r"""Time evolving block decimation (TEBD) for MPS of purification.
See introduction in :mod:`~tenpy.networks.purification_mps`.
Time evolution for finite-temperature ensembles.
This can be used to obtain correlation functions in time.
"""
# Copyright 2018-2019 TeNPy Developers, GNU GPLv3
from . import tebd
from ..linalg import np_conserved as npc
from .truncation import svd_theta, TruncationError
from ..tools.params import get_parameter
from ..tools.math import entropy
from ..linalg import random_matrix as rand_mat
import numpy as np
__all__ = [
'PurificationTEBD', 'PurificationTEBD2', 'Disentangler', 'BackwardDisentangler',
'RenyiDisentangler', 'NormDisentangler', 'DiagonalizeDisentangler',
'GradientDescentDisentangler', 'NoiseDisentangler', 'LastDisentangler',
'CompositeDisentangler', 'MinDisentangler', 'disentanglers_atom_parse_dict', 'get_disentangler'
]
[docs]class PurificationTEBD(tebd.Engine):
r"""Time evolving block decimation (TEBD) for purification MPS.
Parameters
----------
psi : :class:`~tenpy.networs.purification_mps.PurificationMPS`
Initial state to be time evolved. Modified in place.
model : :class:`~tenpy.models.NearestNeighborModel`
The model representing the Hamiltonian for which we want to find the ground state.
TEBD_params : dict
Further optional parameters as described in the following table.
Use ``verbose=1`` to print the used parameters during runtime.
See :func:`run` and :func:`run_GS` for more details.
Attributes
----------
disent_iterations
used_disentangler : :class:`Disentangler`
The disentangler to be used on the auxiliar indices.
Chosen by :func:`get_disentangler`, called with the TEBD parameter ``'disentangle'``.
Defaults to the trivial disentangler for ``TEBD_params['disentangle']=None``.
_disent_iterations : 1D ndarray
Number of iterations performed on all bonds, including trivial bonds; lenght `L`.
_guess_U_disent : list of list of npc.Array
Same index strucuture as `self._U`: for each two-site U of the physical time evolution
the disentangler from the last application. Initialized to identities.
"""
def __init__(self, psi, model, TEBD_params):
super().__init__(psi, model, TEBD_params)
self._disent_iterations = np.zeros(psi.L)
self._guess_U_disent = None # will be set in calc_U
method = get_parameter(self.TEBD_params, 'disentangle', None, 'PurificationTEBD')
self.used_disentangler = get_disentangler(str(method), self)
[docs] def run_imaginary(self, beta):
"""Run imaginary time evolution to cool down to the given `beta`.
Note that we don't change the `norm` attribute of the MPS, i.e. normalization is preserved.
Parameters
----------
beta : float
The inverse temperature `beta` = 1/T, by which we should cool down.
We evolve to the closest multiple of ``TEBD_params['dt']``,
see also :attr:`evolved_time`.
"""
delta_t = get_parameter(self.TEBD_params, 'dt', 0.1, 'PurificationTEBD')
TrotterOrder = 2 # currently, imaginary time evolution works only for second order.
self.calc_U(TrotterOrder, delta_t, type_evo='imag')
self.update_imag(N_steps=int(beta / delta_t + 0.5))
if self.verbose >= 1:
E = np.average(self.model.bond_energies(self.psi))
S = np.average(self.psi.entanglement_entropy())
print("--> time={t:.6f}, E_bond={E:.10f}, S={S:.10f}".format(t=self.evolved_time,
E=E.real,
S=S.real))
@property
def disent_iterations(self):
"""For each bond the total number of iterations performed in any :class:`Disentangler`."""
return self._disent_iterations[self.psi.nontrivial_bonds]
[docs] def calc_U(self, order, delta_t, type_evo='real', E_offset=None):
"""see :meth:`~tenpy.algorithms.tebd.eng.calc_U`"""
super().calc_U(order, delta_t, type_evo, E_offset)
self._guess_U_disent = [[None] * len(Us) for Us in self._U]
[docs] def update_bond(self, i, U_bond):
"""Updates the B matrices on a given bond.
Function that updates the B matrices, the bond matrix s between and the
bond dimension chi for bond i. This would look something like::
| | |
| ... - B1 - s - B2 - ...
| | |
| |-------------|
| | U |
| |-------------|
| | |
Parameters
----------
i : int
Bond index; we update the matrices at sites ``i-1, i``.
U_bond : :class:`~tenpy.linalg.np_conserved.Array`
The bond operator which we apply to the wave function.
We expect labels ``'p0', 'p1', 'p0*', 'p1*'`` for `U_bond`.
Returns
-------
trunc_err : :class:`~tenpy.algorithms.truncation.TruncationError`
The error of the represented state which is introduced by the truncation
during this update step.
"""
i0, i1 = i - 1, i
if self.verbose >= 30:
print("Update sites ({0:d}, {1:d})".format(i0, i1))
# Construct the theta matrix
theta = self.psi.get_theta(i0, n=2) # 'vL', 'vR', 'p0', 'p1', 'q0', 'q1'
theta = npc.tensordot(U_bond, theta, axes=(['p0*', 'p1*'], ['p0', 'p1']))
# ##### new hook compared to tebd.Engine.calc_U
theta, U_disent = self.disentangle(theta)
# ####
theta = theta.combine_legs([('vL', 'p0', 'q0'), ('vR', 'p1', 'q1')], qconj=[+1, -1])
# Perform the SVD and truncate the wavefunction
U, S, V, trunc_err, renormalize = svd_theta(theta,
self.trunc_params,
inner_labels=['vR', 'vL'])
# bring back to right-canonical 'B' form and update matrices
B_R = V.split_legs(1).ireplace_labels(['p1', 'q1'], ['p', 'q'])
# In general, we want to do the following:
# B_L = U.iscale_axis(S, 'vR')
# B_L = B_L.split_legs(0).iscale_axis(self.psi.get_SL(i0)**(-1), 'vL')
# B_L = B_L.ireplace_labels(['p0', 'q0'], ['p', 'q'])
# i.e. with SL = self.psi.get_SL(i0), we have ``B_L = SL**(-1) U S``
# However, the inverse of SL is problematic, as it might contain very small singular
# values. Instead, we calculate ``C == SL**-1 theta == SL**-1 U S V``,
# such that we obtain ``B_L = SL**-1 U S = SL**-1 U S V V^dagger = C V^dagger``
C = self.psi.get_theta(i0, n=2, formL=0.)
