Linear algebra with symmetric tensors#
Basic algebra operations#
def test_syntax_basic_algebra(self):
legs = [yastn.Leg(config_U1, s=-1, t=(-1, 0, 1), D=(1, 2, 3)),
yastn.Leg(config_U1, s=1, t=(-1, 1, 2), D=(4, 5, 6)),
yastn.Leg(config_U1, s=1, t=(-1, 1, 2), D=(7, 8, 9)),
yastn.Leg(config_U1, s=-1, t=(-1, 1, 2), D=(10, 11, 12))]
a = yastn.rand(config=config_U1, legs=legs)
#
# Tensor can be multiplied by scalar
#
tensor = a / 2
tensor = 2. * a
tensor = a * 2.
#
# Tensors can be added or subtracted assuming their structure is
# compatible.
#
b = yastn.ones(config=config_U1, legs=legs)
tensor = a + b
tensor = a - b
#
# Attempting to add/subtract two tensors with different total charge,
# or different dimension of a common charge sector raises exception
#
legs[0] = yastn.Leg(config_U1, s=-1, t=(-1, 0, 1), D=(7, 2, 3))
c = yastn.ones(config=config_U1, legs=legs)
with self.assertRaises(Exception):
tensor = a + c
#
# element-wise exponentiation, absolute value, reciprocal i.e. x -> 1/x,
# square root and its reciprocal x -> 1/sqrt(x)
#
tensor = a.exp(step=1)
tensor = yastn.exp(a, step=1)
tensor = abs(a)
tensor = a.reciprocal(cutoff=1e-12)
tensor = yastn.reciprocal(a, cutoff=1e-12)
tensor = abs(a).sqrt()
tensor = yastn.sqrt(abs(a))
tensor = abs(a).rsqrt(cutoff=1e-12)
tensor = yastn.rsqrt(abs(a), cutoff=1e-12)
#
# Sometimes a composite operation is faster than serial execution of individual
# operations. For example, multiplication by scalar and addition, a + x*b,
# are handled by specialized function
#
tensor = a.apxb(b, x=1)
tensor = yastn.apxb(a, b, x=1)
Tensor contractions#
Basic contractions with yastn.tensordot(), matrix-multiplication operator @, tracing with yastn.trace()
def test_syntax_contraction(self):
# create a set of U1-symmetric tensors
leg1 = yastn.Leg(config_U1, s=-1, t=(-1, 0, 1), D=(1, 2, 3))
leg2 = yastn.Leg(config_U1, s=1, t=(-1, 1, 2), D=(4, 5, 6))
leg3 = yastn.Leg(config_U1, s=1, t=(-1, 1, 2), D=(7, 8, 9))
leg4 = yastn.Leg(config_U1, s=-1, t=(-1, 1, 2), D=(10, 11, 12))
a = yastn.rand(config=config_U1, legs=[leg1, leg2, leg3, leg4])
b = yastn.ones(config=config_U1, legs=[leg1, leg2, leg3, leg4])
c = yastn.rand(config=config_U1, legs=[leg4.conj(), leg3, leg2.conj()])
# Contract a and b by two indices. The a tensor is conjugated, which
# reverses the signature on its indices
# __ _ ___
# 0->-|a*|->-1 1->-|b|->-0 = 0->-|a*b|->-0->2
# 3->-|__|->-2 2->-|_|->-3 1<-3->-|___|->-3
#
# The order of the indices on the resulting tensor is as follows:
# First, the outgoing indices of a (the first argument to tensordot), then
# the outgoing indices of tensor b
tensor = yastn.tensordot(a, b, axes=((1, 2), (1, 2)), conj=(1, 0))
# tensordot can also be invoked also as a function of the tensor itself
#
tensor = a.tensordot(b, axes=((1, 2), (1, 2)), conj=(1, 0))
# If no axes are specified, the outer product of two tensors is returned
tensor = yastn.tensordot(c, c, axes=((), ()) )
assert tensor.get_rank() == 6
