"""Generate specific tensor network states and operators.
"""
import math
import functools
import itertools
import collections
from numbers import Integral
import numpy as np
import opt_einsum as oe
from ..core import make_immutable, ikron
from ..utils import deprecated, unique, concat
from ..gen.operators import (
spin_operator, eye, _gen_mbl_random_factors, ham_heis
)
from ..gen.rand import randn, choice, random_seed_fn, rand_phase
from .tensor_core import (Tensor, new_bond, TensorNetwork, rand_uuid,
tensor_direct_product)
from .array_ops import asarray, sensibly_scale, reshape
from .decomp import eigh
from .tensor_arbgeom import TensorNetworkGen, TensorNetworkGenVector
from .tensor_1d import MatrixProductState, MatrixProductOperator
from .tensor_2d import gen_2d_bonds, TensorNetwork2D
from .tensor_3d import TensorNetwork3D
from .tensor_1d_tebd import LocalHam1D
from .tensor_2d_tebd import LocalHam2D
from .tensor_3d_tebd import LocalHam3D
[docs]@random_seed_fn
def rand_tensor(shape, inds, tags=None, dtype='float64', left_inds=None):
"""Generate a random tensor with specified shape and inds.
Parameters
----------
shape : sequence of int
Size of each dimension.
inds : sequence of str
Names of each dimension.
tags : sequence of str
Labels to tag this tensor with.
dtype : {'float64', 'complex128', 'float32', 'complex64'}, optional
The underlying data type.
left_inds : sequence of str, optional
Which, if any, indices to group as 'left' indices of an effective
matrix. This can be useful, for example, when automatically applying
unitary constraints to impose a certain flow on a tensor network but at
the atomistic (Tensor) level.
Returns
-------
Tensor
"""
data = randn(shape, dtype=dtype)
return Tensor(data=data, inds=inds, tags=tags, left_inds=left_inds)
[docs]@random_seed_fn
def rand_phased(shape, inds, tags=None, dtype=complex):
"""Generate a random tensor with specified shape and inds, and randomly
'phased' (distributed on the unit circle) data, such that
``T.H @ T == T.norm()**2 == T.size``.
Parameters
----------
shape : sequence of int
Size of each dimension.
inds : sequence of str
Names of each dimension.
tags : sequence of str
Labels to tag this tensor with.
dtype : {'complex128', 'complex64'}, optional
The underlying data type - can only be complex.
Returns
-------
Tensor
"""
data = rand_phase(shape, dtype=dtype)
return Tensor(data=data, inds=inds, tags=tags)
def gen_unique_edges(edges):
seen = set()
for node_a, node_b in edges:
if node_b < node_a:
node_a, node_b = node_b, node_a
key = (node_a, node_b)
if key in seen:
continue
yield (node_a, node_b)
seen.add(key)
[docs]def TN_from_edges_and_fill_fn(
fill_fn,
edges,
D,
phys_dim=None,
site_tag_id='I{}',
site_ind_id='k{}',
):
"""Create a tensor network from a sequence of edges defining a graph,
and a 'fill' function that maps shapes to data.
Parameters
----------
fill_fn : callable
A function with signature ``fill_fn(shape) -> array``, used to fill
each tensor.
edges : sequence of tuple[hashable, hashable]
The graph edges, as a sequence of pairs of hashable objects, for
example integers, representing the nodes. You can redundantly specify
``(u, v)`` and ``(v, u)`` and only one edge will be added.
D : int
The bond dimension connecting tensors.
phys_dim : int, optional
If not ``None``, give each tensor a 'physical', free index of this size
at each node.
site_tag_id : str, optional
String with formatter to tag sites.
site_ind_id : str, optional
String with formatter to tag indices (if ``phys_dim`` specified).
Returns
-------
TensorNetworkGen or TensorNetworkGenVector
"""
terms = collections.defaultdict(list)
bonds = collections.defaultdict(rand_uuid)
for node_a, node_b in gen_unique_edges(edges):
bond = bonds[node_a, node_b]
# insert at 0 to exactly match geometry of old TN_rand_reg
terms[node_a].insert(0, bond)
terms[node_b].insert(0, bond)
ts = []
sites = []
for node, inds in sorted(terms.items(), key=lambda x: x[0]):
sites.append(node)
shape = [D] * len(inds)
if phys_dim is not None:
inds.append(site_ind_id.format(node))
shape.append(phys_dim)
data = fill_fn(shape)
tags = site_tag_id.format(node)
ts.append(Tensor(data=data, inds=inds, tags=tags))
tn = TensorNetwork(ts)
if phys_dim is not None:
tn.view_as_(
TensorNetworkGenVector, sites=sites,
site_tag_id=site_tag_id, site_ind_id=site_ind_id
)
else:
tn.view_as_(
TensorNetworkGen, sites=sites,
site_tag_id=site_tag_id
)
return tn
[docs]def TN_from_edges_empty(
edges,
D,
phys_dim=None,
site_tag_id='I{}',
site_ind_id='k{}',
dtype='float64',
):
"""Create a tensor network from a sequence of edges defining a graph,
initialized with empty tensors.
Parameters
----------
edges : sequence of tuple[hashable, hashable]
The graph edges, as a sequence of pairs of hashable objects, for
example integers, representing the nodes. You can redundantly specify
``(u, v)`` and ``(v, u)`` and only one edge will be added.
D : int
The bond dimension connecting tensors.
phys_dim : int, optional
If not ``None``, give each tensor a 'physical', free index of this size
at each node.
site_tag_id : str, optional
String with formatter to tag sites.
site_ind_id : str, optional
String with formatter to tag indices (if ``phys_dim`` specified).
dtype : str, optional
The data type of the tensors.
Returns
-------
TensorNetworkGen or TensorNetworkGenVector
"""
def fill_fn(shape):
return np.empty(shape, dtype=dtype)
return TN_from_edges_and_fill_fn(
edges=edges, D=D, fill_fn=fill_fn, phys_dim=phys_dim,
site_tag_id=site_tag_id, site_ind_id=site_ind_id)
[docs]def TN_from_edges_with_value(
value,
edges,
D,
phys_dim=None,
site_tag_id='I{}',
site_ind_id='k{}',
dtype=None,
):
"""Create a tensor network from a sequence of edges defining a graph,
initialized with a constant value. This uses ``numpy.broadcast_to`` and
therefore essentially no memory.
Parameters
----------
value : scalar
The value to fill the tensors with.
edges : sequence of tuple[hashable, hashable]
The graph edges, as a sequence of pairs of hashable objects, for
example integers, representing the nodes. You can redundantly specify
``(u, v)`` and ``(v, u)`` and only one edge will be added.
D : int
The bond dimension connecting tensors.
phys_dim : int, optional
If not ``None``, give each tensor a 'physical', free index of this size
at each node.
site_tag_id : str, optional
String with formatter to tag sites.
site_ind_id : str, optional
String with formatter to tag indices (if ``phys_dim`` specified).
dtype : str, optional
The data type of the tensors.
Returns
-------
TensorNetworkGen or TensorNetworkGenVector
"""
element = np.array(value, dtype=dtype)
def fill_fn(shape):
return np.broadcast_to(element, shape)
return TN_from_edges_and_fill_fn(
edges=edges, D=D, fill_fn=fill_fn, phys_dim=phys_dim,
site_tag_id=site_tag_id, site_ind_id=site_ind_id)
[docs]def TN_from_edges_rand(
edges,
D,
phys_dim=None,
seed=None,
dtype='float64',
site_tag_id='I{}',
site_ind_id='k{}',
):
"""Create a random tensor network with geometry defined from a sequence
of edges defining a graph.
Parameters
----------
G : sequence of tuple[node, node]
The edges defining a graph, each element should be a pair of nodes
described by hashable objects.
D : int
The bond dimension connecting tensors.
phys_dim : int, optional
If not ``None``, give each tensor a 'physical', free index of this size
to mimic a wavefunction of ``len(G)`` sites.
seed : int, optional
A random seed.
site_tag_id : str, optional
String with formatter to tag sites.
site_ind_id : str, optional
String with formatter to tag indices (if ``phys_dim`` specified).
Returns
-------
TensorNetworkGen or TensorNetworkGenVector
"""
ts = {}
sites = tuple(sorted(set(concat(edges))))
for node in sites:
t = Tensor(tags=site_tag_id.format(node))
if phys_dim is not None:
t.new_ind(site_ind_id.format(node), size=phys_dim)
ts[node] = t
for node_a, node_b in gen_unique_edges(edges):
new_bond(ts[node_a], ts[node_b], size=D)
tn = TensorNetwork(ts.values())
tn.randomize_(seed=seed, dtype=dtype)
if phys_dim is not None:
tn.view_as_(
TensorNetworkGenVector, sites=sites,
site_tag_id=site_tag_id, site_ind_id=site_ind_id
)
else:
tn.view_as_(
TensorNetworkGen, sites=sites,
site_tag_id=site_tag_id
)
return tn
TN_rand_from_edges = deprecated(
TN_from_edges_rand, "TN_rand_from_edges", "TN_from_edges_rand"
)
[docs]def TN_rand_reg(
n,
reg,
D,
phys_dim=None,
seed=None,
dtype='float64',
site_tag_id='I{}',
site_ind_id='k{}',
):
"""Create a random regular tensor network.
Parameters
----------
n : int
The number of tensors.
reg : int
The degree of the tensor network (how many tensors each tensor
connects to).
D : int
The bond dimension connecting tensors.
phys_dim : int, optional
If not ``None``, give each tensor a 'physical', free index of this size
to mimic a wavefunction of ``n`` sites.
seed : int, optional
A random seed.
site_tag_id : str, optional
String with formatter to tag sites.
site_ind_id : str, optional
String with formatter to tag indices (if ``phys_dim`` specified).
Returns
-------
TensorNetwork
"""
import networkx as nx
G = nx.random_degree_sequence_graph([reg] * n, seed=seed)
return TN_from_edges_rand(
G.edges, D=D, phys_dim=phys_dim, seed=seed, dtype=dtype,
site_tag_id=site_tag_id, site_ind_id=site_ind_id)
[docs]def TN2D_from_fill_fn(
fill_fn,
Lx,
Ly,
D,
cyclic=False,
site_tag_id='I{},{}',
row_tag_id='ROW{}',
col_tag_id='COL{}',
):
"""A scalar 2D lattice tensor network with tensors filled by a function.
Parameters
----------
fill_fn : callable
A function with signature ``fill_fn(shape) -> array``, used to fill
each tensor.
Lx : int
Length of side x.
Ly : int
Length of side y.
D : int
The bond dimension connecting sites.
cyclic : bool or (bool, bool), optional
Whether to use periodic boundary conditions. X and Y can be specified
separately using a tuple.
site_tag_id : str, optional
String specifier for naming convention of site tags.
row_tag_id : str, optional
String specifier for naming convention of row tags.
col_tag_id : str, optional
String specifier for naming convention of column tags.
Returns
-------
TensorNetwork2D
"""
try:
cyclic_x, cyclic_y = cyclic
except TypeError:
cyclic_x = cyclic_y = cyclic
ts = []
bonds = collections.defaultdict(rand_uuid)
for i, j in itertools.product(range(Lx), range(Ly)):
directions = ""
inds = []
if j > 0 or cyclic_y:
directions += 'l'
inds.append(bonds[(i, (j - 1) % Ly), (i, j)])
if j < Ly - 1 or cyclic_y:
directions += 'r'
inds.append(bonds[(i, j), (i, (j + 1) % Ly)])
if i < Lx - 1 or cyclic_x:
directions += 'u'
inds.append(bonds[(i, j), ((i + 1) % Lx, j)])
if i > 0 or cyclic_x:
directions += 'd'
inds.append(bonds[((i - 1) % Lx, j), (i, j)])
shape = (D,) * len(inds)
data = fill_fn(shape)
tags = [site_tag_id.format(i, j),
row_tag_id.format(i), col_tag_id.format(j)]
ts.append(Tensor(data=data, inds=inds, tags=tags))
tn = TensorNetwork(ts)
return tn.view_as_(
TensorNetwork2D,
Lx=Lx, Ly=Ly,
site_tag_id=site_tag_id,
row_tag_id=row_tag_id,
col_tag_id=col_tag_id,
)
[docs]def TN2D_empty(
Lx,
Ly,
D,
cyclic=False,
site_tag_id='I{},{}',
row_tag_id='ROW{}',
col_tag_id='COL{}',
dtype='float64',
):
"""A scalar 2D lattice tensor network initialized with empty tensors.
