def calc_x_tp(Kp1, C_tp, Cm1_tp, rp1, lm2, Am1, A, Ap1, lm1_s, lm1_si, r_s, r_si, Vsh):
D = A.shape[2]
Dm1 = A.shape[1]
q = A.shape[0]
x = np.zeros((Dm1, q * D - Dm1), dtype=A.dtype)
V = sp.transpose(Vsh, axes=(0, 2, 1)).conj().copy()
Vri = V.copy()
try:
for Vris in Vri:
Vris[:] = r_si.dot_left(Vris)
except AttributeError:
for Vris in Vri:
Vris[:] = Vris.dot(r_si)
if not C_tp is None:
x += lm1_s.dot(eps_r_op_2s_C12_tp(rp1, C_tp, Vri, Ap1)) #1
if not Cm1_tp is None:
for al in xrange(len(Cm1_tp)):
x += lm1_si.dot(eps_l_noop(lm2, Am1, Cm1_tp[al][0]).dot(eps_r_noop(r_s, Cm1_tp[al][1], V))) #2
if not Kp1 is None:
x += lm1_s.dot(eps_r_noop(Kp1, A, Vri))
return x
def calc_Vsh_l(A, lm1_sqrt, sanity_checks=False):
D = A.shape[2]
Dm1 = A.shape[1]
q = A.shape[0]
if q * Dm1 - D <= 0:
return None
L = sp.zeros((D, q, Dm1), dtype=A.dtype, order='C')
for s in xrange(q):
L[:, s, :] = lm1_sqrt.dot(A[s]).conj().T
L = L.reshape((D, q * Dm1))
V = ns.nullspace_qr(L)
if sanity_checks:
if not sp.allclose(L.dot(V), 0):
log.warning("Sanity Fail in calc_Vsh_l!: LV != 0")
if not sp.allclose(V.conj().T.dot(V), sp.eye(V.shape[1])):
log.warning("Sanity Fail in calc_Vsh_l!: V H(V) != eye")
V = V.reshape((q, Dm1, q * Dm1 - D))
Vsh = sp.transpose(V.conj(), axes=(0, 2, 1))
Vsh = sp.asarray(Vsh, order='C')
if sanity_checks:
M = eps_l_noop(lm1_sqrt, A, V)
if not sp.allclose(M, 0):
log.warning("Sanity Fail in calc_Vsh_l!: Bad Vsh")
return Vsh
def calc_x_tp(Kp1, C_tp, Cm1_tp, rp1, lm2, Am1, A, Ap1, lm1_s, lm1_si, r_s,
r_si, Vsh):
D = A.shape[2]
Dm1 = A.shape[1]
q = A.shape[0]
x = np.zeros((Dm1, q * D - Dm1), dtype=A.dtype)
V = sp.transpose(Vsh, axes=(0, 2, 1)).conj().copy()
Vri = V.copy()
try:
for Vris in Vri:
Vris[:] = r_si.dot_left(Vris)
except AttributeError:
for Vris in Vri:
Vris[:] = Vris.dot(r_si)
if not C_tp is None:
x += lm1_s.dot(eps_r_op_2s_C12_tp(rp1, C_tp, Vri, Ap1)) #1
if not Cm1_tp is None:
for al in xrange(len(Cm1_tp)):
x += lm1_si.dot(
eps_l_noop(lm2, Am1, Cm1_tp[al][0]).dot(
eps_r_noop(r_s, Cm1_tp[al][1], V))) #2
if not Kp1 is None:
x += lm1_s.dot(eps_r_noop(Kp1, A, Vri))
return x
def calc_Vsh_l(A, lm1_sqrt, sanity_checks=False):
D = A.shape[2]
Dm1 = A.shape[1]
q = A.shape[0]
if q * Dm1 - D <= 0:
return None
L = sp.zeros((D, q, Dm1), dtype=A.dtype, order='C')
for s in xrange(q):
L[:,s,:] = lm1_sqrt.dot(A[s]).conj().T
L = L.reshape((D, q * Dm1))
V = ns.nullspace_qr(L)
if sanity_checks:
if not sp.allclose(L.dot(V), 0):
log.warning("Sanity Fail in calc_Vsh_l!: LV != 0")
if not sp.allclose(V.conj().T.dot(V), sp.eye(V.shape[1])):