# A shorthand notation for the specific contraction
# _ _ __
# 0-<-|a|-<-2 |c|-<-1 = 0-<-|ac|-<-2
# 1->-|_|->-3 0->-|_|->-2 1->-| |-<-1->3
# |__|->-2->4
t0 = yastn.tensordot(a, c, axes=(a.ndim - 1, 0))
#
# is the @ operator. For rank-2 tensor it is thus equivalent to matrix multiplication
t1 = a @ c
assert yastn.norm(t0 - t1) < tol
#
# Utility functions simplifying execution of contractions
t2 = yastn.ncon([a, c], ((-0, -1, -2, 1), (1, -3, -4)))
t3 = yastn.einsum('ijkx,xlm->ijklm', a, c)
assert yastn.norm(t0 - t2) < tol
assert yastn.norm(t0 - t3) < tol
# Another special case of tensor contraction is a dot product of vectorized tensors
# __ _
# |a*|-<-0 0-<-|b| = scalar
# | |->-1 1->-| |
# | |->-2 2->-| |
# |__|-<-3 3-<-|_|
tensor = a.tensordot(b, axes=((0, 1, 2, 3), (0, 1, 2, 3)), conj=(1, 0))
assert isinstance(tensor,yastn.Tensor)
#
# such single element symmetric Tensor can be converted to a single-element
# tensor of the backend type, or even further to python scalar
number = tensor.to_number()
python_scalar = tensor.item()
assert isinstance(python_scalar,float)
# A shorthand function for computing dot products is vdot
number = yastn.vdot(a, b)
number = a.vdot(b)
# Trace over certain indices can be computed using identically named function.
# In this case, a2_ijil = a2_jl
a2 = yastn.tensordot(a, a, axes=((0, 1), (0, 1)), conj=(1, 0))
tensor = a2.trace(axes=(0, 2))
assert tensor.get_rank()==2
#
# More pairs of indices can be traced at once a_ijij = scalar
tensor = a2.trace(axes=((0, 1), (2, 3)))
number = tensor.to_number()
Higher-level interface ncon (or equivalently einsum) composing simple contractions
def test_ncon_einsum_syntax():
# create a set of U1-symmetric tensors
a = yastn.rand(config=config_U1, s=[-1, 1, -1], n=0,
D=((20, 10), (3, 3), (1, 1)), t=((1, 0), (1, 0), (1, 0)))
b = yastn.rand(config=config_U1, s=[1, 1, 1], n=1,
D=((4, 4), (2, 2), (20, 10)), t=((1, 0), (1, 0), (1, 0)))
c = yastn.rand(config=config_U1, s=[1, 1, 1, -1], n=1,
D=((20, 10), (30, 20), (10, 5), (10, 5)), t=((1, 0), (1, 0), (1, 0), (1, 0)))
d = yastn.rand(config=config_U1, s=[1, 1, -1, -1], n=0,
D=((30, 20), (10, 5), (20, 10), (10, 5)), t=((1, 0), (1, 0), (1, 0), (1, 0)))
# Perform basic contraction of two tensors - equivalent to a single tensordot call
# _ _ _ _
# (-1) 1--|a|--0 (1) (1) 2--|b|--0 (-0) = (-1) 1--|a|--|b|--0 (-0)
# (-3) 2--|_| |_|--1 (-2) (-3) 3<-2--|_| |_|--1->2 (-2)
#
# The uncontracted indices, labeled by negative integers, are ordered according in
# descending fashion on resulting tensor
e = yastn.ncon([a, b], [[1, -1, -3], [-0, -2, 1]])
assert e.get_shape() == (8, 6, 4, 2)
# The same can be obtained using einsum function, which tries to mimic the syntax of np.einsum
e1 = yastn.einsum('xbd,acx->abcd', a, b)
assert yastn.norm(e1 - e) < tol