Parameters
----------
Lx : int
Length of side x.
Ly : int
Length of side y.
D : int
The bond dimension connecting sites.
cyclic : bool or (bool, bool), optional
Whether to use periodic boundary conditions. X and Y can be specified
separately using a tuple.
site_tag_id : str, optional
String specifier for naming convention of site tags.
row_tag_id : str, optional
String specifier for naming convention of row tags.
col_tag_id : str, optional
String specifier for naming convention of column tags.
dtype : str, optional
The data type of the tensors.
Returns
-------
TensorNetwork2D
"""
def fill_fn(shape):
return np.empty(shape, dtype=dtype)
return TN2D_from_fill_fn(
fill_fn, Lx=Lx, Ly=Ly, D=D, cyclic=cyclic,
site_tag_id=site_tag_id, row_tag_id=row_tag_id, col_tag_id=col_tag_id,
)
[docs]def TN2D_with_value(
value,
Lx,
Ly,
D,
cyclic=False,
site_tag_id='I{},{}',
row_tag_id='ROW{}',
col_tag_id='COL{}',
dtype=None,
):
"""A scalar 2D lattice tensor network with every element set to ``value``.
This uses ``numpy.broadcast_to`` and therefore essentially no memory.
Parameters
----------
value : scalar
The value to fill the tensors with.
Lx : int
Length of side x.
Ly : int
Length of side y.
D : int
The bond dimension connecting sites.
cyclic : bool or (bool, bool), optional
Whether to use periodic boundary conditions. X and Y can be specified
separately using a tuple.
site_tag_id : str, optional
String specifier for naming convention of site tags.
row_tag_id : str, optional
String specifier for naming convention of row tags.
col_tag_id : str, optional
String specifier for naming convention of column tags.
dtype : str, optional
The data type of the tensors.
Returns
-------
TensorNetwork2D
"""
element = np.array(value, dtype=dtype)
def fill_fn(shape):
return np.broadcast_to(element, shape)
return TN2D_from_fill_fn(
fill_fn, Lx=Lx, Ly=Ly, D=D, cyclic=cyclic,
site_tag_id=site_tag_id, row_tag_id=row_tag_id, col_tag_id=col_tag_id,
)
[docs]@random_seed_fn
def TN2D_rand(
Lx,
Ly,
D,
cyclic=False,
site_tag_id='I{},{}',
row_tag_id='ROW{}',
col_tag_id='COL{}',
dtype='float64',
):
"""A random scalar 2D lattice tensor network.
Parameters
----------
Lx : int
Length of side x.
Ly : int
Length of side y.
D : int
The bond dimension connecting sites.
cyclic : bool or (bool, bool), optional
Whether to use periodic boundary conditions. X and Y can be specified
separately using a tuple.
site_tag_id : str, optional
String specifier for naming convention of site tags.
row_tag_id : str, optional
String specifier for naming convention of row tags.
col_tag_id : str, optional
String specifier for naming convention of column tags.
dtype : dtype, optional
Data type of the random arrays.
seed : int, optional
A random seed.
Returns
-------
TensorNetwork2D
"""
def fill_fn(shape):
return randn(shape, dtype=dtype)
return TN2D_from_fill_fn(
fill_fn, Lx=Lx, Ly=Ly, D=D, cyclic=cyclic,
site_tag_id=site_tag_id, row_tag_id=row_tag_id, col_tag_id=col_tag_id,
)
[docs]def TN3D_from_fill_fn(
fill_fn,
Lx,
Ly,
Lz,
D,
cyclic=False,
site_tag_id='I{},{},{}',
x_tag_id='X{}',
y_tag_id='Y{}',
z_tag_id='Z{}',
):
"""A scalar 3D lattice tensor network with tensors filled by a function.
Parameters
----------
fill_fn : callable
A function with signature ``fill_fn(shape) -> array``, used to fill
each tensor.
Lx : int
Length of side x.
Ly : int
Length of side y.
Lz : int
Length of side z.
D : int
The bond dimension connecting sites.
cyclic : bool or (bool, bool, bool), optional
Whether to use periodic boundary conditions. X, Y and Z can be
specified separately using a tuple.
site_tag_id : str, optional
String formatter specifying how to label each site.
dtype : dtype, optional
Data type of the random arrays.
Returns
-------
TensorNetwork3D
"""
try:
cyclic_x, cyclic_y, cyclic_z = cyclic
except TypeError:
cyclic_x = cyclic_y = cyclic_z = cyclic
ts = []
bonds = collections.defaultdict(rand_uuid)
for i, j, k in itertools.product(range(Lx), range(Ly), range(Lz)):
directions = ""
inds = []
if k > 0 or cyclic_z:
directions += 'b'
inds.append(bonds[(i, j, (k - 1) % Lz), (i, j, k)])
if k < Lz - 1 or cyclic_z:
directions += 'a'
inds.append(bonds[(i, j, k), (i, j, (k + 1) % Lz)])
if j > 0 or cyclic_y:
directions += 'l'
inds.append(bonds[(i, (j - 1) % Ly, k), (i, j, k)])
if j < Ly - 1 or cyclic_y:
directions += 'r'
inds.append(bonds[(i, j, k), (i, (j + 1) % Ly, k)])
if i < Lx - 1 or cyclic_x:
directions += 'u'
inds.append(bonds[(i, j, k), ((i + 1) % Lx, j, k)])
if i > 0 or cyclic_x:
directions += 'd'
inds.append(bonds[((i - 1) % Lx, j, k), (i, j, k)])
shape = (D,) * len(inds)
data = fill_fn(shape)
tags = [site_tag_id.format(i, j, k), x_tag_id.format(i),
y_tag_id.format(j), z_tag_id.format(k)]
ts.append(Tensor(data=data, inds=inds, tags=tags))
tn = TensorNetwork(ts)
return tn.view_as_(
TensorNetwork3D,
Lx=Lx, Ly=Ly, Lz=Lz,
site_tag_id=site_tag_id,
x_tag_id=x_tag_id,
y_tag_id=y_tag_id,
z_tag_id=z_tag_id,
)
[docs]def TN3D_empty(
Lx,
Ly,
Lz,
D,
cyclic=False,
site_tag_id='I{},{},{}',
x_tag_id='X{}',
y_tag_id='Y{}',
z_tag_id='Z{}',
dtype='float64',
):
"""A scalar 3D lattice tensor network initialized with empty tensors.
Parameters
----------
Lx : int
Length of side x.
Ly : int
Length of side y.
Lz : int
Length of side z.
D : int
The bond dimension connecting sites.
cyclic : bool or (bool, bool, bool), optional
Whether to use periodic boundary conditions. X, Y and Z can be
specified separately using a tuple.
site_tag_id : str, optional
String formatter specifying how to label each site.
dtype : dtype, optional
Data type of the random arrays.
seed : int, optional
Random seed.
Returns
-------
TensorNetwork3D
"""
def fill_fn(shape):
return np.empty(shape, dtype=dtype)
return TN3D_from_fill_fn(
fill_fn, Lx, Ly, Lz, D,
cyclic=cyclic, site_tag_id=site_tag_id,
x_tag_id=x_tag_id, y_tag_id=y_tag_id, z_tag_id=z_tag_id,
)
[docs]def TN3D_with_value(
value,
Lx,
Ly,
Lz,
D,
cyclic=False,
site_tag_id='I{},{},{}',
x_tag_id='X{}',
y_tag_id='Y{}',
z_tag_id='Z{}',
dtype=None,
):
"""A scalar 2D lattice tensor network with every element set to ``value``.
This uses ``numpy.broadcast_to`` and therefore essentially no memory.
Parameters
----------
value : scalar
The value to fill the tensors with.
Lx : int
Length of side x.
Ly : int
Length of side y.
Lz : int
Length of side z.
D : int
The bond dimension connecting sites.
cyclic : bool or (bool, bool, bool), optional
Whether to use periodic boundary conditions. X, Y and Z can be
specified separately using a tuple.
site_tag_id : str, optional
String formatter specifying how to label each site.
dtype : dtype, optional
Data type of the random arrays.
seed : int, optional
Random seed.
Returns
-------
TensorNetwork3D
"""
element = np.array(value, dtype=dtype)
def fill_fn(shape):
return np.broadcast_to(element, shape)
return TN3D_from_fill_fn(
fill_fn, Lx, Ly, Lz, D,
cyclic=cyclic, site_tag_id=site_tag_id,
x_tag_id=x_tag_id, y_tag_id=y_tag_id, z_tag_id=z_tag_id,
)
[docs]@random_seed_fn
def TN3D_rand(
Lx,
Ly,
Lz,
D,
cyclic=False,
site_tag_id='I{},{},{}',
x_tag_id='X{}',
y_tag_id='Y{}',
z_tag_id='Z{}',
dtype='float64',
):
"""A random scalar 3D lattice tensor network.
Parameters
----------
Lx : int
Length of side x.
Ly : int
Length of side y.
Lz : int
Length of side z.
D : int
The bond dimension connecting sites.
cyclic : bool or (bool, bool, bool), optional
Whether to use periodic boundary conditions. X, Y and Z can be
specified separately using a tuple.
site_tag_id : str, optional
String formatter specifying how to label each site.
dtype : dtype, optional
Data type of the random arrays.
seed : int, optional
Random seed.
Returns
-------
TensorNetwork
"""
def fill_fn(shape):
return randn(shape, dtype=dtype)
return TN3D_from_fill_fn(
fill_fn, Lx, Ly, Lz, D,
cyclic=cyclic, site_tag_id=site_tag_id,
x_tag_id=x_tag_id, y_tag_id=y_tag_id, z_tag_id=z_tag_id,
)
# ---------------------------- classical models ----------------------------- #
[docs]@functools.lru_cache(128)
def classical_ising_S_matrix(beta, j=1.0):
"""The interaction term for the classical ising model.
"""
S = np.array(
[[math.exp(+j * beta), math.exp(-j * beta)],
[math.exp(-j * beta), math.exp(+j * beta)]])
make_immutable(S)
return S
[docs]@functools.lru_cache(128)
def classical_ising_H_matrix(beta, h=0.0):
"""The magnetic field term for the classical ising model.
"""
H = np.array([math.exp(-beta * h), math.exp(beta * h)])
make_immutable(H)
return H
[docs]@functools.lru_cache(128)
def classical_ising_sqrtS_matrix(beta, j=1.0, asymm=None):
"""The sqrt factorized interaction term for the classical ising model.
If ``j`` is negative you can supply ``asymm='l'`` or ``'r'`` to
keep the matrix real, but it must be paired with the opposite in a tensor
network.
"""
if (j < 0.0) and (asymm is not None):
Slr = eigh(classical_ising_S_matrix(beta=beta, j=j))
S_1_2 = {
'l': Slr[0], 'lT': Slr[0].T,
'r': Slr[-1], 'rT': Slr[-1].T,
}[asymm]
else:
S_1_2 = np.array(
[[math.cosh(j * beta)**0.5 + math.sinh(j * beta)**0.5,
math.cosh(j * beta)**0.5 - math.sinh(j * beta)**0.5],
[math.cosh(j * beta)**0.5 - math.sinh(j * beta)**0.5,
math.cosh(j * beta)**0.5 + math.sinh(j * beta)**0.5]]
) / 2**0.5
make_immutable(S_1_2)
return S_1_2
[docs]@functools.lru_cache(128)
def classical_ising_T_matrix(
beta,
j=1.0,
h=0.0,
directions='lrud',
asymm=None,
):
"""The single effective TN site for the classical ising model.
"""
try:
js = tuple(j)
except TypeError:
js = (j,) * len(directions)
try:
asymms = tuple(asymm)
except TypeError:
asymms = (asymm,) * len(directions)
arrays = (
[
classical_ising_sqrtS_matrix(beta=beta, j=j, asymm=a)
for j, a in zip(js, asymms)
] +
[classical_ising_H_matrix(beta, h)]
)
lhs = ",".join(f'i{x}' for x in directions)
eq = lhs + ",i->" + directions
return oe.contract(eq, *arrays)
[docs]def HTN2D_classical_ising_partition_function(
Lx,
Ly,
beta,
h=0.0,
j=1.0,
ind_id='s{},{}',
cyclic=False,
):
"""Hyper tensor network representation of the 2D classical ising model
partition function. The indices will be shared by 4 or 5 tensors depending
on whether ``h`` is non-zero. As opposed to the 'normal' tensor network,
here each classical spin is still a single index, which is easier to
contract exactly.
