=2))) # [vR*] wR* [vR]
class SimpleHeff0(scipy.sparse.linalg.LinearOperator):
"""Class for the effective Hamiltonian.
Basically the same as d_dmrg.SimpleHeff1, but acts on the zero-site wave function::
.--vL* vR*--.
| |
| |
(LP)----------(RP)
| |
| |
.--vL vR--.
"""
def __init__(self, LP, RP, prefactor=1.):
self.LP = LP # vL wL* vL*
self.RP = RP # vR* wR* vR
chi1, chi2 = LP.shape[0], RP.shape[2]
self.theta_shape = (chi1, chi2) # vL vR
self.shape = (chi1 * chi2, chi1 * chi2)
self.dtype = LP.dtype
def _matvec(self, theta):
"""Calculate |theta'> = H_eff |theta>."""
x = np.reshape(theta, self.theta_shape) # vL vR
x = np.tensordot(self.LP, x, axes=(2, 0)) # vL wL* [vL*], [vL] vR
x = np.tensordot(x, self.RP, axes=([1, 2], [1, 0])) # vL [wL*] [vL*] , [vR*] [wR*] vR
x = np.reshape(x, self.shape[0])
return x
def _adjoint(self):
"""Define self as hermitian."""
return self
def trace(self):
"""The trace of the operator.
Only needed for expm_multiply in scipy version > 1.9.0 to avoid warnings,
but cheap to calculate anyways.
"""
return np.inner(np.trace(self.LP, axis1=0, axis2=2), # [vL] wL* [vL*]
np.trace(self.RP, axis1=0, axis2=2)) # [vR*] wR* [vR]
def example_TDVP_tf_ising_lightcone(L, g, tmax, dt, one_site=True, chi_max=50):
# compare this code to c_tebd.example_TEBD_tf_ising_lightcone - it's almost the same.
print("finite TEBD, real time evolution, transverse field Ising")
print("L={L:d}, g={g:.2f}, tmax={tmax:.2f}, dt={dt:.3f}".format(L=L, g=g, tmax=tmax, dt=dt))
# find ground state with TEBD or DMRG
# E, psi, model = example_TEBD_gs_tf_ising_finite(L, g)
from .d_dmrg import example_DMRG_tf_ising_finite
E, psi, model = example_DMRG_tf_ising_finite(L, g)
i0 = L // 2
# apply sigmax on site i0
SxB = np.tensordot(model.sigmaz, psi.Bs[i0], axes=(1, 1)) # i [i*], vL [i] vR
psi.Bs[i0] = np.transpose(SxB, [1, 0, 2]) # vL i vR
E = np.sum(psi.bond_expectation_value(model.H_bonds))
print("E after applying Sz = {E:.13f}".format(E=E))
eng = SimpleTDVPEngine(psi, model, chi_max=chi_max, eps=1.e-7)
S = [psi.entanglement_entropy()]
Nsteps = int(tmax / dt + 0.5)
for n in range(Nsteps):
if abs((n * dt + 0.1) % 0.2 - 0.1) < 1.e-10:
print("t = {t:.2f}, chi =".format(t=n * dt), psi.get_chi())
if one_site:
eng.sweep_one_site(dt)
else:
eng.sweep_two_site(dt)
S.append(psi.entanglement_entropy())
import matplotlib.pyplot as plt
plt.figure()
plt.imshow(S[::-1],
vmin=0.,
aspect='auto',
interpolation='nearest',
extent=(0, L - 1., -0.5 * dt, (Nsteps + 0.5) * dt))
plt.xlabel('site $i$')
plt.ylabel('time $t/J$')
plt.ylim(0., tmax)
plt.colorbar().set_label('entropy $S$')
E = np.sum(psi.bond_expectation_value(model.H_bonds))
print("final E = {E:.13f}".format(E=E))