# Network composed of several tensors can be contracted by a single ncon call,
# including traces and conjugations
# _ _ _ _ __ __
# (-2) 1--|a|--0 (4) (4) 0--|c|--2 (1) = (-2) 2<-1--|a|--|c|--|d*|--|b*|--0->3 (-3)
# (-0) 2--|_| |_|--3 (1) (-0) 0<-2--|_| |_| |__| |__|--1->1 (-1)
# |
# 1 (3)
# 0 (3)
# __ |_
# (-3) 0--|b*|--2(5) (5) 2--|d*|--1 (2)
# (-1) 1--|__| |__|--3 (2)
#
f = yastn.ncon([a, b, c, d], [[4, -2, -0], [-3, -1, 5], [4, 3, 1, 1], [3, 2, 5, 2]],
conjs=(0, 1, 0, 1))
assert f.get_shape() == (2, 4, 6, 8)
# In yastn.einsum(subscripts, *operands, order='Alphabetic')
#
# order specifies the order of contraction, otherwise alphabetic order of contracted indices is used
# character '*' can be used in einsum subscripts to conjugate respective tensor
# spaces in subscripts are ignored
f1 = yastn.einsum('nCA, *DBo, nmkk, *mlol -> ABCD', a, b, c, d, order='klmno')
assert yastn.norm(f1 - f) < tol
Transposition#
def test_transpose_syntax(self):
#
# define rank-6 U1-symmetric tensor
#
a = yastn.ones(config=config_U1, s=(-1, -1, -1, 1, 1, 1),
t=[(0, 1), (0, 1), (0, 1), (0, 1), (0, 1), (0, 1)],
D=[(2, 3), (4, 5), (6, 7), (6, 5), (4, 3), (2, 1)])
#
# for each leg its dense dimension is given by the sum of dimensions
# of individual sectors. Hence, the (dense) shape of this tensor
# is (2+3, 4+5, 6+7, 6+5, 4+3, 2+1)
assert a.get_shape() == (5, 9, 13, 11, 7, 3)
#
# permute the legs of the tensor and check the shape is changed
# accordingly
b = a.transpose(axes=(0, 2, 4, 1, 3, 5))
assert b.get_shape() == (5, 13, 7, 9, 11, 3)
#
# if axes is not provided, reverse the order
# This can be also done using a shorthand self.T
a.transpose().get_shape() == (3, 7, 11, 13, 9, 5)
a.T.get_shape() == (3, 7, 11, 13, 9, 5)
#
# sometimes, instead of writing explicit permutation of all legs
# it is more convenient to only specify pairs of legs to switched.
# In this example, we reverse the permutation done previously thus
# ending up with tensor numerically identical to a.
#
c = b.move_leg(source=(1, 2), destination=(2, 4))
assert c.get_shape() == a.get_shape()
assert yastn.norm(a - c) < tol
Fusion (reshaping)#
Following example showcases fusion, in particular its hard mode. In this case, the tensor data is reshuffled/resized in memory.
def test_fuse_hard(self):
# define a rank-5 U1-symmetric tensor
a = yastn.rand(config=config_U1, s=(-1, 1, 1, -1, 1,),
t=((0, 1), (0, 1), (0, 1), (0, 1), (0, 1)),
D=((1, 2), (3, 4), (5, 6), (7, 8), (9, 10)))
# this tensor has 10 non-zero blocks, the largest one being
# of shape (2, 4, 6, 8, 9) with total number of 3456 elements
a.print_blocks_shape()
#
# charges shape
# (0, 0, 0, 0, 0) (1, 3, 5, 7, 9)
# (0, 0, 0, 1, 1) (1, 3, 5, 8, 10)
# (0, 0, 1, 1, 0) (1, 3, 6, 8, 9)
# (0, 1, 0, 1, 0) (1, 4, 5, 8, 9)
# (1, 0, 0, 0, 1) (2, 3, 5, 7, 10)
# (1, 0, 1, 0, 0) (2, 3, 6, 7, 9)
# (1, 0, 1, 1, 1) (2, 3, 6, 8, 10)
# (1, 1, 0, 0, 0) (2, 4, 5, 7, 9)
# (1, 1, 0, 1, 1) (2, 4, 5, 8, 10)
# (1, 1, 1, 1, 0) (2, 4, 6, 8, 9)
# Lets fuse last three legs of the tensor a into a new leg
b = a.fuse_legs(axes=(0, 1, (2, 3, 4)), mode='hard')
# resulting tensor has just four non-zero blocks, with the largest
# one holding 9176 elements
b.print_blocks_shape()