Parameters
----------
Lx : int
Length of side x.
Ly : int
Length of side y.
beta : float
The inverse temperature.
h : float, optional
The magnetic field strength.
j : float, optional
The interaction strength, positive being *ferromagnetic*.
cyclic : bool or (bool, bool), optional
Whether to use periodic boundary conditions. X and Y can be specified
separately using a tuple.
ind_id : str, optional
How to label the indices i.e. ``ind_id.format(i, j)``, each of which
corresponds to a single classical spin.
Returns
-------
TensorNetwork
See Also
--------
TN2D_classical_ising_partition_function
"""
try:
cyclic_x, cyclic_y = cyclic
except TypeError:
cyclic_x = cyclic_y = cyclic
if callable(j):
j_factory = j
else:
def j_factory(node_a, node_b):
return j
ts = []
for ni, nj in itertools.product(range(Lx), range(Ly)):
if ni < Lx - 1 or cyclic_x:
node_a, node_b = (ni, nj), ((ni + 1) % Lx, nj)
inds = ind_id.format(*node_a), ind_id.format(*node_b)
data = classical_ising_S_matrix(
beta=beta, j=j_factory(node_a, node_b))
ts.append(Tensor(data, inds=inds))
if nj < Ly - 1 or cyclic_y:
node_a, node_b = (ni, nj), (ni, (nj + 1) % Ly)
inds = ind_id.format(*node_a), ind_id.format(*node_b)
data = classical_ising_S_matrix(
beta=beta, j=j_factory(node_a, node_b))
ts.append(Tensor(data, inds=inds))
if h != 0.0:
data = classical_ising_H_matrix(beta=beta, h=h)
ts.append(Tensor(data, inds=(ind_id.format(ni, nj),)))
return TensorNetwork(ts)
[docs]def HTN3D_classical_ising_partition_function(
Lx,
Ly,
Lz,
beta,
j=1.0,
h=0.0,
cyclic=False,
ind_id='s{},{},{}',
):
"""Hyper tensor network representation of the 3D classical ising model
partition function. The indices will be shared by 6 or 7 tensors depending
on whether ``h`` is non-zero. As opposed to the 'normal' tensor network,
here each classical spin is still a single index, which is easier to
contract exactly.
Parameters
----------
Lx : int
Length of side x.
Ly : int
Length of side y.
Lz : int
Length of side z.
beta : float
The inverse temperature.
j : float, optional
The interaction strength, positive being *ferromagnetic*.
h : float, optional
The magnetic field strength.
cyclic : bool or (bool, bool, bool), optional
Whether to use periodic boundary conditions. X, Y and Z can be
specified separately using a tuple.
ind_id : str, optional
How to label the indices i.e. ``ind_id.format(i, j, k)``, each of which
corresponds to a single classical spin.
Returns
-------
TensorNetwork
See Also
--------
TN3D_classical_ising_partition_function
"""
try:
cyclic_x, cyclic_y, cyclic_z = cyclic
except TypeError:
cyclic_x = cyclic_y = cyclic_z = cyclic
if callable(j):
j_factory = j
else:
def j_factory(node_a, node_b):
return j
ts = []
for ni, nj, nk in itertools.product(range(Lx), range(Ly), range(Lz)):
if ni < Lx - 1 or cyclic_x:
node_a, node_b = (ni, nj, nk), ((ni + 1) % Lx, nj, nk)
inds = (ind_id.format(*node_a), ind_id.format(*node_b))
data = classical_ising_S_matrix(
beta=beta, j=j_factory(node_a, node_b))
ts.append(Tensor(data, inds=inds))
if nj < Ly - 1 or cyclic_y:
node_a, node_b = (ni, nj, nk), (ni, (nj + 1) % Ly, nk)
inds = (ind_id.format(*node_a), ind_id.format(*node_b))
data = classical_ising_S_matrix(
beta=beta, j=j_factory(node_a, node_b))
ts.append(Tensor(data, inds=inds))
if nk < Lz - 1 or cyclic_z:
node_a, node_b = (ni, nj, nk), (ni, nj, (nk + 1) % Lz)
inds = (ind_id.format(*node_a), ind_id.format(*node_b))
data = classical_ising_S_matrix(
beta=beta, j=j_factory(node_a, node_b))
ts.append(Tensor(data, inds=inds))
if h != 0.0:
data = classical_ising_H_matrix(beta=beta, h=h)
ts.append(Tensor(data, inds=(ind_id.format(ni, nj, nk),)))
return TensorNetwork(ts)
[docs]def TN2D_classical_ising_partition_function(
Lx,
Ly,
beta,
j=1.0,
h=0.0,
cyclic=False,
site_tag_id='I{},{}',
row_tag_id='ROW{}',
col_tag_id='COL{}',
):
"""The tensor network representation of the 2D classical ising model
partition function.
Parameters
----------
Lx : int
Length of side x.
Ly : int
Length of side y.
beta : float
The inverse temperature.
j : float, optional
The interaction strength, positive being *ferromagnetic*.
h : float, optional
The magnetic field strength.
cyclic : bool or (bool, bool), optional
Whether to use periodic boundary conditions. X and Y can be specified
separately using a tuple.
site_tag_id : str, optional
String specifier for naming convention of site tags.
row_tag_id : str, optional
String specifier for naming convention of row tags.
col_tag_id : str, optional
String specifier for naming convention of column tags.
Returns
-------
TensorNetwork2D
See Also
--------
HTN2D_classical_ising_partition_function
"""
try:
cyclic_x, cyclic_y = cyclic
except TypeError:
cyclic_x = cyclic_y = cyclic
if callable(j):
j_factory = j
else:
def j_factory(node_a, node_b):
return j
ts = []
bonds = collections.defaultdict(rand_uuid)
for ni, nj in itertools.product(range(Lx), range(Ly)):
directions = ""
inds = []
js = ()
asymms = ()
for inbounds, pair, direction in [
(nj > 0 or cyclic_y, ((ni, (nj - 1) % Ly), (ni, nj)), 'l'),
(nj < Ly - 1 or cyclic_y, ((ni, nj), (ni, (nj + 1) % Ly)), 'r'),
(ni < Lx - 1 or cyclic_x, ((ni, nj), ((ni + 1) % Lx, nj)), 'u'),
(ni > 0 or cyclic_x, (((ni - 1) % Lx, nj), (ni, nj)), 'd'),
]:
if inbounds:
js += (j_factory(*pair),)
directions += direction
# this is logic for handling negative j without imag tensors
# i.e. add the left factor if the first instance of bond, right
# factor if second. If j > 0.0 this doesn't matter anyhow
asymms += ('l' if pair not in bonds else 'rT',)
inds.append(bonds[pair])
ts.append(Tensor(
data=classical_ising_T_matrix(
beta=beta, directions=directions, j=js, h=h, asymm=asymms,
),
inds=inds,
tags=[site_tag_id.format(ni, nj),
row_tag_id.format(ni),
col_tag_id.format(nj)]))
tn = TensorNetwork(ts)
return tn.view_as_(
TensorNetwork2D,
Lx=Lx, Ly=Ly,
site_tag_id=site_tag_id,
row_tag_id=row_tag_id,
col_tag_id=col_tag_id,
)
[docs]def TN3D_classical_ising_partition_function(
Lx,
Ly,
Lz,
beta,
j=1.0,
h=0.0,
cyclic=False,
site_tag_id='I{},{},{}',
x_tag_id='X{}',
y_tag_id='Y{}',
z_tag_id='Z{}',
):
"""Tensor network representation of the 3D classical ising model
partition function.
Parameters
----------
Lx : int
Length of side x.
Ly : int
Length of side y.
Lz : int
Length of side z.
beta : float
The inverse temperature.
j : float, optional
The interaction strength, positive being *ferromagnetic*.
h : float, optional
The magnetic field strength.
cyclic : bool or (bool, bool, bool), optional
Whether to use periodic boundary conditions. X, Y and Z can be
specified separately using a tuple.
site_tag_id : str, optional
String formatter specifying how to label each site.
Returns
-------
TensorNetwork
See Also
--------
HTN3D_classical_ising_partition_function
"""
try:
cyclic_x, cyclic_y, cyclic_z = cyclic
except TypeError:
cyclic_x = cyclic_y = cyclic_z = cyclic
if callable(j):
j_factory = j
else:
def j_factory(node_a, node_b):
return j
ts = []
bonds = collections.defaultdict(rand_uuid)
for ni, nj, nk in itertools.product(range(Lx), range(Ly), range(Lz)):
directions = ""
inds = []
js = ()
asymms = ()
for inbounds, pair, direction in [
(nk > 0 or cyclic_z,
((ni, nj, (nk - 1) % Lz), (ni, nj, nk)), 'b'),
(nk < Lz - 1 or cyclic_z,
((ni, nj, nk), (ni, nj, (nk + 1) % Lz)), 'a'),
(nj > 0 or cyclic_y,
((ni, (nj - 1) % Ly, nk), (ni, nj, nk)), 'l'),
(nj < Ly - 1 or cyclic_y,
((ni, nj, nk), (ni, (nj + 1) % Ly, nk)), 'r'),
(ni < Lx - 1 or cyclic_x,
((ni, nj, nk), ((ni + 1) % Lx, nj, nk)), 'u'),
(ni > 0 or cyclic_x,
(((ni - 1) % Lx, nj, nk), (ni, nj, nk)), 'd'),
]:
if inbounds:
js += (j_factory(*pair),)
directions += direction
# this is logic for handling negative j without imag tensors
# i.e. add the left factor if the first instance of bond, right
# factor if second. If j > 0.0 this doesn't matter anyhow
asymms += ('l' if pair not in bonds else 'rT',)
inds.append(bonds[pair])
ts.append(Tensor(
data=classical_ising_T_matrix(
beta=beta, directions=directions, j=js, h=h, asymm=asymms,
),
inds=inds,
tags=[
site_tag_id.format(ni, nj, nk),
x_tag_id.format(ni),
y_tag_id.format(nj),
z_tag_id.format(nk),
],
))
tn = TensorNetwork(ts)
return tn.view_as_(
TensorNetwork3D,
Lx=Lx, Ly=Ly, Lz=Lz,
site_tag_id=site_tag_id,
x_tag_id=x_tag_id,
y_tag_id=y_tag_id,
z_tag_id=z_tag_id,
)
[docs]def HTN_classical_partition_function_from_edges(
edges,
beta,
j=1.0,
h=0.0,
site_ind_id="s{}",
site_tag_id="I{}",
bond_tag_id="B{},{}",
):
"""Build a hyper tensor network representation of a classical ising model
partition function by specifying graph edges. There will be a single
tensor *per interaction* rather than per site, as well as a single tensor
for each site, if ``h != 0.0``.
Parameters
----------
edges : sequence of tuple[hashable, hashable]
The graph edges, as a sequence of pairs of hashable objects, for
example integers, representing the nodes. You can redundantly specify
``(u, v)`` and ``(v, u)`` and only one edge will be added.
beta : float, optional
The inverse temperature.
j : float, or callable, optional
The interaction strength, positive being *ferromagnetic*. If a
callable should have the signature ``j(node_a, node_b)`` and return
a float.
h : float, or callable, optional
The magnetic field strength. If a callable should have the
signature ``h(node)`` and return a float.
site_ind_id : str, optional
A string formatter for naming tensor indices like
``site_ind_id.format(node)``.
site_tag_id : str, optional
A string formatter for naming tensor tags like
``site_tag_id.format(node)``.
bond_tag_id : str, optional
A string formatter for naming tensor tags like
``bond_tag_id.format(node_a, node_b)``.