# here, C is the same as theta, but without the `S` on the very left
# (Note: this requires no inverse if the MPS is initially in 'B' canonical form)
C = npc.tensordot(U_bond, C, axes=(['p0*', 'p1*'], ['p0', 'p1'])) # apply U as for theta
if U_disent is not None:
C = npc.tensordot(U_disent, C, axes=[['q0*', 'q1*'], ['q0', 'q1']])
B_L = npc.tensordot(C.combine_legs(('vR', 'p1', 'q1'), pipes=theta.legs[1]),
V.conj(),
axes=['(vR.p1.q1)', '(vR*.p1*.q1*)'])
B_L.ireplace_labels(['vL*', 'p0', 'q0'], ['vR', 'p', 'q'])
B_L /= renormalize # re-normalize to <psi|psi> = 1
self.psi.set_SR(i0, S)
self.psi.set_B(i0, B_L, form='B')
self.psi.set_B(i1, B_R, form='B')
self._trunc_err_bonds[i] = self._trunc_err_bonds[i] + trunc_err
return trunc_err
[docs] def update_bond_imag(self, i, U_bond):
"""Update a bond with a (possibly non-unitary) `U_bond`.
Similar as :meth:`update_bond`; but after the SVD just keep the `A, S, B` canonical form.
In that way, one can sweep left or right without using old singular values,
thus preserving the canonical form during imaginary time evolution.
Parameters
----------
i : int
Bond index; we update the matrices at sites ``i-1, i``.
U_bond : :class:`~tenpy.linalg.np_conserved.Array`
The bond operator which we apply to the wave function.
We expect labels ``'p0', 'p1', 'p0*', 'p1*'``.
Returns
-------
trunc_err : :class:`~tenpy.algorithms.truncation.TruncationError`
The error of the represented state which is introduced by the truncation
during this update step.
"""
i0, i1 = i - 1, i
if self.verbose >= 100:
print("Update sites ({0:d}, {1:d})".format(i0, i1))
# Construct the theta matrix
theta = self.psi.get_theta(i0, n=2) # 'vL', 'vR', 'p0', 'q0', 'p1', 'q1'
theta = npc.tensordot(U_bond, theta, axes=(['p0*', 'p1*'], ['p0', 'p1']))
theta = theta.combine_legs([('vL', 'p0', 'q0'), ('vR', 'p1', 'q1')], qconj=[+1, -1])
# Perform the SVD and truncate the wavefunction
U, S, V, trunc_err, renormalize = svd_theta(theta,
self.trunc_params,
inner_labels=['vR', 'vL'])
# Split legs and update matrices
B_R = V.split_legs(1).ireplace_labels(['p1', 'q1'], ['p', 'q'])
A_L = U.split_legs(0).ireplace_labels(['p0', 'q0'], ['p', 'q'])
self.psi.set_SR(i0, S)
self.psi.set_B(i0, A_L, form='A')
self.psi.set_B(i1, B_R, form='B')
self._trunc_err_bonds[i] = self._trunc_err_bonds[i] + trunc_err
return trunc_err
[docs] def disentangle(self, theta):
r"""Disentangle `theta` before splitting with svd.
For the purification we write :math:`\rho_P = Tr_Q{|\psi_{P,Q}><\psi_{P,Q}|}`. Thus, we
can actually apply any unitary to the auxiliar `Q` space of :math:`|\psi>` without
changing the result.
.. note ::
We have to apply the *same* unitary to the 'bra' and 'ket' used for expectation values
/ correlation functions!
The behaviour of this function is set by :attr:`used_disentangler`,
which in turn is obtained from ``get_disentangler(TEBD_params['disentangle'])``,
see :func:`get_disentangler` for details on the syntax.
Parameters
----------
theta : :class:`~tenpy.linalg.np_conserved.Array`
Wave function to disentangle, with legs ``'vL', 'vR', 'p0', 'p1', 'q0', 'q1'``.
Returns
-------
theta_disentangled : :class:`~tenpy.linalg.np_conserved.Array`
Disentangled `theta`; ``npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']])``.
U : :class:`~tenpy.linalg.conserved.Array`
The unitary used to disentangle `theta`, with labels ``'q0', 'q1', 'q0*', 'q1*'``.
If no unitary was found/applied, it might also be ``None``.
"""
theta, U = self.used_disentangler(theta)
U_idx_dt, i = self._update_index
if U_idx_dt is not None:
self._guess_U_disent[U_idx_dt][i] = U # save result as guess for `LastDisentangler`
return theta, U
[docs] def disentangle_global(self, pair=None):
"""Try global disentangling by determining the maximally entangled pairs of sites.
Caclulate the mutual information (in the auxiliar space) between two sites and determine
where it is maximal. Disentangle these two sites with :meth:`disentangle`
"""
max_range = get_parameter(self.TEBD_params, 'disent_gl_maxrange', 10, 'PurificationTEBD')
if pair is None:
coords, mutinf = self.psi.mutinf_two_site(max_range, legs='q')
# TODO: recalculate mutinf only as necessary and do multiple steps at once...
sorted = np.argsort(mutinf)
pair = coords[sorted[-1]]
i, j = pair
# for i, j in coords[sorted[-1:]]:
if self.verbose > 10:
print('disentangle global pair ' + repr((i, j)))
self._disentangle_two_site(i, j)
return i, j # TODO
# done
[docs] def disentangle_global_nsite(self, n=2):
"""Perform a sweep through the system and disentangle with :meth:`disentangle_n_site`.