#
# (0, 0, 0) (1, 3, 1147)
# (0, 1, -1) (1, 4, 360)
# (1, 0, 1) (2, 3, 1208)
# (1, 1, 0) (2, 4, 1147)
# We can also fuse more than a single group of indices.
# Their order can be permuted as well
c0 = a.fuse_legs(axes=(0, (3, 4), (2, 1)), mode='hard')
# Fusion can be applied successively - fusing fused spaces together.
# This results in a rank-2 tensor, equivalent to block-sparse matrix
c1 = c0.fuse_legs(axes=((0, 1), 2), mode='hard')
# Resulting matrix has three blocks and the largest block holds 13604 elements
assert c1.s == (-1, 1)
c1.print_blocks_shape()
#
# (0, 0) (283, 15)
# (1, 1) (358, 38)
# (2, 2) (144, 24)
# The fusion can be simply reverted, proceeding step-by-step in reverse
# order of the applied fusions.
# NOTE: Unfusing an index *does not* permute the resulting indices into
# into their original order
#
# fusion step 1: 0 1 2 3 4 -> permute -> 0 3 4 2 1 -> fuse -> 0 (3 4) (2 1) = 0 1 2
# fusion step 2: 0 1 2 -> fuse -> (0 1) 2 = 0 1
#
# unfuse step 2: 0 1 -> unfuse -> (0 1) 1->2 = 0 1 2
c0_0 = c1.unfuse_legs(axes=0)
assert yastn.norm(c0_0 - c0) < tol
# unfuse step 1: 0 1 2 -> unfuse -> 0 (3->1 4->2) (2->3 1->4) = 0 1 2 3 4
#
# Hence, to retrieve original tensor, we have to permute its indices
a_0 = c0_0.unfuse_legs(axes=(1, 2))
a_0 = a_0.transpose(axes=(0, 4, 3, 1, 2))
assert yastn.norm(a - a_0) < tol
Conjugation of symmetric tensors#
def test_conj_Z2xU1(self):
#
# create random complex-valued symmetric tensor with symmetry Z2 x U1
#
legs = [yastn.Leg(config_Z2xU1, s=1, t=((0, 2), (1, 1), (1, 2)), D=(1, 2, 3)),
yastn.Leg(config_Z2xU1, s=-1, t=((0, 0), (0, -1), (1, 0)), D=(4, 5, 6))]
a = yastn.rand(config=config_Z2xU1, legs=legs, n=(1, 2), dtype="complex128")
#
# conjugate tensor a: verify that signature and total charge
# has been reversed.
#
b = a.conj()
assert b.get_tensor_charge() == (1, -2)
assert b.get_signature() == (-1, 1)
#
# Interpreting these tensors a,b as vectors, following contraction
# _ _
# | |-<-0 0-<-| |
# |a|->-1 1->-|b|
#
# is equivalent to computing the square of Frobenius norm of a.
# Result is chargeless single-element tensor equivalent to scalar.
#
norm_F = yastn.tensordot(a, b, axes=((0, 1), (0, 1)))
assert norm_F.get_tensor_charge() == (0, 0)
assert norm_F.get_signature() == ()
assert abs(a.norm()**2 - norm_F.real().to_number()) < tol
#
# only complex-conjugate elements of the blocks of tensor a, leaving
# the structure i.e. signature and total charge intact.
#
c = a.conj_blocks()
assert c.get_tensor_charge() == a.get_tensor_charge()
assert c.get_signature() == a.get_signature()
#
# flip signature of the tensor c and its total charge, but do not
# complex-conjugate elements of its block
#
d = c.flip_signature()
assert d.get_tensor_charge() == b.get_tensor_charge()
assert d.get_signature() == b.get_signature()
#
# conj() is equivalent to flip_signature().conj_blocks() (or in the
# opposite order). Hence, tensor b and tensor d should be numerically
# identical
#
assert yastn.norm(b - d)<tol