Returns
-------
TensorNetwork
"""
if callable(j):
j_factory = j
else:
def j_factory(node_a, node_b):
return j
ts = []
for node_a, node_b in gen_unique_edges(edges):
data = classical_ising_S_matrix(beta=beta, j=j_factory(node_a, node_b))
inds = [site_ind_id.format(node_a),
site_ind_id.format(node_b)]
tags = [bond_tag_id.format(node_a, node_b),
site_tag_id.format(node_a),
site_tag_id.format(node_b)]
ts.append(Tensor(data=data, inds=inds, tags=tags))
if h != 0.0:
if callable(h):
h_factory = h
else:
def h_factory(node):
return h
for node in unique(concat(edges)):
data = classical_ising_H_matrix(beta, h=h_factory(node))
inds = [site_ind_id.format(node)]
tags = [site_tag_id.format(node)]
ts.append(Tensor(data=data, inds=inds, tags=tags))
return TensorNetwork(ts)
[docs]def TN_classical_partition_function_from_edges(
edges,
beta,
j=1.0,
h=0.0,
site_tag_id="I{}",
bond_ind_id="b{},{}",
):
"""Build a regular tensor network representation of a classical ising model
partition function by specifying graph edges. There will be a single
tensor per site.
Parameters
----------
edges : sequence of tuple[hashable, hashable]
The graph edges, as a sequence of pairs of hashable objects, for
example integers, representing the nodes. You can redundantly specify
``(u, v)`` and ``(v, u)`` and only one edge will be added.
beta : float, optional
The inverse temperature.
j : float, or callable, optional
The interaction strength, positive being *ferromagnetic*. If a
callable should have the signature ``j(node_a, node_b)`` and return
a float.
h : float, or callable, optional
The magnetic field strength. If a callable should have the
signature ``h(node)`` and return a float.
site_tag_id : str, optional
A string formatter for naming tensor tags like
``site_ind_id.format(node)``.
bond_ind_id : str, optional
A string formatter for naming the indices bewteen tensors like
``bond_ind_id.format(node_a, node_b)``.
Returns
-------
TensorNetwork
"""
if callable(j):
j_factory = j
else:
def j_factory(node_a, node_b):
return j
to_contract = collections.defaultdict(list)
ts = []
for node_a, node_b in gen_unique_edges(edges):
j_ab = j_factory(node_a, node_b)
bond_ab = bond_ind_id.format(node_a, node_b)
# left tensor factor
data = classical_ising_sqrtS_matrix(beta=beta, j=j_ab, asymm='l')
inds = [f's{node_a}', bond_ab]
tags = [site_tag_id.format(node_a)]
ts.append(Tensor(data=data, inds=inds, tags=tags))
# right tensor factor
data = classical_ising_sqrtS_matrix(beta=beta, j=j_ab, asymm='r')
inds = [bond_ab, f's{node_b}']
tags = [site_tag_id.format(node_b)]
ts.append(Tensor(data=data, inds=inds, tags=tags))
to_contract[f's{node_a}'].append(bond_ab)
to_contract[f's{node_b}'].append(bond_ab)
sites = tuple(sorted(set(concat(edges))))
if h != 0.0:
if callable(h):
h_factory = h
else:
def h_factory(node):
return h
for node in sites:
data = classical_ising_H_matrix(beta, h=h_factory(node))
inds = [f's{node}']
tags = [site_tag_id.format(node)]
ts.append(Tensor(data=data, inds=inds, tags=tags))
to_contract[f's{node}'].extend(())
tn = TensorNetwork(ts)
for ind, output_inds in to_contract.items():
tn.contract_ind(ind, output_inds=output_inds)
tn.view_as_(TensorNetworkGen, sites=sites, site_tag_id=site_tag_id)
return tn
@functools.lru_cache(128)
def dimer_data(d, cover_count=1, dtype=float):
shape = [2] * d
x = np.zeros(shape, dtype=dtype)
index_sum = np.indices(shape).sum(axis=0)
x[index_sum == cover_count] = 1
make_immutable(x)
return x
[docs]def TN_dimer_covering_from_edges(
edges,
cover_count=1,
site_tag_id="I{}",
bond_ind_id="b{},{}",
dtype=float,
):
"""Make a tensor network from sequence of graph edges that counts the
number of ways to cover the graph exactly with dimers. See
https://arxiv.org/abs/1805.10598 for the construction.
Parameters
----------
edges : sequence of tuple
The edges, each item should be a pair of hashable objects describing
nodes linked.
cover_count : int, optional
The exact number of times each node must be 'covered'. For example
1 for a standard dimer covering or 2 for 'ice rules'.
site_tag_id : str, optional
A string formatter for naming tensor tags like
``site_ind_id.format(node)``.
bond_ind_id : str, optional
A string formatter for naming the indices bewteen tensors like
``bond_ind_id.format(node_a, node_b)``.
Returns
-------
TensorNetwork
"""
nodes2inds = collections.defaultdict(list)
for ni, nj in edges:
bond = bond_ind_id.format(ni, nj)
nodes2inds[ni].append(bond)
nodes2inds[nj].append(bond)
ts = []
for node, inds in nodes2inds.items():
data = dimer_data(len(inds), cover_count=cover_count, dtype=dtype)
tag = site_tag_id.format(node)
ts.append(Tensor(data, inds=inds, tags=tag))
tn = TensorNetwork(ts)
sites = tuple(sorted(nodes2inds))
tn.view_as_(TensorNetworkGen, sites=sites, site_tag_id=site_tag_id)
return tn
# --------------------------------------------------------------------------- #
# Weighted Model Counting #
# --------------------------------------------------------------------------- #
def clause_negmask(clause):
return int("".join('0' if x > 0 else '1' for x in clause), 2)
[docs]@functools.lru_cache(128)
def or_clause_data(ndim, m=0, dtype=float, q=2):
"""Get the array representing satisfiability of ``ndim`` clauses with
unsatisfied condition encoded in ``m``.
"""
shape = [q] * ndim
t = np.ones(shape, dtype=dtype)
t[np.unravel_index(m, shape)] = 0
return t
[docs]def or_clause_tensor(ndim, m, inds, tags=None):
"""Get the tensor representing satisfiability of ``ndim`` clauses with
unsatisfied condition encoded in ``m`` labelled by ``inds`` and ``tags``.
"""
data = or_clause_data(ndim, m=m)
return Tensor(data=data, inds=inds, tags=tags)
[docs]def or_clause_mps_tensors(ndim, m, inds, tags=None):
"""Get the set of MPS tensors representing satisfiability of ``ndim``
clauses with unsatisfied condition encoded in ``m`` labelled by ``inds``
and ``tags``.
"""
mps = (
MPS_computational_state('+' * ndim, tags=tags) * (2**(ndim / 2)) -
MPS_computational_state(f'{m:0>{ndim}b}', tags=tags)
)
mps.reindex_({
mps.site_ind(i): ind
for i, ind in enumerate(inds)
})
return mps.tensors
[docs]@functools.lru_cache(2**10)
def or_clause_parafac_data(ndim, m):
"""Get the set of PARAFAC arrays representing satisfiability of ``ndim``
clauses with unsatisfied condition encoded in ``m``.
"""
inds = [f'k{i}' for i in range(ndim)]
bond = 'b'
pfc_ones = np.ones((2, 1))
pfc_up = np.array([[1.], [0.]])
pfc_dn = np.array([[0.], [1.]])
ts_ones = [Tensor(data=pfc_ones, inds=[ix, bond]) for ix in inds]
bmask = f'{m:0>{ndim}b}'
ts_mask = [Tensor(data=(pfc_dn if b == '1' else pfc_up), inds=[ix, bond])
for ix, b in zip(inds, bmask)]
# just need to multiply a single mask tensor by -1
ts_mask[0] *= -1
ts = [tensor_direct_product(t1, t2, sum_inds=(ix,))
for ix, t1, t2 in zip(inds, ts_ones, ts_mask)]
return tuple(t.data for t in ts)
[docs]def clause_parafac_tensors(ndim, m, inds, tags=None):
"""Get the set of PARAFAC tensors representing satisfiability of ``ndim``
clauses with unsatisfied condition encoded in ``m`` labelled by ``inds``
and ``tags``.
"""
bond = rand_uuid()
return [Tensor(x, inds=[ix, bond], tags=tags)
for x, ix in zip(or_clause_parafac_data(ndim, m), inds)]
[docs]def HTN_from_cnf(fname, mode='parafac'):
"""Create a hyper tensor network from a '.cnf' or '.wcnf' file - i.e. a
model counting or weighted model counting instance specification.
Parameters
----------
fname : str
Path to a '.cnf' or '.wcnf' file.
mode : {'parafac', 'mps', 'dense', int}, optional
How to represent the clauses:
* 'parafac' - `N` rank-2 tensors connected by a single hyper index.
You could further call :meth:`hyperinds_resolve` for more options
to convert the hyper index into a (decomposed) COPY-tensor.
* 'mps' - `N` rank-3 tensors connected along a 1D line.
* 'dense' - contract the hyper index.
* int - use the 'parafac' mode, but only if the length of a clause
is larger than this threshold.
Returns
-------
htn : TensorNetwork
"""
ts = []
weights = {}
weighted = set()
clause_counter = 1
with open(fname, 'r') as f:
for line in f:
args = line.split()
# global info, don't need
if args[0] == 'p':
# num_vars = int(args[2])
# num_clauses = int(args[3])
continue
# translate mc2021 style weight to normal
if args[:3] == ['c', 'p', 'weight']:
args = ('w', *args[3:5])
# variable weight
if args[0] == 'w':
sgn_var, w = args[1:]
sgn_var = int(sgn_var)
sgn = '-' if sgn_var < 0 else '+'
var = str(abs(sgn_var))
w = float(w)
weights[var, sgn] = w
weighted.add(var)
continue
# ignore empty lines, other comments and info line
if (not args) or (args == ['0']) or (args[0][0] in 'c%'):
continue
# clause tensor
clause = tuple(map(int, filter(None, args[:-1])))
# encode the OR statement with possible negations as int
m = clause_negmask(clause)
inds = [str(abs(var)) for var in clause]
tag = f'CLAUSE{clause_counter}'
if (
# parafac mode
(mode == 'parafac' and len(inds) > 2) or
# parafac above cutoff size mode
(isinstance(mode, int) and len(inds) > mode)
):
ts.extend(clause_parafac_tensors(len(inds), m, inds, tag))
elif mode == 'mps' and len(inds) > 2:
ts.extend(or_clause_mps_tensors(len(inds), m, inds, tag))
else:
# dense
ts.append(or_clause_tensor(len(inds), m, inds, tag))
clause_counter += 1
for var in sorted(weighted):
wp_specified = (var, '+') in weights
wm_specified = (var, '-') in weights
if wp_specified and wm_specified:
wp, wm = weights[var, '+'], weights[var, '-']
elif wp_specified:
wp = weights[var, '+']
wm = 1 - wp
elif wm_specified:
wm = weights[var, '-']
wp = 1 - wm
ts.append(Tensor([wm, wp], inds=[var], tags=[f'VAR{var}']))
return TensorNetwork(ts, virtual=True)
# --------------------------------------------------------------------------- #
# MPSs #
# --------------------------------------------------------------------------- #
[docs]@random_seed_fn
def MPS_rand_state(L, bond_dim, phys_dim=2, normalize=True, cyclic=False,
dtype='float64', trans_invar=False, **mps_opts):
"""Generate a random matrix product state.
Parameters
----------
L : int
The number of sites.
bond_dim : int
The bond dimension.
phys_dim : int, optional
The physical (site) dimensions, defaults to 2.
normalize : bool, optional
Whether to normalize the state.
cyclic : bool, optional
Generate a MPS with periodic boundary conditions or not, default is
open boundary conditions.
dtype : {float, complex} or numpy dtype, optional
Data type of the tensor network.
trans_invar : bool (optional)
Whether to generate a translationally invariant state,
requires cyclic=True.
mps_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductState`.
"""
if trans_invar and not cyclic:
raise ValueError("State cannot be translationally invariant with open "
"boundary conditions.")
# check for site varying physical dimensions
if isinstance(phys_dim, Integral):
phys_dims = itertools.repeat(phys_dim)
else:
phys_dims = itertools.cycle(phys_dim)
cyc_dim = (bond_dim,) if cyclic else ()
def gen_shapes():
yield (*cyc_dim, bond_dim, next(phys_dims))
for _ in range(L - 2):
yield (bond_dim, bond_dim, next(phys_dims))
yield (bond_dim, *cyc_dim, next(phys_dims))
def gen_data(shape):
return randn(shape, dtype=dtype)
if trans_invar:
array = sensibly_scale(gen_data(next(gen_shapes())))
arrays = (array for _ in range(L))
else:
arrays = map(sensibly_scale, map(gen_data, gen_shapes()))
rmps = MatrixProductState(arrays, **mps_opts)
if normalize == 'left':
rmps.left_canonize(normalize=True)
elif normalize == 'right':
rmps.left_canonize(normalize=True)
elif normalize:
rmps /= (rmps.H @ rmps)**0.5
return rmps
[docs]def MPS_product_state(arrays, cyclic=False, **mps_opts):
"""Generate a product state in MatrixProductState form, i,e,
with bond dimension 1, from single site vectors described by ``arrays``.