log.warning("Sanity Fail in calc_Vsh_l!: V H(V) != eye")
V = V.reshape((q, Dm1, q * Dm1 - D))
Vsh = sp.transpose(V.conj(), axes=(0, 2, 1))
Vsh = sp.asarray(Vsh, order='C')
if sanity_checks:
M = eps_l_noop(lm1_sqrt, A, V)
if not sp.allclose(M, 0):
log.warning("Sanity Fail in calc_Vsh_l!: Bad Vsh")
return Vsh
def restore_LCF_l(A, lm1, Gm1, sanity_checks=False, zero_tol=1E-15):
if Gm1 is None:
GhGm1 = lm1
else:
GhGm1 = Gm1.conj().T.dot(lm1.dot(Gm1))
M = eps_l_noop(GhGm1, A, A)
G, Gi, new_D = herm_fac_with_inv(M,
zero_tol=zero_tol,
return_rank=True,
sanity_checks=sanity_checks)
if Gm1 is None:
Gm1 = G
if sanity_checks:
if new_D == A.shape[2]:
eye = sp.eye(A.shape[2])
else:
eye = mm.simple_diag_matrix(np.append(np.zeros(A.shape[2] - new_D),
np.ones(new_D)),
dtype=A.dtype)
if not sp.allclose(G.dot(Gi), eye, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_LCF_l!: Bad GT!")
for s in xrange(A.shape[0]):
A[s] = Gm1.dot(A[s]).dot(Gi)
if new_D == A.shape[2]:
l = mm.eyemat(A.shape[2], A.dtype)
else:
l = mm.simple_diag_matrix(np.append(np.zeros(A.shape[2] - new_D),
np.ones(new_D)),
dtype=A.dtype)
if sanity_checks:
lm1_ = mm.eyemat(A.shape[1], A.dtype)
l_ = eps_l_noop(lm1_, A, A)
if not sp.allclose(l_, l.A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_LCF_l!: l is bad")
log.warning(la.norm(l_ - l))
return l, G, Gi
def calc_K_3s_l(Km1, Cm2, lm3, r, A, Am2Am1A):
K = eps_l_noop(Km1, A, A)
Hl = eps_l_op_3s_AAA123_C456(lm3, Am2Am1A, Cm2)
op_expect = mm.adot_noconj(Hl, r)
K += Hl
return K, op_expect
def calc_K_3s_l(Km1, Cm2, lm3, r, A, Am2Am1A):
K = eps_l_noop(Km1, A, A)
Hl = eps_l_op_3s_AAA123_C456(lm3, Am2Am1A, Cm2)
op_expect = mm.adot_noconj(Hl, r)
K += Hl
return K, op_expect
def calc_K_l_tp(Km1, lm2, r, Am1, A, Cm1_tp):
K = eps_l_noop(Km1, A, A)
Hl = eps_l_op_2s_C34_tp(lm2, Am1, A, Cm1_tp)
op_expect = mm.adot_noconj(Hl, r)
K += Hl
return K, op_expect
def calc_K_l_tp(Km1, lm2, r, Am1, A, Cm1_tp):
K = eps_l_noop(Km1, A, A)
Hl = eps_l_op_2s_C34_tp(lm2, Am1, A, Cm1_tp)
op_expect = mm.adot_noconj(Hl, r)
K += Hl
return K, op_expect
def calc_BB_Y_2s_tp(C_tp, Vlh, Vrh_p1, l_s_m1, r_s_p1):
Vl = sp.transpose(Vlh, axes=(0, 2, 1)).conj().copy()
Vr_p1 = sp.transpose(Vrh_p1, axes=(0, 2, 1)).conj().copy()
Y = 0
for al in xrange(len(C_tp)):
Y += eps_l_noop(l_s_m1, Vl, C_tp[al][0]).dot(eps_r_noop(r_s_p1, C_tp[al][1], Vr_p1))
etaBB_sq = mm.adot(Y, Y)
return Y, etaBB_sq
def calc_BB_Y_2s_tp(C_tp, Vlh, Vrh_p1, l_s_m1, r_s_p1):
Vl = sp.transpose(Vlh, axes=(0, 2, 1)).conj().copy()
Vr_p1 = sp.transpose(Vrh_p1, axes=(0, 2, 1)).conj().copy()