Parameters
----------
n: int
maximal number of sites to disentangle at once.
"""
for i in range(0, self.psi.L - n + 1): # sweep left to right
self._update_index = None, i
theta = self.psi.get_theta(i, n=n)
self.disentangle_n_site(i, n, theta) # works recursively
for i in range(self.psi.L - n, -1, -1): # sweep right to left
self._update_index = None, i
theta = self.psi.get_theta(i, n=n)
self.disentangle_n_site(i, n, theta) # works recursively
self._update_index = None
[docs] def disentangle_n_site(self, i, n, theta):
r"""Generalization of :meth:`disentangle` to `n` sites.
Simply group left and right `n`/2 physical legs, adjust labels, and
apply :meth:`disentangle` to disentangle the central bond.
Recursively proceed to disentangle left and right parts afterwards.
Scales (for even `n`) as :math:`O(\chi^3 d^n d^{n/2})`.
"""
assert (n >= 2)
n1 = n // 2
n2 = n - n1
p = ['p' + str(j) for j in range(n)] # labels of theta to be separated
q = ['q' + str(j) for j in range(n)]
pL, pR = p[:n1], p[n1:]
qL, qR = q[:n1], q[n1:]
theta = theta.combine_legs([pL, qL, pR, qR], qconj=[+1, -1, +1, -1], new_axes=[1, 2, 3, 4])
_, p0, q0, p1, q1, _ = theta.get_leg_labels() # keep the labels for later
theta.ireplace_labels([p0, q0, p1, q1], ['p0', 'q0', 'p1', 'q1'])
theta, _ = self.disentangle(theta) # apply two-site disentangling
theta = theta.combine_legs([('vL', 'p0', 'q0'), ('vR', 'p1', 'q1')], qconj=[+1, -1])
# Perform the SVD and truncate the wavefunction
U, S, V, trunc_err, renormalize = svd_theta(theta,
self.trunc_params,
inner_labels=['vR', 'vL'])
self.psi.set_SL(i + n1, S) # update S
if n1 == 1:
# save U as left B in psi
U = U.split_legs(0).ireplace_labels(['p0', 'q0'], ['(p)', '(q)'])
U = U.split_legs(['(p)', '(q)'])
self.psi.set_B(i, U, form='A') # TODO: might want to do this inversion-free?
else:
# disentangle left n1-site wave function recursively
theta_L = U.iscale_axis(S, 1).split_legs(0)
theta_L = theta_L.ireplace_labels(['p0', 'q0'], [p0, q0]).split_legs([p0, q0])
self.disentangle_n_site(i, n1, theta_L)
if n2 == 1:
# save V as right B in psi
V = V.split_legs(1).ireplace_labels(['p1', 'q1'], ['(p)', '(q)'])
V = V.split_legs(['(p)', '(q)'])
self.psi.set_B(i + n1, V, form='B')
else:
# disentangle right n2-site wave function recursively
theta_R = V.iscale_axis(S, 0).split_legs(1)
theta_R = theta_R.ireplace_labels(['p1', 'q1'], [p1, q1]).split_legs([p1, q1])
theta_R.ireplace_labels(pR, p[:n2]).ireplace_labels(qR, q[:n2])
self.disentangle_n_site(i + n1, n2, theta_R)
def _disentangle_two_site(self, i, j):
"""swap until i and j are next to each other and use :meth:`disentangle`; swap back."""
on_way = get_parameter(self.TEBD_params, 'disent_gl_on_swap', False, 'PurificationTEBD')
if not self.psi.finite:
raise NotImplementedError # adjust: what's the shortest path?
assert (i < j)
for j0 in range(j, i + 1, -1): # j0 = current site of `j`
# originial leg `j` is at j0
self._update_index = None, j0
self._swap_disentangle_bond(j0, swap=True, disentangle=False) # swap j0-1, j0
# originial leg is at `j0-1`
# disentangle i, i+1
self._update_index = None, i + 1
self._swap_disentangle_bond(i + 1, swap=False, disentangle=True)
for j0 in range(i + 1, j): # j0 = current site of `j`
# originial leg `j` is at j0
self._update_index = None, j0 + 1
self._swap_disentangle_bond(j0 + 1, disentangle=on_way) # swap j0, j0+1
# originial leg is at `j0+1`
self._update_index = None # done
def _swap_disentangle_bond(self, i, swap=True, disentangle=False):
"""swap sites (i-1, i) (if swap = True) """
# very similar to update_bond
i0, i1 = i - 1, i
if self.verbose >= 30:
print("Update sites ({0:d}, {1:d}), swap={2!s}, disentangle={3!s}".format(
i0, i1, swap, disentangle))
# Construct the theta matrix
theta = self.psi.get_theta(i0, n=2) # 'vL', 'vR', 'p0', 'p1', 'q0', 'q1'
if swap:
theta.ireplace_labels(['p0', 'q0', 'p1', 'q1'], ['p1', 'q1', 'p0', 'q0'])
if disentangle:
theta, U_disent = self.disentangle(theta)
theta = theta.combine_legs([('vL', 'p0', 'q0'), ('vR', 'p1', 'q1')], qconj=[+1, -1])
# Perform the SVD and truncate the wavefunction
U, S, V, trunc_err, renormalize = svd_theta(theta,
self.trunc_params,
inner_labels=['vR', 'vL'])
# bring back to right-canonical 'B' form and update matrices
B_R = V.split_legs(1).ireplace_labels(['p1', 'q1'], ['p', 'q'])
C = self.psi.get_theta(i0, n=2, formL=0.)
if swap:
C.ireplace_labels(['p0', 'q0', 'p1', 'q1'], ['p1', 'q1', 'p0', 'q0'])
if disentangle and U_disent is not None:
C = npc.tensordot(U_disent, C, axes=[['q0*', 'q1*'], ['q0', 'q1']])
B_L = npc.tensordot(C.combine_legs(('vR', 'p1', 'q1'), pipes=theta.legs[1]),
V.conj(),
axes=['(vR.p1.q1)', '(vR*.p1*.q1*)'])
B_L.ireplace_labels(['vL*', 'p0', 'q0'], ['vR', 'p', 'q'])
B_L /= renormalize # re-normalize to <psi|psi> = 1
self.psi.set_SR(i0, S)
self.psi.set_B(i0, B_L, form='B')
self.psi.set_B(i1, B_R, form='B')
self._trunc_err_bonds[i] = self._trunc_err_bonds[i] + trunc_err
return trunc_err
[docs]class PurificationTEBD2(PurificationTEBD):
"""Similar as PurificationTEBD, but perform sweeps instead of brickwall.