"""
cyc_dim = (1,) if cyclic else ()
def gen_array_shapes():
yield (*cyc_dim, 1, -1)
for _ in range(len(arrays) - 2):
yield (1, 1, -1)
yield (*cyc_dim, 1, -1)
mps_arrays = (asarray(array).reshape(*shape)
for array, shape in zip(arrays, gen_array_shapes()))
return MatrixProductState(mps_arrays, shape='lrp', **mps_opts)
[docs]def MPS_computational_state(binary, dtype='float64', cyclic=False, **mps_opts):
"""A computational basis state in Matrix Product State form.
Parameters
----------
binary : str or sequence of int
String specifying the state, e.g. ``'00101010111'`` or ``[0, 0, 1]``.
cyclic : bool, optional
Generate a MPS with periodic boundary conditions or not, default open
boundary conditions.
mps_opts
Supplied to MatrixProductState constructor.
"""
array_map = {
'0': np.array([1., 0.], dtype=dtype),
'1': np.array([0., 1.], dtype=dtype),
'+': np.array([2**-0.5, 2**-0.5], dtype=dtype),
'-': np.array([2**-0.5, -2**-0.5], dtype=dtype),
}
def gen_arrays():
for s in binary:
yield array_map[str(s)]
return MPS_product_state(tuple(gen_arrays()), cyclic=cyclic, **mps_opts)
[docs]def MPS_neel_state(L, down_first=False, dtype='float64', **mps_opts):
"""Generate the neel state in Matrix Product State form.
Parameters
----------
L : int
The number of spins.
down_first : bool, optional
Whether to start with '1' (down) or '0' (up) first.
mps_opts
Supplied to MatrixProductState constructor.
"""
binary_str = "01" * (L // 2) + (L % 2 == 1) * "0"
if down_first:
binary_str = "1" + binary_str[:-1]
return MPS_computational_state(binary_str, dtype=dtype, **mps_opts)
[docs]def MPS_ghz_state(L, dtype='float64', **mps_opts):
"""Build the chi=2 OBC MPS representation of the GHZ state.
Parameters
----------
L : int
Number of qubits.
dtype : {'float64', 'complex128', 'float32', 'complex64'}, optional
The underlying data type.
mps_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductState`.
"""
def gen_arrays():
yield 2**-0.5 * np.array([[1., 0.],
[0., 1.]]).astype(dtype)
for i in range(1, L - 1):
yield np.array([[[1., 0.],
[0., 0.]],
[[0., 0.],
[0., 1.]]]).astype(dtype)
yield np.array([[1., 0.],
[0., 1.]]).astype(dtype)
return MatrixProductState(gen_arrays(), **mps_opts)
[docs]def MPS_w_state(L, dtype='float64', **mps_opts):
"""Build the chi=2 OBC MPS representation of the W state.
Parameters
----------
L : int
Number of qubits.
dtype : {'float64', 'complex128', 'float32', 'complex64'}, optional
The underlying data type.
mps_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductState`.
"""
def gen_arrays():
yield (np.array([[1., 0.],
[0., 1.]]) / L ** 0.5).astype(dtype)
for i in range(1, L - 1):
yield np.array([[[1., 0.],
[0., 1.]],
[[0., 0.],
[1., 0.]]]).astype(dtype)
yield np.array([[0., 1.],
[1., 0.]]).astype(dtype)
return MatrixProductState(gen_arrays(), **mps_opts)
[docs]@random_seed_fn
def MPS_rand_computational_state(L, dtype='float64', **mps_opts):
"""Generate a random computation basis state, like '01101001010'.
Parameters
----------
L : int
The number of qubits.
seed : int, optional
The seed to use.
dtype : {float, complex} or numpy dtype, optional
Data type of the tensor network.
mps_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductState`.
"""
cstr = (choice(('0', '1')) for _ in range(L))
return MPS_computational_state(cstr, dtype=dtype, **mps_opts)
[docs]def MPS_zero_state(L, bond_dim=1, phys_dim=2, cyclic=False,
dtype='float64', **mps_opts):
"""The all-zeros MPS state, of given bond-dimension.
Parameters
----------
L : int
The number of sites.
bond_dim : int, optional
The bond dimension, defaults to 1.
phys_dim : int, optional
The physical (site) dimensions, defaults to 2.
cyclic : bool, optional
Generate a MPS with periodic boundary conditions or not, default is
open boundary conditions.
dtype : {float, complex} or numpy dtype, optional
Data type of the tensor network.
mps_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductState`.
"""
cyc_dim = (bond_dim,) if cyclic else ()
def gen_arrays():
yield np.zeros((*cyc_dim, bond_dim, phys_dim), dtype=dtype)
for _ in range(L - 2):
yield np.zeros((bond_dim, bond_dim, phys_dim), dtype=dtype)
yield np.zeros((bond_dim, *cyc_dim, phys_dim), dtype=dtype)
return MatrixProductState(gen_arrays(), **mps_opts)
[docs]def MPS_sampler(L, dtype=complex, squeeze=True, **mps_opts):
"""A product state for sampling tensor network traces. Seen as a vector it
has the required property that ``psi.H @ psi == d`` always for hilbert
space size ``d``.
"""
arrays = [rand_phase(2, dtype=dtype) for _ in range(L)]
psi = MPS_product_state(arrays, **mps_opts)
if squeeze:
psi.squeeze_()
return psi
# --------------------------------------------------------------------------- #
# MPOs #
# --------------------------------------------------------------------------- #
[docs]def MPO_identity(L, phys_dim=2, dtype='float64', cyclic=False, **mpo_opts):
"""Generate an identity MPO of size ``L``.
Parameters
----------
L : int
The number of sites.
phys_dim : int, optional
The physical (site) dimensions, defaults to 2.
dtype : {float, complex} or numpy dtype, optional
Data type of the tensor network.
cyclic : bool, optional
Generate a MPO with periodic boundary conditions or not, default is
open boundary conditions.
mpo_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductOperator`.
"""
II = np.identity(phys_dim, dtype=dtype)
cyc_dim = (1,) if cyclic else ()
def gen_arrays():
yield II.reshape(*cyc_dim, 1, phys_dim, phys_dim)
for _ in range(L - 2):
yield II.reshape(1, 1, phys_dim, phys_dim)
yield II.reshape(1, *cyc_dim, phys_dim, phys_dim)
return MatrixProductOperator(gen_arrays(), **mpo_opts)
[docs]def MPO_identity_like(mpo, **mpo_opts):
"""Return an identity matrix operator with the same physical index and
inds/tags as ``mpo``.
"""
return MPO_identity(L=mpo.L, phys_dim=mpo.phys_dim(), dtype=mpo.dtype,
site_tag_id=mpo.site_tag_id, cyclic=mpo.cyclic,
upper_ind_id=mpo.upper_ind_id,
lower_ind_id=mpo.lower_ind_id, **mpo_opts)
[docs]def MPO_zeros(L, phys_dim=2, dtype='float64', cyclic=False, **mpo_opts):
"""Generate a zeros MPO of size ``L``.
Parameters
----------
L : int
The number of sites.
phys_dim : int, optional
The physical (site) dimensions, defaults to 2.
dtype : {float, complex} or numpy dtype, optional
Data type of the tensor network.
cyclic : bool, optional
Generate a MPO with periodic boundary conditions or not, default is
open boundary conditions.
mpo_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductOperator`.
Returns
-------
MatrixProductOperator
"""
cyc_dim = (1,) if cyclic else ()
def gen_arrays():
yield np.zeros((*cyc_dim, 1, phys_dim, phys_dim), dtype=dtype)
for _ in range(L - 2):
yield np.zeros((1, 1, phys_dim, phys_dim), dtype=dtype)
yield np.zeros((1, *cyc_dim, phys_dim, phys_dim), dtype=dtype)
return MatrixProductOperator(gen_arrays(), **mpo_opts)
[docs]def MPO_zeros_like(mpo, **mpo_opts):
"""Return a zeros matrix product operator with the same physical index and
inds/tags as ``mpo``.
Parameters
----------
mpo : MatrixProductOperator
The MPO to copy the shape of.
Returns
-------
MatrixProductOperator
"""
return MPO_zeros(L=mpo.L, phys_dim=mpo.phys_dim(),
dtype=mpo.dtype, site_tag_id=mpo.site_tag_id,
upper_ind_id=mpo.upper_ind_id, cyclic=mpo.cyclic,
lower_ind_id=mpo.lower_ind_id, **mpo_opts)
[docs]def MPO_product_operator(
arrays,
cyclic=False,
**mpo_opts,
):
"""Return an MPO of bond dimension 1 representing the product of raw
operators given in ``arrays``.
Parameters
----------
arrays : sequence of 2D array_like
The operators to form a tensor product of.
cyclic : bool, optional
Whether to generate a cyclic MPO or not.
mpo_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductOperator`.
Returns
-------
MatrixProductOperator
"""
cyc_dim = (1,) if cyclic else ()
def gen_arrays():
array_i, *arrays_mid, array_f = arrays
yield reshape(array_i, (*cyc_dim, 1, *array_i.shape))
for array_m in arrays_mid:
yield reshape(array_m, (1, 1, *array_m.shape))
yield reshape(array_f, (*cyc_dim, 1, *array_f.shape))
return MatrixProductOperator(gen_arrays(), shape='lrud', **mpo_opts)
[docs]@random_seed_fn
def MPO_rand(L, bond_dim, phys_dim=2, normalize=True, cyclic=False,
herm=False, dtype='float64', **mpo_opts):
"""Generate a random matrix product state.
Parameters
----------
L : int
The number of sites.
bond_dim : int
The bond dimension.
phys_dim : int, optional
The physical (site) dimensions, defaults to 2.
normalize : bool, optional
Whether to normalize the operator such that ``trace(A.H @ A) == 1``.
cyclic : bool, optional
Generate a MPO with periodic boundary conditions or not, default is
open boundary conditions.
dtype : {float, complex} or numpy dtype, optional
Data type of the tensor network.
herm : bool, optional
Whether to make the matrix hermitian (or symmetric if real) or not.
mpo_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductOperator`.
"""
cyc_shp = (bond_dim,) if cyclic else ()
shapes = [(*cyc_shp, bond_dim, phys_dim, phys_dim),
*((bond_dim, bond_dim, phys_dim, phys_dim),) * (L - 2),
(bond_dim, *cyc_shp, phys_dim, phys_dim)]
def gen_data(shape):
data = randn(shape, dtype=dtype)
if not herm:
return data
trans = (0, 2, 1) if len(shape) == 3 else (0, 1, 3, 2)
return data + data.transpose(*trans).conj()
arrays = map(sensibly_scale, map(gen_data, shapes))
rmpo = MatrixProductOperator(arrays, **mpo_opts)
if normalize:
rmpo /= (rmpo.H @ rmpo)**0.5
return rmpo
[docs]@random_seed_fn
def MPO_rand_herm(L, bond_dim, phys_dim=2, normalize=True,
dtype='float64', **mpo_opts):
"""Generate a random hermitian matrix product operator.
See :class:`~quimb.tensor.tensor_gen.MPO_rand`.
"""
return MPO_rand(L, bond_dim, phys_dim=phys_dim, normalize=normalize,
dtype=dtype, herm=True, **mpo_opts)
# ---------------------------- MPO hamiltonians ----------------------------- #
[docs]def maybe_make_real(X):
"""Check if ``X`` is real, if so, convert to contiguous array.
"""
if np.allclose(X.imag, np.zeros_like(X)):
return np.ascontiguousarray(X.real)
return X
[docs]def spin_ham_mpo_tensor(one_site_terms, two_site_terms, S=1 / 2,
left_two_site_terms=None, which=None, cyclic=False):
"""Generate tensor(s) for a spin hamiltonian MPO.
Parameters
----------
one_site_terms : sequence of (scalar, operator)
The terms that act on a single site, each ``operator`` can be a string
suitable to be sent to :func:`spin_operator` or an actual 2d-array.
two_site_terms : sequence of (scalar, operator operator)
The terms that act on two neighbouring sites, each ``operator`` can be
a string suitable to be sent to :func:`spin_operator` or an actual
2d-array.
S : fraction, optional
What size spin to use, defaults to spin-1/2.
left_two_site_terms : sequence of (scalar, operator operator), optional
If the interaction to the left of this site has different spin terms
then the equivalent list of terms for that site.
which : {None, 'L', 'R', 'A'}, optional
If ``None``, generate the middle tensor, if 'L' a left-end tensor, if
'R' a right-end tensor and if 'A' all three.
cyclic : bool, optional
Whether to use periodic boundary conditions - default False.
Returns
-------
numpy.ndarray[, numpy.ndarray, numpy.ndarray]
The middle, left, right or all three MPO tensors.
"""
# assume same interaction type everywhere
if left_two_site_terms is None:
left_two_site_terms = two_site_terms
# local dimension
D = int(2 * S + 1)
# bond dimension to right
B = len(two_site_terms) + 2
# bond dimension to left
BL = len(left_two_site_terms) + 2
H = np.zeros((BL, B, D, D), dtype=complex)
# add one-body terms
for factor, s in one_site_terms:
if isinstance(s, str):
s = spin_operator(s, S=S)
H[-1, 0, :, :] += factor * s
# add two-body terms
for i, (factor, s1, _) in enumerate(two_site_terms):
if isinstance(s1, str):
s1 = spin_operator(s1, S=S)
H[-1, 1 + i, :, :] = factor * s1
for i, (_, _, s2) in enumerate(left_two_site_terms):
if isinstance(s2, str):
s2 = spin_operator(s2, S=S)
H[i + 1, 0, :, :] = s2
H[0, 0, :, :] = eye(D)
H[-1, -1, :, :] = eye(D)
H = maybe_make_real(H)
make_immutable(H)
if which in {None, 'M'}:
return H
if cyclic:
# need special conditions for first MPO matrix
HL = np.zeros_like(H)
HL[0, :, :, :] = H[-1, :, :, :]
HL[1:-1, -1, :, :] = H[1:-1, 0, :, :]
HR = H
else:
HL = H[-1, :, :, :]
HR = H[:, 0, :, :]
if which == 'L':
return HL
elif which == 'R':
return HR
elif which == 'A':
return HL, H, HR
class _TermAdder:
"""Simple class to allow ``SpinHam1D`` syntax like
``builder[i, j] += (1/2, 'Z', 'X')``. This object is temporarily created
by the getitem call, accumulates the new term, then has its the new
combined list of terms extracted in the setitem call.
"""
def __init__(self, terms, nsite):
self.terms = terms
self.nsite = nsite
def __iadd__(self, new):
if len(new) != self.nsite + 1:
raise ValueError(
"New terms should be of the form")
if self.terms is None:
self.terms = [new]
else:
self.terms += [new]
return self
[docs]class SpinHam1D:
"""Class for easily building custom spin hamiltonians in MPO or LocalHam1D
form. Currently limited to nearest neighbour interactions (and single site
terms). It is possible to set 'default' translationally invariant terms,
but also terms acting on specific sites only (which take precedence).
It is also possible to build a sparse matrix version of the hamiltonian
(obviously for small sizes only).
Parameters
----------
S : float, optional
The type of spin, defaults to 1/2.
cyclic : bool, optional
Whether to use periodic boundary conditions - default is False.
Examples
--------
Initialize the spin hamiltonian builder:
>>> builder = SpinHam1D(S=3 / 2)
Add some two-site terms:
>>> builder += 0.5, '+', '-'
>>> builder += 0.5, '-', '+'
>>> builder += 1.0, 'Z', 'Z'
Add a single site term:
>>> builder -= 0.3, 'Z'
Build a MPO version of the hamiltonian for use with DMRG:
>>> mpo_ham = builder.build_mpo(100)
>>> mpo_ham
<MatrixProductOperator(tensors=100, L=100, max_bond=5)>
Build a LocalHam1D version of the hamiltonian for use with TEBD:
>>> builder.build_local_ham(100)
<LocalHam1D(L=100, cyclic=False)>
You can also set terms for specific sites (this overides any of the
'default', translationally invariant terms set as above):
>>> builder[10, 11] += 0.75, '+', '-'
>>> builder[10, 11] += 0.75, '-', '+'
>>> builder[10, 11] += 1.5, 'Z', 'Z'
Or specific one-site terms (which again overides any default
single site terms set above):
>>> builder[10] += 3.7, 'Z'
>>> builder[11] += 0.0, 'I' # '0' term turns off field
"""
def __init__(self, S=1 / 2, cyclic=False):
self.S = S
self.one_site_terms = []
self.two_site_terms = []
self.cyclic = cyclic
# Holders for any non-translationally invariant terms
self.var_one_site_terms = {}
self.var_two_site_terms = {}
[docs] def add_term(self, factor, *operators):
"""Add another term to the expression to be built.
Parameters
----------
factor : scalar
Scalar factor to multiply this term by.
*operators : str or array
The operators to use. Can specify one or two for single or two site
terms respectively. Can use strings, which are supplied to
:func:`~quimb.spin_operator`, or actual arrays as long as they have
the correct dimension.
"""
if factor == 0.0:
# no need to add zero terms
return
if len(operators) == 1:
self.one_site_terms.append((factor, *operators))
elif len(operators) == 2:
self.two_site_terms.append((factor, *operators))
else:
raise NotImplementedError("3-body+ terms are not supported yet.")
[docs] def sub_term(self, factor, *operators):
"""Subtract a term - simple alias that flips sign of ``factor``.
"""
self.add_term(-factor, *operators)
def __iadd__(self, term):
self.add_term(*term)
return self
def __isub__(self, term):
self.sub_term(*term)
return self
def __getitem__(self, sites):
"""Part of the machinery that allows terms to be added to specific
sites like::
>>> builder[i] += 1/2, 'X'
>>> builder[45, 46] += 1/2, 'Z', 'Z'
"""
if isinstance(sites, Integral):
return _TermAdder(self.var_one_site_terms.get(sites, None), 1)
i, j = sorted(sites)
if j - i != 1:
raise NotImplementedError("Can only add nearest neighbour terms.")
return _TermAdder(self.var_two_site_terms.get(sites, None), 2)
def __setitem__(self, sites, value):
"""Part of the machinery that allows terms to be added to specific
sites like::
>>> builder[i] += 1/2, 'X'
>>> builder[45, 46] += 1/2, 'Z', 'Z'
Could also be called directly with a list of terms like::
>>> builder[13, 14] = [(1, 'Z', 'Z'), (0.5, 'X', 'Y')]
Which would overide any terms set so far.
"""
if isinstance(value, _TermAdder):
terms = value.terms
else:
terms = value
if isinstance(sites, Integral):
self.var_one_site_terms[sites] = terms
else:
i, j = sorted(sites)
if j - i != 1:
raise ValueError("Can only add nearest neighbour terms.")
self.var_two_site_terms[sites] = terms
[docs] def build_mpo(self, L, upper_ind_id='k{}', lower_ind_id='b{}',
site_tag_id='I{}', tags=None, bond_name=""):
"""Build an MPO instance of this spin hamiltonian of size ``L``. See
also ``MatrixProductOperator``.
"""
# cache the default term
t_defs = {}
def get_default_term(which):
try:
return t_defs[which]
except KeyError:
t_defs['L'], t_defs[None], t_defs['R'] = spin_ham_mpo_tensor(
self.one_site_terms, self.two_site_terms,
S=self.S, which='A', cyclic=self.cyclic)
return t_defs[which]
def gen_tensors():
for i in range(L):
which = {0: 'L', L - 1: 'R'}.get(i, None)
ij_L = (i - 1, i)
ij_R = (i, i + 1)
# check for site/bond specific terms
var_one = i in self.var_one_site_terms
var_two = (
(ij_L in self.var_two_site_terms) or
(ij_R in self.var_two_site_terms)
)
if not (var_one or var_two):
yield get_default_term(which)
else:
t1s = self.var_one_site_terms.get(i, self.one_site_terms)
t2s = self.var_two_site_terms.get(ij_R,
self.two_site_terms)
t2s_L = self.var_two_site_terms.get(ij_L,
self.two_site_terms)
yield spin_ham_mpo_tensor(t1s, t2s, S=self.S,
left_two_site_terms=t2s_L,
which=which, cyclic=self.cyclic)
return MatrixProductOperator(arrays=gen_tensors(), bond_name=bond_name,
upper_ind_id=upper_ind_id,
lower_ind_id=lower_ind_id,
site_tag_id=site_tag_id, tags=tags)
[docs] def build_sparse(self, L, **ikron_opts):
"""Build a sparse matrix representation of this Hamiltonian.
Parameters
----------
L : int, optional
The number of spins to build the matrix for.
ikron_opts
Supplied to :func:`~quimb.core.ikron`.
Returns
-------
H : matrix
"""
ikron_opts.setdefault('sparse', True)
D = int(2 * self.S + 1)
dims = [D] * L
terms = []
for i in range(L):
t1s = self.var_one_site_terms.get(i, self.one_site_terms)
for factor, s in t1s:
if isinstance(s, str):
s = spin_operator(s, S=self.S, sparse=True)
terms.append(
ikron(factor * s, dims, i, **ikron_opts)
)
if (i + 1 == L) and (not self.cyclic):
break
t2s = self.var_two_site_terms.get((i, i + 1), self.two_site_terms)
for factor, s1, s2 in t2s:
if isinstance(s1, str):
s1 = spin_operator(s1, S=self.S, sparse=True)
if isinstance(s2, str):
s2 = spin_operator(s2, S=self.S, sparse=True)
terms.append(
ikron([factor * s1, s2], dims, [i, i + 1], **ikron_opts)
)
return sum(terms)
def _get_spin_op(self, factor, *ss):
if len(ss) == 1:
s, = ss
if isinstance(s, str):
s = spin_operator(s, S=self.S)
return factor * s
if len(ss) == 2:
s1, s2 = ss
if isinstance(s1, str):
s1 = spin_operator(s1, S=self.S)
if isinstance(s2, str):
s2 = spin_operator(s2, S=self.S)
return factor * (s1 & s2)
def _sum_spin_ops(self, terms):
H = sum(self._get_spin_op(*term) for term in terms)
H = maybe_make_real(H)
make_immutable(H)
return H
[docs] def build_local_ham(self, L=None, **local_ham_1d_opts):
"""Build a nearest neighbour interactor instance of this spin
hamiltonian of size ``L``. See also
:class:`~quimb.tensor.tensor_1d_tebd.LocalHam1D`.
Parameters
----------
L : int, optional
The number of spins, if the hamiltonian only has two-site terms
this is optional.
Returns
-------
LocalHam1D
"""
H1s, H2s = {}, {}
# add default two site term
if self.two_site_terms:
H2s[None] = self._sum_spin_ops(self.two_site_terms)
# add specific two site terms
if self.var_two_site_terms:
for sites, terms in self.var_two_site_terms.items():
H2s[sites] = self._sum_spin_ops(terms)
# add default one site term
if self.one_site_terms:
H1s[None] = self._sum_spin_ops(self.one_site_terms)
# add specific one site terms
if self.var_one_site_terms:
for site, terms in self.var_one_site_terms.items():
H1s[site] = self._sum_spin_ops(terms)
return LocalHam1D(H2=H2s, H1=H1s, L=L,
cyclic=self.cyclic, **local_ham_1d_opts)
SpinHam = deprecated(SpinHam1D, 'SpinHam', 'SpinHam1D')
def _ham_ising(j=1.0, bx=0.0, *, S=1 / 2, cyclic=False):
H = SpinHam1D(S=1 / 2, cyclic=cyclic)
H += j, 'Z', 'Z'
H -= bx, 'X'
return H
[docs]def MPO_ham_ising(L, j=1.0, bx=0.0, *, S=1 / 2, cyclic=False, **mpo_opts):
r"""Ising Hamiltonian in MPO form.
.. math::
H_\mathrm{Ising} =
J \sum_{i} \sigma^Z_i \sigma^Z_{i + 1} -
B_x \sum_{i} \sigma^X_i
Note the default convention of antiferromagnetic interactions and spin
operators not Pauli matrices.
Parameters
----------
L : int
The number of sites.
j : float, optional
The ZZ interaction strength. Positive is antiferromagnetic.
bx : float, optional
The X-magnetic field strength.