Instead of the A-B pattern of even/odd bonds used in TEBD, perform sweeps similar as in DMRG
for real-time evolution (similar as :meth:`~tenpy.algorithms.tebd.Engine.update_imag` does for
imaginary time evolution).
"""
[docs] def update(self, N_steps):
"""Evolve by ``N_steps * U_param['dt']``.
Parameters
----------
N_steps : int
The number of steps for which the whole lattice should be updated.
Returns
-------
trunc_err : :class:`~tenpy.algorithms.truncation.TruncationError`
The error of the represented state which is introduced due to the truncation during
this sequence of update steps.
"""
trunc_err = TruncationError()
order = self._U_param['order']
assert (order == 2 and self.psi.finite)
for i in range(N_steps):
trunc_err += self.update_step(0, False)
trunc_err += self.update_step(0, True)
self.evolved_time = self.evolved_time + N_steps * self._U_param['tau']
self.trunc_err = self.trunc_err + trunc_err # not += : make a copy!
# (this is done to avoid problems of users storing self.trunc_err after each `update`)
return trunc_err
[docs] def update_step(self, U_idx_dt, odd):
"""Updates bonds in unit cell.
Depending on the choice of `odd`, perform a sweep to the left or right,
updating once per site with a time step given by U_idx_dt.
Parameters
----------
U_idx_dt : int
Time step index in ``self._U``,
evolve with ``Us[i] = self.U[U_idx_dt][i]`` at bond ``(i-1,i)``.
odd : bool/int
Indication of whether to update even (``odd=False,0``) or even (``odd=True,1``) sites
Returns
-------
trunc_err : :class:`~tenpy.algorithms.truncation.TruncationError`
The error of the represented state which is introduced due to the truncation
during this sequence of update steps.
"""
Us = self._U[U_idx_dt]
trunc_err = TruncationError()
if odd:
sweep = range(1, self.psi.L) # start with 1: only finite!
else:
sweep = range(self.psi.L - 1, 0, -1)
for i_bond in sweep:
if Us[i_bond] is None:
if self.verbose >= 10:
print("Skip U_bond element:", i_bond)
continue # handles finite vs. infinite boundary conditions
if self.verbose >= 10:
print("Apply U_bond element", i_bond)
self._update_index = (U_idx_dt, i_bond)
trunc_err += self.update_bond(i_bond, Us[i_bond])
self._update_index = None
return trunc_err
[docs]class Disentangler:
r"""Prototype for a disentangler. Trivial, does nothing.
In purification, we write :math:`\rho_P = Tr_Q{|\psi_{P,Q}><\psi_{P,Q}|}`. Thus, we
can actually apply any unitary to the auxiliar `Q` space of :math:`|\psi>` without
changing the physical expectation values.
.. note ::
We have to apply the *same* unitary to the 'bra' and 'ket' used for expectation values
/ correlation functions!
However, the unitary can strongly influence the entanglement structure of :math:`|\psi>`.
Therefore, the :class:`PurificationTEBD` includes a hook in
:meth:`PurificationTEBD.update_bond` (and similar methods) to find and apply a disentangling
unitary to the auxiliar indices of a two-site wave function by calling (``__call__`` method)
a `Disentangler`.
This class is a 'trivial' disentangler which does *nothing* to the two-site wave function;
derived classes use different strategies to find various disentanglers.
Parameters
----------
parent : :class:`~tenpy.algorithms.tebd.Engine`
The parent class calling the disentangler.
Attributes
----------
parent : :class:`~tenpy.algorithms.tebd.Engine`
The parent class calling the disentangler.
"""
def __init__(self, parent):
self.parent = parent
def __call__(self, theta):
"""Find and apply a unitary to disentangle `theta`.
Parameters
----------
theta : :class:`~tenpy.linalg.np_conserved.Array`
Wave function to disentangle, with legs ``'vL', 'vR', 'p0', 'p1', 'q0', 'q1'``.
Returns
-------
theta_disentangled : :class:`~tenpy.linalg.np_conserved.Array`
Disentangled `theta`; ``npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']])``.
U : :class:`~tenpy.linalg.conserved.Array` | None
The unitary used to disentangle `theta`, with labels ``'q0', 'q1', 'q0*', 'q1*'``.
If no unitary was found/applied, it might also be ``None``.
"""
# do nothing
return theta, None
[docs]class BackwardDisentangler(Disentangler):
"""Disentangle with backward time evolution.
See [Karrasch2013]_ for details; only useful during real-time evolution.
For the infinite temperature state, ``theta = delta_{p0, q0}*delta_{p1, q1}``.
Thus, an application of `U_bond` to ``p0, p1`` can be reverted completely by applying
``U_bond^{dagger}`` to ``q0, q1``, resulting in the same state.
This works also for finite temperatures, since `exp(-beta H)` and `exp(-i H t)` commute.
Once we apply an operator to measure correlation function, the disentangling
breaks down, yet for a local operator only in it's light-cone.
Arguments and return values are the same as for :class:`Disentangler`.
"""
def __call__(self, theta):
eng = self.parent
if eng._U_param['type_evo'] == 'imag':
return theta, None # doesn't work for this...