S : {1/2, 1, 3/2, ...}, optional
The underlying spin of the system, defaults to 1/2.
cyclic : bool, optional
Generate a MPO with periodic boundary conditions or not, default is
open boundary conditions.
mpo_opts or local_ham_1d_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductOperator`.
Returns
-------
MatrixProductOperator
"""
H = _ham_ising(j=j, bx=bx, S=S, cyclic=cyclic)
return H.build_mpo(L, **mpo_opts)
[docs]def ham_1d_ising(L=None, j=1.0, bx=0.0, *, S=1 / 2,
cyclic=False, **local_ham_1d_opts):
r"""Ising Hamiltonian in
:class:`~quimb.tensor.tensor_1d_tebd.LocalHam1D` form.
.. math::
H_\mathrm{Ising} =
J \sum_{i} \sigma^Z_i \sigma^Z_{i + 1} -
B_x \sum_{i} \sigma^X_i
Note the default convention of antiferromagnetic interactions and spin
operators not Pauli matrices.
Parameters
----------
L : int
The number of sites.
j : float, optional
The ZZ interaction strength. Positive is antiferromagnetic.
bx : float, optional
The X-magnetic field strength.
S : {1/2, 1, 3/2, ...}, optional
The underlying spin of the system, defaults to 1/2.
cyclic : bool, optional
Generate a hamiltonian with periodic boundary conditions or not,
default is open boundary conditions.
mpo_opts or local_ham_1d_opts
Supplied to :class:`~quimb.tensor.tensor_1d.LocalHam1D`.
Returns
-------
LocalHam1D
"""
H = _ham_ising(j=j, bx=bx, S=S, cyclic=cyclic)
return H.build_local_ham(L=L, **local_ham_1d_opts)
NNI_ham_ising = deprecated(ham_1d_ising, 'NNI_ham_ising', 'ham_1d_ising')
def _ham_XY(j=1.0, bz=0.0, *, S=1 / 2, cyclic=False):
H = SpinHam1D(S=S, cyclic=cyclic)
try:
jx, jy = j
except (TypeError, ValueError):
jx = jy = j
if jx == jy:
# easy way to enforce realness
H += jx / 2, '+', '-'
H += jx / 2, '-', '+'
else:
H += jx, 'X', 'X'
H += jy, 'Y', 'Y'
H -= bz, 'Z'
return H
[docs]def MPO_ham_XY(L, j=1.0, bz=0.0, *, S=1 / 2, cyclic=False, **mpo_opts):
r"""XY-Hamiltonian in MPO form.
.. math::
H_\mathrm{XY} =
\sum_{i} (
J_X \sigma^X_i \sigma^X_{i + 1} +
J_Y \sigma^Y_i \sigma^Y_{i + 1}
)
- B_x \sum_{i} \sigma^Z_i
Note the default convention of antiferromagnetic interactions and spin
operators not Pauli matrices.
Parameters
----------
L : int
The number of sites.
j : float or (float, float), optional
The XX and YY interaction strength. Positive is antiferromagnetic.
bz : float, optional
The Z-magnetic field strength.
S : {1/2, 1, 3/2, ...}, optional
The underlying spin of the system, defaults to 1/2.
cyclic : bool, optional
Generate a MPO with periodic boundary conditions or not, default is
open boundary conditions.
mpo_opts or local_ham_1d_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductOperator`.
Returns
-------
MatrixProductOperator
"""
H = _ham_XY(j=j, bz=bz, S=S, cyclic=cyclic)
return H.build_mpo(L, **mpo_opts)
[docs]def ham_1d_XY(L=None, j=1.0, bz=0.0, *, S=1 / 2,
cyclic=False, **local_ham_1d_opts):
r"""XY-Hamiltonian in
:class:`~quimb.tensor.tensor_1d_tebd.LocalHam1D` form.
.. math::
H_\mathrm{XY} =
\sum_{i} (
J_X \sigma^X_i \sigma^X_{i + 1} +
J_Y \sigma^Y_i \sigma^Y_{i + 1}
)
- B_Z \sum_{i} \sigma^Z_i
Note the default convention of antiferromagnetic interactions and spin
operators not Pauli matrices.
Parameters
----------
L : int
The number of sites.
j : float or (float, float), optional
The XX and YY interaction strength. Positive is antiferromagnetic.
bz : float, optional
The Z-magnetic field strength.
S : {1/2, 1, 3/2, ...}, optional
The underlying spin of the system, defaults to 1/2.
cyclic : bool, optional
Generate a hamiltonian with periodic boundary conditions or not,
default is open boundary conditions.
local_ham_1d_opts
Supplied to :class:`~quimb.tensor.tensor_1d.LocalHam1D`.
Returns
-------
LocalHam1D
"""
H = _ham_XY(j=j, bz=bz, S=S, cyclic=cyclic)
return H.build_local_ham(L=L, **local_ham_1d_opts)
NNI_ham_XY = deprecated(ham_1d_XY, 'NNI_ham_XY', 'ham_1d_XY')
def _ham_heis(j=1.0, bz=0.0, *, S=1 / 2, cyclic=False):
H = SpinHam1D(S=S, cyclic=cyclic)
try:
jx, jy, jz = j
except (TypeError, ValueError):
jx = jy = jz = j
if jx == jy:
# easy way to enforce realness
H += jx / 2, '+', '-'
H += jx / 2, '-', '+'
else:
H += jx, 'X', 'X'
H += jy, 'Y', 'Y'
H += jz, 'Z', 'Z'
H -= bz, 'Z'
return H
[docs]def MPO_ham_heis(L, j=1.0, bz=0.0, *, S=1 / 2, cyclic=False, **mpo_opts):
r"""Heisenberg Hamiltonian in MPO form.
.. math::
H_\mathrm{Heis} =
\sum_{i} (
J_X \sigma^X_i \sigma^X_{i + 1} +
J_Y \sigma^Y_i \sigma^Y_{i + 1} +
J_Z \sigma^Z_i \sigma^Z_{i + 1}
)
- B_Z \sum_{i} \sigma^Z_i
Note the default convention of antiferromagnetic interactions and spin
operators not Pauli matrices.
Parameters
----------
L : int
The number of sites.
j : float or (float, float, float), optional
The XX, YY and ZZ interaction strength. Positive is antiferromagnetic.
bz : float, optional
The Z-magnetic field strength.
S : {1/2, 1, 3/2, ...}, optional
The underlying spin of the system, defaults to 1/2.
cyclic : bool, optional
Generate a MPO with periodic boundary conditions or not, default is
open boundary conditions.
mpo_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductOperator`.
Returns
-------
MatrixProductOperator
"""
H = _ham_heis(j=j, bz=bz, S=S, cyclic=cyclic)
return H.build_mpo(L, **mpo_opts)
[docs]def ham_1d_heis(L=None, j=1.0, bz=0.0, *, S=1 / 2,
cyclic=False, **local_ham_1d_opts):
r"""Heisenberg Hamiltonian in
:class:`~quimb.tensor.tensor_1d_tebd.LocalHam1D` form.
.. math::
H_\mathrm{Heis} =
\sum_{i} (
J_X \sigma^X_i \sigma^X_{i + 1} +
J_Y \sigma^Y_i \sigma^Y_{i + 1} +
J_Z \sigma^Z_i \sigma^Z_{i + 1}
)
- B_Z \sum_{i} \sigma^Z_i
Note the default convention of antiferromagnetic interactions and spin
operators not Pauli matrices.
Parameters
----------
L : int
The number of sites.
j : float or (float, float, float), optional
The XX, YY and ZZ interaction strength. Positive is antiferromagnetic.
bz : float, optional
The Z-magnetic field strength.
S : {1/2, 1, 3/2, ...}, optional
The underlying spin of the system, defaults to 1/2.
cyclic : bool, optional
Generate a hamiltonian with periodic boundary conditions or not,
default is open boundary conditions.
local_ham_1d_opts
Supplied to :class:`~quimb.tensor.tensor_1d_tebd.LocalHam1D`.
Returns
-------
LocalHam1D
"""
H = _ham_heis(j=j, bz=bz, S=S, cyclic=cyclic)
return H.build_local_ham(L=L, **local_ham_1d_opts)
NNI_ham_heis = deprecated(ham_1d_heis, 'NNI_ham_heis', 'ham_1d_heis')
[docs]def MPO_ham_XXZ(L, delta, jxy=1.0, *, S=1 / 2, cyclic=False, **mpo_opts):
r"""XXZ-Hamiltonian in MPO form.
.. math::
H_\mathrm{XXZ} =
\sum_{i} (
J_X \sigma^X_i \sigma^X_{i + 1} +
J_Y \sigma^Y_i \sigma^Y_{i + 1} +
\Delta \sigma^Z_i \sigma^Z_{i + 1}
)
- B_Z \sum_{i} \sigma^Z_i
Note the default convention of antiferromagnetic interactions and spin
operators not Pauli matrices.
Parameters
----------
L : int
The number of sites.
delta : float
The ZZ-interaction strength. Positive is antiferromagnetic.
jxy : float, optional
The X- and Y- interaction strength, defaults to 1. Positive is
antiferromagnetic.
S : {1/2, 1, 3/2, ...}, optional
The underlying spin of the system, defaults to 1/2.
cyclic : bool, optional
Generate a MPO with periodic boundary conditions or not, default is
open boundary conditions.
mpo_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductOperator`.
Returns
-------
MatrixProductOperator
"""
return MPO_ham_heis(L, j=(jxy, jxy, delta), S=S, cyclic=cyclic, **mpo_opts)
[docs]def ham_1d_XXZ(L=None, delta=None, jxy=1.0, *,
S=1 / 2, cyclic=False, **local_ham_1d_opts):
r"""XXZ-Hamiltonian in
:class:`~quimb.tensor.tensor_1d_tebd.LocalHam1D` form.
.. math::
H_\mathrm{XXZ} =
\sum_{i} (
J_X \sigma^X_i \sigma^X_{i + 1} +
J_Y \sigma^Y_i \sigma^Y_{i + 1} +
\Delta \sigma^Z_i \sigma^Z_{i + 1}
)
- B_Z \sum_{i} \sigma^Z_i
Note the default convention of antiferromagnetic interactions and spin
operators not Pauli matrices.
Parameters
----------
L : int
The number of sites.
delta : float
The ZZ-interaction strength. Positive is antiferromagnetic.
jxy : float, optional
The X- and Y- interaction strength, defaults to 1. Positive is
antiferromagnetic.
S : {1/2, 1, 3/2, ...}, optional
The underlying spin of the system, defaults to 1/2.
cyclic : bool, optional
Generate a hamiltonian with periodic boundary conditions or not,
default is open boundary conditions.
local_ham_1d_opts
Supplied to :class:`~quimb.tensor.tensor_1d_tebd.LocalHam1D`.
Returns
-------
LocalHam1D
"""
if delta is None:
raise ValueError("You need to specify ``delta``.")
return ham_1d_heis(L, j=(jxy, jxy, delta), S=S,
cyclic=cyclic, **local_ham_1d_opts)
NNI_ham_XXZ = deprecated(ham_1d_XXZ, 'NNI_ham_XXZ', 'ham_1d_XXZ')
def _ham_bilinear_biquadratic(theta, *, S=1 / 2, cyclic=False):
H = SpinHam1D(S=S, cyclic=cyclic)
H += np.cos(theta), 'X', 'X'
H += np.cos(theta), 'Y', 'Y'
H += np.cos(theta), 'Z', 'Z'
XX = spin_operator('X', S=S) @ spin_operator('X', S=S)
YY = spin_operator('Y', S=S) @ spin_operator('Y', S=S)
ZZ = spin_operator('Z', S=S) @ spin_operator('Z', S=S)
H += np.sin(theta), XX, XX
H += np.sin(theta), XX, YY
H += np.sin(theta), XX, ZZ
H += np.sin(theta), YY, XX
H += np.sin(theta), YY, YY
H += np.sin(theta), YY, ZZ
H += np.sin(theta), ZZ, XX
H += np.sin(theta), ZZ, YY
H += np.sin(theta), ZZ, ZZ
return H
[docs]def MPO_ham_bilinear_biquadratic(L=None, theta=0, *, S=1 / 2, cyclic=False,
compress=True, **mpo_opts):
""" Hamiltonian of one-dimensional bilinear biquadratic chain in MPO form,
see PhysRevB.93.184428.
Parameters
----------
L : int
The number of sites.
theta : float or (float, float), optional
The parameter for linear and non-linear term of interaction strength,
defaults to 0.
S : {1/2, 1, 3/2, ...}, optional
The underlying spin of the system, defaults to 1/2.
cyclic : bool, optional
Generate a hamiltonian with periodic boundary conditions or not,
default is open boundary conditions.
mpo_opts
Supplied to :class:`~quimb.tensor.tensor_1d.MatrixProductOperator`.
Returns
-------
MatrixProductOperator
"""
H = _ham_bilinear_biquadratic(theta, S=S, cyclic=cyclic)
H_mpo = H.build_mpo(L, **mpo_opts)
if compress is True:
H_mpo.compress(cutoff=1e-12, cutoff_mode='rel' if cyclic else 'sum2')
return H_mpo
[docs]def ham_1d_bilinear_biquadratic(L=None, theta=0, *, S=1 / 2,
cyclic=False, **local_ham_1d_opts):
""" Hamiltonian of one-dimensional bilinear biquadratic chain in LocalHam1D
form, see PhysRevB.93.184428.
Parameters
----------
L : int
The number of sites.
theta : float or (float, float), optional
The parameter for linear and non-linear term of interaction strength,
defaults to 0.
S : {1/2, 1, 3/2, ...}, optional
The underlying spin of the system, defaults to 1/2.
cyclic : bool, optional
Generate a hamiltonian with periodic boundary conditions or not,
default is open boundary conditions.
local_ham_1d_opts
Supplied to :class:`~quimb.tensor.tensor_1d_tebd.LocalHam1D`.
Returns
-------
LocalHam1D
"""
H = _ham_bilinear_biquadratic(theta, S=S, cyclic=cyclic)
return H.build_local_ham(L=L, **local_ham_1d_opts)
NNI_ham_bilinear_biquadratic = deprecated(ham_1d_bilinear_biquadratic,
'NNI_ham_bilinear_biquadratic',
'ham_1d_bilinear_biquadratic')
def _ham_mbl(L, dh, j=1.0, seed=None, S=1 / 2, *, cyclic=False,
dh_dist='s', dh_dim=1, beta=None):
# start with the heisenberg builder
H = _ham_heis(j, S=S, cyclic=cyclic)
dhds, rs = _gen_mbl_random_factors(L, dh, dh_dim, dh_dist, seed, beta)
# generate noise, potentially in all directions, each with own strength
for i in range(L):
dh_r_xyzs = zip(dhds, rs[:, i], 'XYZ')
for dh, r, xyz in dh_r_xyzs:
H[i] += dh * r, xyz
return H
[docs]def MPO_ham_mbl(L, dh, j=1.0, seed=None, S=1 / 2, *, cyclic=False,
dh_dist='s', dh_dim=1, beta=None, **mpo_opts):
r"""The many-body-localized spin hamiltonian in MPO form.
.. math::
H_\mathrm{MBL} =
\sum_{i} (
J_X \sigma^X_i \sigma^X_{i + 1} +
J_Y \sigma^Y_i \sigma^Y_{i + 1} +
J_Z \sigma^Z_i \sigma^Z_{i + 1}
)
- \sum_{i} h_i \sigma^Z_i
Note the default convention of antiferromagnetic interactions and spin
operators not Pauli matrices.
Parameters
----------
L : int
Number of spins.
dh : float
Random noise strength.
j : float, or (float, float, float), optional
Interaction strength(s) e.g. 1 or (1., 1., 0.5). Positive is
antiferromagnetic.
seed : int, optional
Random number to seed the noise with.
S : {1/2, 1, 3/2, ...}, optional
The underlying spin of the system, defaults to 1/2.
cyclic : bool, optional
Whether to use periodic boundary conditions - default is False.
dh_dist : {'s', 'g', 'qp'}, optional
Whether to use sqaure, guassian or quasiperiodic noise.
beta : float, optional
Frequency of the quasirandom noise, only if ``dh_dist='qr'``.
mpo_opts
Supplied to :class:`MatrixProductOperator`.
Returns
-------
MatrixProductOperator
"""
H = _ham_mbl(L, dh=dh, j=j, seed=seed, S=S, cyclic=cyclic,
dh_dist=dh_dist, dh_dim=dh_dim, beta=beta)
return H.build_mpo(L, **mpo_opts)
[docs]def ham_1d_mbl(L, dh, j=1.0, seed=None, S=1 / 2, *, cyclic=False,
dh_dist='s', dh_dim=1, beta=None, **local_ham_1d_opts):
r"""The many-body-localized spin hamiltonian in
:class:`~quimb.tensor.tensor_1d_tebd.LocalHam1D` form.
.. math::
H_\mathrm{MBL} =
\sum_{i} (
J_X \sigma^X_i \sigma^X_{i + 1} +
J_Y \sigma^Y_i \sigma^Y_{i + 1} +
J_Z \sigma^Z_i \sigma^Z_{i + 1}
)
- \sum_{i} h_i \sigma^Z_i
Note the default convention of antiferromagnetic interactions and spin
operators not Pauli matrices.
Parameters
----------
L : int
Number of spins.
dh : float
Random noise strength.
j : float, or (float, float, float), optional
Interaction strength(s) e.g. 1 or (1., 1., 0.5). Positive is
antiferromagnetic.
seed : int, optional
Random number to seed the noise with.
S : {1/2, 1, 3/2, ...}, optional
The underlying spin of the system, defaults to 1/2.
cyclic : bool, optional
Whether to use periodic boundary conditions - default is False.
dh_dist : {'s', 'g', 'qp'}, optional
Whether to use sqaure, guassian or quasiperiodic noise.
beta : float, optional
Frequency of the quasirandom noise, only if ``dh_dist='qr'``.
local_ham_1d_opts
Supplied to :class:`LocalHam1D`.
Returns
-------
LocalHam1D
"""
H = _ham_mbl(L, dh=dh, j=j, seed=seed, S=S, cyclic=cyclic,
dh_dist=dh_dist, dh_dim=dh_dim, beta=beta)
return H.build_local_ham(L, **local_ham_1d_opts)
NNI_ham_mbl = deprecated(ham_1d_mbl, 'NNI_ham_mbl', 'ham_1d_mbl')
[docs]def ham_2d_ising(Lx, Ly, j=1.0, bx=0.0, **local_ham_2d_opts):
r"""Ising Hamiltonian in
:class:`~quimb.tensor.tensor_2d_tebd.LocalHam2D` form.
.. math::
H_\mathrm{Ising} =
J \sum_{<ij>} \sigma^Z_i \sigma^Z_{j} -
B_x \sum_{i} \sigma^X_i
for nearest neighbors :math:`<ij>`. Note the default convention of
antiferromagnetic interactions and spin operators not Pauli matrices.
Parameters
----------
Lx : int
The number of rows.
Ly : int
The number of columns.
j : float, optional
The ZZ interaction strength. Positive is antiferromagnetic.
bx : float, optional
The X-magnetic field strength.
local_ham_2d_opts
Supplied to :class:`~quimb.tensor.tensor_2d.LocalHam2D`.
Returns
-------
LocalHam2D
"""
H2 = ham_heis(2, j=(0, 0, j), cyclic=False)
if bx == 0.0:
H1 = None
else:
H1 = -bx * spin_operator('X').real
return LocalHam2D(Lx, Ly, H2=H2, H1=H1, **local_ham_2d_opts)
[docs]def ham_2d_heis(Lx, Ly, j=1.0, bz=0.0, **local_ham_2d_opts):
r"""Heisenberg Hamiltonian in
:class:`~quimb.tensor.tensor_2d_tebd.LocalHam2D`. form.
.. math::
H_\mathrm{Heis} =
\sum_{<ij>} (
J_X \sigma^X_i \sigma^X_{j} +
J_Y \sigma^Y_i \sigma^Y_{j} +
J_Z \sigma^Z_i \sigma^Z_{j}
)
- B_Z \sum_{i} \sigma^Z_{i}
for nearest neighbors :math:`<ij>`. Note the default convention of
antiferromagnetic interactions and spin operators not Pauli matrices.
Parameters
----------
Lx : int
The number of rows.
Ly : int
The number of columns.
j : float or (float, float, float), optional
The XX, YY and ZZ interaction strength. Positive is antiferromagnetic.
bz : float, optional
The Z-magnetic field strength.
local_ham_2d_opts
Supplied to :class:`~quimb.tensor.tensor_2d_tebd.LocalHam2D`.
Returns
-------
LocalHam2D
"""
H2 = ham_heis(2, j=j, cyclic=False)
if bz == 0.0:
H1 = None
else:
H1 = -bz * spin_operator('Z').real
return LocalHam2D(Lx, Ly, H2=H2, H1=H1, **local_ham_2d_opts)
[docs]def ham_2d_j1j2(Lx, Ly, j1=1.0, j2=0.5, bz=0.0, **local_ham_2d_opts):
r"""Heisenberg Hamiltonian in
:class:`~quimb.tensor.tensor_2d_tebd.LocalHam2D`. form.
.. math::
H_\mathrm{Heis} =
\sum_{<ij>} (
J_{1,X} \sigma^X_i \sigma^X_{j} +
J_{1,Y} \sigma^Y_i \sigma^Y_{j} +
J_{1,Z} \sigma^Z_i \sigma^Z_{j}
)
+
\sum_{<<ij>>} (
J_{2,X} \sigma^X_i \sigma^X_{j} +
J_{2,Y} \sigma^Y_i \sigma^Y_{j} +
J_{2,Z} \sigma^Z_i \sigma^Z_{j}
)
- B_Z \sum_{i} \sigma^Z_{i}
for nearest neighbors :math:`<ij>` and diagonal next nearest neighbors
:math:`<<ij>>`. Note the default convention of antiferromagnetic
interactions and spin operators not Pauli matrices.
Parameters
----------
Lx : int
The number of rows.
Ly : int
The number of columns.
j2 : float or (float, float, float), optional
The nearest neighbor XX, YY and ZZ interaction strength. Positive is
antiferromagnetic.
j2 : float or (float, float, float), optional
The diagonal next nearest nearest XX, YY and ZZ interaction strength.
Positive is antiferromagnetic.
bz : float, optional
The Z-magnetic field strength.
local_ham_2d_opts
Supplied to :class:`~quimb.tensor.tensor_2d_tebd.LocalHam2D`.
Returns
-------
LocalHam2D
"""
# default nearest neighbor interaction
H2 = {None: ham_heis(2, j=j1, cyclic=False)}
# diagonal next nearest neighbor interactions
for bond in gen_2d_bonds(Lx, Ly, steppers=[
lambda i, j: (i + 1, j - 1),
lambda i, j: (i + 1, j + 1),
]):
H2[bond] = ham_heis(2, j=j2, cyclic=False)
# magnetic field
if bz == 0.0:
H1 = None
else:
H1 = -bz * spin_operator('Z').real
return LocalHam2D(Lx, Ly, H2=H2, H1=H1, **local_ham_2d_opts)
[docs]def ham_3d_heis(Lx, Ly, Lz, j=1.0, bz=0.0, **local_ham_3d_opts):
r"""Heisenberg Hamiltonian in
:class:`~quimb.tensor.tensor_3d_tebd.LocalHam3D`. form.
.. math::
H_\mathrm{Heis} =
\sum_{<ij>} (
J_X \sigma^X_i \sigma^X_{j} +
J_Y \sigma^Y_i \sigma^Y_{j} +
J_Z \sigma^Z_i \sigma^Z_{j}
)
- B_Z \sum_{i} \sigma^Z_{i}
for nearest neighbors :math:`<ij>`. Note the default convention of
antiferromagnetic interactions and spin operators not Pauli matrices.
Parameters
----------
Lx : int
The number of x-planes.
Ly : int
The number of y-planes.
Ly : int
The number of z-planes.
j : float or (float, float, float), optional
The XX, YY and ZZ interaction strength. Positive is antiferromagnetic.
bz : float, optional
The Z-magnetic field strength.
local_ham_3d_opts
Supplied to :class:`~quimb.tensor.tensor_3d_tebd.LocalHam3D`.
Returns
-------
LocalHam3D
"""
H2 = ham_heis(2, j=j, cyclic=False)
if bz == 0.0:
H1 = None
else:
H1 = -bz * spin_operator('Z').real
return LocalHam3D(Lx, Ly, Lz, H2=H2, H1=H1, **local_ham_3d_opts)