U_idx_dt, i = eng._update_index
U = eng._U[U_idx_dt][i].conj()
U.ireplace_labels(['p0*', 'p1*', 'p0', 'p1'], ['q0', 'q1', 'q0*', 'q1*'])
theta = npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']])
return theta, U
[docs]class RenyiDisentangler(Disentangler):
"""Iterative find `U` which minimized the second Renyi entropy.
See [Hauschild2018]_
Reads of the following `TEBD_params` as break criteria for the iteration:
================ ====== ======================================================
key type description
================ ====== ======================================================
disent_eps float Break, if the change in the Renyi entropy ``S(n=2)``
per iteration is smaller than this value.
---------------- ------ ------------------------------------------------------
disent_max_iter float Maximum number of iterations to perform.
================ ====== ======================================================
Arguments and return values are the same as for :meth:`disentangle`.
"""
def __init__(self, parent):
self.max_iter = get_parameter(parent.TEBD_params, 'disent_max_iter', 20,
'PurificationTEBD')
self.eps = get_parameter(parent.TEBD_params, 'disent_eps', 1.e-10, 'PurificationTEBD')
self.parent = parent
def __call__(self, theta):
"""Find optimal `U` which minimizes the second Renyi entropy."""
U_idx_dt, i = self.parent._update_index
U = npc.outer(
npc.eye_like(theta, 'q0').iset_leg_labels(['q0', 'q0*']),
npc.eye_like(theta, 'q1').iset_leg_labels(['q1', 'q1*']))
Sold = np.inf
S0 = None
for j in range(self.max_iter):
S, U = self.iter(theta, U)
if S0 is None:
S0 = S
if abs(Sold - S) < self.eps:
break
Sold, S = S, Sold
theta = npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']])
self.parent._disent_iterations[i] += j # save the number of iterations performed
if self.parent.verbose >= 10:
print("disentangle renyi: {j:d} iterations, Sold-S = {DS:.3e}".format(j=j,
DS=S0 - Sold))
return theta, U
[docs] def iter(self, theta, U):
r"""Given `theta` and `U`, find another `U` which reduces the 2nd Renyi entropy.
Temporarily view the different `U` as independt and mimizied one of them -
this corresponds to a linearization of the cost function.
Defining `Utheta` as the application of `U` to `theata`, and combining the `p` legs of
`theta` with ``'vL', 'vR'``, this function contracts::
| .----theta----.
| | | | |
| | q0 q1 |
| | |
| | q1* |
| | | |
| | .-Utheta*-.
| | | |
| | .-Utheta--.
| | | |
| | q0* | |
| | | | |
| .----Utheta*-.
The trace yields the second Renyi entropy `S2`. Further, we calculate the unitary `U`
with maximum overlap with this network.
Parameters
----------
theta : :class:`~tenpy.linalg.np_conserved.Array`
Two-site wave function to be disentangled.
U : :class:`~tenpy.linalg.np_conserved.Array`
The previous guess for `U`; with legs ``'q0', 'q1', 'q0*', 'q1*'``.
Returns
-------
S2 : float
Renyi entopy (n=2), :math:`S2 = \frac{1}{1-2} \log tr(\rho_L^2)` of `U theta`.
new_U : :class:`~tenpy.linalg.np_conserved.Array`
Unitary with legs ``'q0', 'q1', 'q0*', 'q1*'``, which should disentangle `theta`.
"""
U_theta = npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']])
# same legs as theta: 'vL', 'p0', 'q0', 'p1', 'q1', 'vR'
# contract diagram from bottom to top
dS = npc.tensordot(U_theta,
U_theta.conj(),
axes=[['p1', 'q1', 'vR'], ['p1*', 'q1*', 'vR*']])
# dS has legs 'vL', 'p0', 'q0', 'vL*', 'p0*', 'q0*'
dS = npc.tensordot(U_theta.conj(), dS, axes=[['vL*', 'p0*', 'q0*'], ['vL', 'p0', 'q0']])
# dS has legs 'vL', 'p0', 'q0', 'vR', 'p1', 'q1'
dS = npc.tensordot(theta,
dS,
axes=[['vL', 'p0', 'vR', 'p1'], ['vL*', 'p0*', 'vR*', 'p1*']])
S2 = npc.inner(U, dS, axes=[['q0', 'q1', 'q0*', 'q1*'], ['q0*', 'q1*', 'q0', 'q1']])
# dS has legs 'q0', 'q1', 'q0*', 'q1*'
dS = dS.combine_legs([['q0', 'q1'], ['q0*', 'q1*']], qconj=[+1, -1])
# Find unitary which maximizes `trace(U dS)`.
W, Y, VH = npc.svd(dS)
new_U = npc.tensordot(W, VH, axes=[1, 0]).conj() # == V W^dagger.
# this yields trace(U dS) = trace(Y), which is maximal.
return -np.log(S2.real), new_U.split_legs([0, 1])
[docs]class NormDisentangler(Disentangler):
"""Find optimal `U` for which the truncation of U|theta> has maximal overlap with U|theta>.
Reads of the following `TEBD_params` as break criteria for the iteration:
================ ========= ======================================================
key type description
================ ========= ======================================================
disent_eps float Break, if the change in the Renyi entropy ``S(n=2)``
per iteration is smaller than this value.
---------------- --------- ------------------------------------------------------
disent_max_iter float Maximum number of iterations to perform.
---------------- --------- ------------------------------------------------------
disent_trunc_par dict Truncation parameters; defaults to `trunc_params`.
---------------- --------- ------------------------------------------------------
disent_norm_chi iterable To find the optimal U it can help to increase `chi_max`
of `disent_trunc_par` slowly, the default is
``range(1, disent_trunc_par['chi_max']+1)``.
However, that's **very** slow for large `chi_max`,
so we allow to change it. (In fact, it makes the
disentangler *scale* worse than the rest of TEBD.)
================ ========= ======================================================
Arguments and return values are the same as for :meth:`disentangle`.
"""
def __init__(self, parent):
self.max_iter = get_parameter(parent.TEBD_params, 'disent_max_iter', 20,
'PurificationTEBD')
self.eps = get_parameter(parent.TEBD_params, 'disent_eps', 1.e-10, 'PurificationTEBD')
self.trunc_par = get_parameter(parent.TEBD_params, 'disent_trunc_par', parent.trunc_params,
'PurificationTEBD')
self.chi_max = get_parameter(self.trunc_par, 'chi_max', 100, 'PurificationTEBD')
self.trunc_cut = get_parameter(self.trunc_par, 'trunc_cut', None, 'PurificationTEBD')
self.chi_range = get_parameter(self.trunc_par, 'disent_norm_chi',
range(1, self.chi_max + 1), 'PurificationTEBD')
self.parent = parent
def __call__(self, theta):
_, i = self.parent._update_index
U = npc.outer(
npc.eye_like(theta, 'q0').iset_leg_labels(['q0', 'q0*']),
npc.eye_like(theta, 'q1').iset_leg_labels(['q1', 'q1*']))
err = None
trunc_par = self.trunc_par.copy()
for chi_opt in self.chi_range:
trunc_par['chi_max'] = chi_opt
for j in range(self.max_iter):
err2, U = self.iter(theta, U, trunc_par)
if err is not None and abs(err.eps - err2.eps) <= err.eps * self.eps:
break
err = err2
if self.trunc_cut is not None:
if err2.eps < self.trunc_cut * self.trunc_cut:
break
theta = npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']])
self.parent._disent_iterations[i] += j # save the number of iterations performed
if self.parent.verbose >= 10:
print("disentangle norm: {j:d} iterations, err={err!s}".format(j=j, err=err))
return theta, U
[docs] def iter(self, theta, U, trunc_params):
r"""Given `theta` and `U`, find `U2` maximizing ``<theta|U2 truncate(U |theta>)``.
Finds unitary `U2` which maximizes Tr(U
Parameters
----------
theta : :class:`~tenpy.linalg.np_conserved.Array`
Two-site wave function to be disentangled.
U : :class:`~tenpy.linalg.np_conserved.Array`
The previous guess for `U`; with legs ``'q0', 'q1', 'q0*', 'q1*'``.
trunc_params : dict
The truncation parameters (similar as `self.trunc_params`) used to truncate `U|theta>`.
Returns
-------
trunc_err : TruncationError
Norm error discarded during the truncation of ``U|theta>``.
new_U : :class:`~tenpy.linalg.np_conserved.Array`
Unitary with legs ``'q0', 'q1', 'q0*', 'q1*'``.
Chosen such that ``new_U|theta>`` has maximal overlap with the truncated ``U|theta>``.
"""
U_theta = npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']])
lambda_ = U_theta.combine_legs([['vL', 'p0', 'q0'], ['vR', 'p1', 'q1']], qconj=[+1, -1])
X, Y, Z, err, _ = svd_theta(lambda_, trunc_params)
lambda_ = npc.tensordot(X.scale_axis(Y), Z, axes=1).split_legs()
dS = npc.tensordot(theta,
lambda_.conj(),
axes=[['vL', 'vR', 'p0', 'p1'], ['vL*', 'vR*', 'p0*', 'p1*']])
# dS has legs 'q0', 'q1', 'q0*', 'q1*'
dS = dS.combine_legs([['q0', 'q1'], ['q0*', 'q1*']], qconj=[+1, -1])
# Find unitary U2 which maximizes `trace(U dS)`.
W, Y, VH = npc.svd(dS)
new_U = npc.tensordot(W, VH, axes=[1, 0]).conj() # == V W^dagger.
# this yields trace(U dS) = trace(Y), which is maximal.
return err, new_U.split_legs([0, 1])
[docs]class GradientDescentDisentangler(Disentangler):
"""Gradient-descent optimization, similar to :class:`RenyiDisentangler`.
Reads of the following `TEBD_params`:
================ ====== ======================================================
key type description
================ ====== ======================================================
disent_eps float Break, if the change in the Renyi entropy ``S(n=2)``
per iteration is smaller than this value.
---------------- ------ ------------------------------------------------------
disent_max_iter float Maximum number of iterations to perform.
---------------- ------ ------------------------------------------------------
disent_n float Renyi index of the entropy to be used.
``n=1`` for von-Neumann entropy.
================ ====== ======================================================
Arguments and return values are the same as for :class:`Disentangler`.
"""
def __init__(self, parent):
self.max_iter = get_parameter(parent.TEBD_params, 'disent_max_iter', 20,
'PurificationTEBD')
self.eps = get_parameter(parent.TEBD_params, 'disent_eps', 1.e-10, 'PurificationTEBD')
self.n = get_parameter(parent.TEBD_params, 'disent_n', 1., 'PurificationTEBD')
self.stepsizes = get_parameter(parent.TEBD_params, 'disent_stepsizes', [0.2, 1., 2.],
'PurificationTEBD')
self.parent = parent
def __call__(self, theta):
U_idx_dt, i = self.parent._update_index
Utot = None
Sold = np.inf
S0 = None
for j in range(self.max_iter):
S, theta, U = self.iter(theta)
if Utot is None:
Utot = U
else:
Utot = npc.tensordot(U, Utot, axes=[['q0*', 'q1*'], ['q0', 'q1']])
if S0 is None:
S0 = S
if abs(Sold - S) < self.eps:
break
Sold, S = S, Sold
theta = npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']])
self.parent._disent_iterations[i] += j # save the number of iterations performed
if self.parent.verbose >= 10:
print("disentangle renyi: {j:d} iterations, Sold-S = {DS:.3e}".format(j=j,
DS=S0 - Sold))
return theta, U
[docs] def iter(self, theta):
r"""Given `theta`, find a unitary `U` towards minimizing the n-th Renyi entropy.
This function calulates the gradiant :math:`dS = \partial S(U theta, n) /\partial U`.
and then ``U(t) = exp(-t*dS)``, where we choose the `t` from stepsizes which
minimizes the entropy of ``U(t) theta``.
When ``R[i]`` is the derivative :math:`\partial S(Y, n) \partial Y_i` of the (n-th Renyi)
entropy, ``dS`` is given by::
| .----X--R--Z----.
| | | | |
| | q0 q1 |
| | |
| | q0* q1* |
| | | | |
| .----X*-Y--Z*---.
Parameters
----------
theta : :class:`~tenpy.linalg.np_conserved.Array`
Two-site wave function to be disentangled
Returns
-------
S : float
n-th Renyi entopy of new_theta
theta : :class:`~tenpy.linalg.np_conserved.Array`
The *disentangled* wave function ``new_U theta``.
new_U : :class:`~tenpy.linalg.np_conserved.Array`
Unitary with legs ``'q0', 'q1', 'q0*', 'q1*'``, which was used to disentangle `theta`.
"""
theta2 = theta.combine_legs([('vL', 'p0', 'q0'), ('vR', 'p1', 'q1')], qconj=[+1, -1])
X, Y, Z = npc.svd(theta2, inner_labels=['vR', 'vL'])
n = self.n
if n == 1:
r = Y * np.log(Y) * 2
r[Y < 1.e-14] = 0.
# S = -np.inner(Y**2, np.log(Y**2))
else:
Y[Y < 1.e-20] = 1.e-20
tr_pn = np.sum(Y**(2 * n))
ss = Y**(2 * (n - 1))
r = Y * ss * (n / (n - 1.) / tr_pn) # TODO: why?
# r = Y*ss *(1 - n.) # TODO: why not?
# S = np.log(tr_pn)/(1 - n)
XrZ = npc.tensordot(X.scale_axis(r, 'vR'), Z, axes=['vR', 'vL']).split_legs()
dS = npc.tensordot(theta,
XrZ.conj(),
axes=[['vL', 'p0', 'p1', 'vR'], ['vL*', 'p0*', 'p1*', 'vR*']])
dS = dS.combine_legs([['q0', 'q1'], ['q0*', 'q1*']], qconj=[1, -1])
dS = dS - dS.conj().transpose(['(q0.q1)', '(q0*.q1*)']) # project: anti-hermitian part
new_Ss = []
new_thetas = []
new_Us = []
for t in self.stepsizes:
U = npc.expm((-t) * dS).split_legs() # dS anti-hermitian => exp(-tdS) unitary
new_theta = npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']])
new_Ss.append(self._entropy_theta(new_theta, n))
new_thetas.append(new_theta)
new_Us.append(U)
a = np.argmin(new_Ss)
return new_Ss[a], new_thetas[a], new_Us[a]
def _entropy_theta(self, theta):
"""Calculate entropy of theta via SVD."""
theta = theta.combine_legs([('vL', 'p0', 'q0'), ('vR', 'p1', 'q1')], qconj=[+1, -1])
_, S, _ = npc.svd(theta)
return entropy(S**2, self.n)
[docs]class NoiseDisentangler(Disentangler):
"""Apply a little bit of random noise. Useful as pre-step to :class:`RenyiDisentangler`.
Arguments and return values are the same as for :class:`Disentangler`.
"""
def __init__(self, parent):
self.a = get_parameter(parent.TEBD_params, 'disent_noiselevel', 0.01, 'PurificationTEBD')
def __call__(self, theta):
a = self.a
leg = theta.make_pipe(['q0', 'q1'])
if a is None:
U = npc.Array.from_func_square(rand_mat.CUE, leg).split_legs()
else:
U = npc.Array.from_func_square(rand_mat.U_close_1, leg, func_args=[a]).split_legs()
U.iset_leg_labels(['q0', 'q1', 'q0*', 'q1*'])
theta = npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']])
return theta, U
[docs]class LastDisentangler(Disentangler):
"""Use the last total 'U' used in :meth:`disentangle` for the same _update_index as guess.
Useful as a starting point in a :class:`CompositeDisentangler` to reduce the number of
iterations for a following disentangler.
"""
def __call__(self, theta):
# result was saved in :meth:`PurificationTEBD.disentangle`
U = None
U_idx_dt, i = self.parent._update_index
if U_idx_dt is not None:
U = self.parent._guess_U_disent[U_idx_dt][i]
if U is not None:
theta = npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']])
return theta, U
[docs]class DiagonalizeDisentangler(Disentangler):
"""Disentangle by diagonalizing the two-site density matrix in the auxiliar space.
See :arxiv:`1704.01974`.
Problem: Sorting by eigenvalues breaks the charge conservation!
Instead we just sort within the charge blocks.
For non-trivial charges, this might increase the entropy!
Arguments and return values are the same as for :class:`Disentangler`.
"""
def __call__(self, theta):
rho = npc.tensordot(theta,
theta.conj(),
axes=(['vL', 'vR', 'p0', 'p1'], ['vL*', 'vR*', 'p0*', 'p1*']))
# eigh sorts only within the charge blocks...
E, V = npc.eigh(rho.combine_legs((['q0', 'q1'], ['q0*', 'q1*']), qconj=[+1, -1]))
# the phase of the eigenvectors is not well defined. Thus, even if V is the identity,
# we might actually increase the entanglement due to the random phases!
# Try to get rid of them by choosing the phase of the maximal element.
V_flat = V.to_ndarray()
phases = V_flat[np.argmax(np.abs(V_flat), axis=0), np.arange(len(V_flat))] # max values
phases = phases / np.abs(phases) # divided by absolute value
V.iscale_axis(np.conj(phases), 'eig')
V.ireplace_label('eig', '(q0*.q1*)')
V = V.split_legs()
Vd = V.conj()
theta1 = npc.tensordot(Vd, theta, axes=(['q0*', 'q1*'], ['q0', 'q1']))
return theta1, Vd
[docs]class CompositeDisentangler(Disentangler):
"""Concatenate multiple disentanglers.
Applies multiple disentanglers, one after another (in iteration order).
Parameters
----------
disentanglers : list of :class:`Disentangler`
The disentanglers to be used.
Attributes
----------
disentanglers : list of :class:`Disentangler`
The disentanglers to be used.
"""
def __init__(self, disentanglers):
self.disentanglers = disentanglers
def __call__(self, theta):
Utot = None
for disent in self.disentanglers:
theta, U = disent(theta)
if Utot is None:
Utot = U
elif U is not None: # neither Utot nor U are None: multiply together
Utot = npc.tensordot(U, Utot, axes=[['q0*', 'q1*'], ['q0', 'q1']])
return theta, Utot
[docs]class MinDisentangler(Disentangler):
"""Chose the disentangler giving the smallest entropy.
Apply each of the disentanglers to the given `theta`, use the result with smallest entropy.
Reads the TEBD_param ``'disent_min_n'`` which selects the :func:`~tenpy.tools.math.entropy`
to be used for comparison.
Parameters
----------
disentanglers : list of :class:`Disentangler`
The disentanglers to be used.
parent : :class:`~tenpy.algorithms.tebd.Engine`
The parent class calling the disentangler.
Attributes
----------
n : float
Selects the entropy to be used for comparison.
disentanglers : list of :class:`Disentangler`
The disentanglers to be used.
"""
def __init__(self, disentanglers, parent):
self.disentanglers = disentanglers
self.n = get_parameter(parent.TEBD_params, 'disent_min_n', 1., 'PurificationTEBD')
def __call__(self, theta):
theta_min, U_min = self.disentanglers[0](theta)
S_min = self._entropy_theta(theta_min)
for disent in self.disentanglers[1:]:
theta2, U2 = disent(theta)
S2 = self._entropy_theta(theta2)
if S2 < S_min:
S_min = S2
theta_min = theta2
U_min = U2
return theta_min, U_min
def _entropy_theta(self, theta):
"""Calculate entropy of theta via SVD."""
theta = theta.combine_legs([('vL', 'p0', 'q0'), ('vR', 'p1', 'q1')], qconj=[+1, -1])
_, S, _ = npc.svd(theta)
return entropy(S**2, self.n)
"""Dictionary to translate the 'disentangle' TEBD parameter into a :class:`Disentangler`.
If you define your own disentanglers, you can dynamically append them to this dictionary.
CompositeDisentangler and MinDisentangler separate: they have non-default constructor and
special syntax.
"""
disentanglers_atom_parse_dict = {
'None': Disentangler,
'backwards': BackwardDisentangler,
'renyi': RenyiDisentangler,
'norm': NormDisentangler,
'graddesc': GradientDescentDisentangler,
'noise': NoiseDisentangler,
'last': LastDisentangler,
'diag': DiagonalizeDisentangler
}
[docs]def get_disentangler(method, parent):
"""Parse the parameter `method` and construct a :class:`Disentangler` instance.
Parameters
----------
method : str | ``None``
The method to be used, of the form 'method1-method2-min(method3,method4-method5)'.
The usage should be clear from the examples, the precise rule follows:
We parse the full `method` string as a `composite`, and define
``composite := min_atom ['-' min_atom ...]``,
``min_atom := { 'min(' composite [',' composite ...] ')' } | atom``, and
``atom := {any key of `disentanglers_atom_parse_dict`}``.
parent : :class:`~tenpy.algorithms.tebd.Engine`
The parent class calling the disentangler.
Returns
-------
disentangler : :class:`Disentangler`
Disentangler instance, which can be called to disentangle a 2-site `theta`
with the specified `method`.
Examples
--------
>>> get_disentangler(None, p)
Disentangler(p)
>>> get_disentangler('last-renyi', p)
Disentangler([LastDisentangler(p), RenyiDisentangler(p)], p)
>>> get_disentangler('min(None,noise-renyi,min(backwards,last)-graddesc)')
MinDisentangler([Disentangler,
CompositeDisentangler([NoiseDisentangler(p), RenyiDisentangler(p)], p),
CompositeDisentangler([MinDisentangler([BackwardDisentangler(p),
LastDisentangler(p)]),
GradientDescentDisentangler(p)], p), p)
"""
try:
disent, unparsed = _parse_composite(str(method), parent)
if len(unparsed) > 0:
raise _ParseError
except _ParseError:
raise
# raise ValueError("Error while parsing disentangle method: " + repr(method))
return disent
def _parse_composite(unparsed, parent):
disentanglers = []
while True:
disent, unparsed = _parse_min_atom(unparsed, parent)
disentanglers.append(disent)
if len(unparsed) == 0 or unparsed[0] != '-':
break # end of composite
# else: unparsed[0] == '-'
unparsed = unparsed[1:]
# -> continue with while loop
if len(disentanglers) == 1:
# just a min_atom
return disentanglers[0], unparsed
return CompositeDisentangler(disentanglers), unparsed
def _parse_min_atom(unparsed, parent):
if unparsed.startswith('min('):
disentanglers = []
unparsed = unparsed[4:]
while True:
disent, unparsed = _parse_composite(unparsed, parent)
disentanglers.append(disent)
if len(unparsed) == 0 or unparsed[0] != ',':
break # parsed the expected part
# else: unparsed[0] == ','
unparsed = unparsed[1:]
# -> continue with while loop
if len(unparsed) == 0 or unparsed[0] != ')':
raise _ParseError
# else: unparsed[0] == ')'
return MinDisentangler(disentanglers, parent), unparsed[1:]
else: # expect atom
return _parse_atom(unparsed, parent)
def _parse_atom(unparsed, parent):
for key, disent in disentanglers_atom_parse_dict.items():
if unparsed.startswith(key):
return disent(parent), unparsed[len(key):]
raise _ParseError
class _ParseError(ValueError):
pass