Y = 0
for al in range(len(C_tp)):
Y += eps_l_noop(l_s_m1, Vl, C_tp[al][0]).dot(eps_r_noop(r_s_p1, C_tp[al][1], Vr_p1))
etaBB_sq = mm.adot(Y, Y)
return Y, etaBB_sq
def restore_LCF_l(A, lm1, Gm1, sanity_checks=False, zero_tol=1E-15):
if Gm1 is None:
GhGm1 = lm1
else:
GhGm1 = Gm1.conj().T.dot(lm1.dot(Gm1))
M = eps_l_noop(GhGm1, A, A)
G, Gi, new_D = herm_fac_with_inv(M, zero_tol=zero_tol, return_rank=True,
sanity_checks=sanity_checks)
if Gm1 is None:
Gm1 = G
if sanity_checks:
if new_D == A.shape[2]:
eye = sp.eye(A.shape[2])
else:
eye = mm.simple_diag_matrix(np.append(np.zeros(A.shape[2] - new_D),
np.ones(new_D)), dtype=A.dtype)
if not sp.allclose(G.dot(Gi), eye, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_LCF_l!: Bad GT!")
for s in xrange(A.shape[0]):
A[s] = Gm1.dot(A[s]).dot(Gi)
if new_D == A.shape[2]:
l = mm.eyemat(A.shape[2], A.dtype)
else:
l = mm.simple_diag_matrix(np.append(np.zeros(A.shape[2] - new_D),
np.ones(new_D)), dtype=A.dtype)
if sanity_checks:
lm1_ = mm.eyemat(A.shape[1], A.dtype)
l_ = eps_l_noop(lm1_, A, A)
if not sp.allclose(l_, l.A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_LCF_l!: l is bad")
log.warning(la.norm(l_ - l))
return l, G, Gi
def calc_K_l(Km1, Cm1, lm2, r, A, Am1A):
"""Calculates the K_left using the recursive definition.
This is the "bra-vector" K_left, which means (K_left.dot(r)).trace() = <K_left|r>.
In other words, K_left ~ <K_left| and K_left.conj().T ~ |K_left>.
"""
K = eps_l_noop(Km1, A, A)
Hl = eps_l_op_2s_AA12_C34(lm2, Am1A, Cm1)
op_expect = mm.adot_noconj(Hl, r)
K += Hl
return K, op_expect
def calc_K_l(Km1, Cm1, lm2, r, A, Am1A):
"""Calculates the K_left using the recursive definition.
This is the "bra-vector" K_left, which means (K_left.dot(r)).trace() = <K_left|r>.
In other words, K_left ~ <K_left| and K_left.conj().T ~ |K_left>.
"""
K = eps_l_noop(Km1, A, A)
Hl = eps_l_op_2s_AA12_C34(lm2, Am1A, Cm1)
op_expect = mm.adot_noconj(Hl, r)
K += Hl
return K, op_expect
def restore_LCF_l_seq(A, l, G0=None, sanity_checks=False, sc_data=''):
"""Transforms a sequence of A[n]'s to obtain l[n] = eye(D).
Implements the condition for left-orthonormalization.
Uses a reduced QR (RQ) decomposition to avoid inverting anything explicity.
Parameters
----------
A : sequence of ndarray
The parameter tensors for a sequence of sites [None, A1, A2,..., AN].
The first entry is ignored so that the indices match up with l.
l : sequence of ndarray or objects with array interface
The matrices [l0, l1, l2,..., lN], where l0 will not be changed, but is
used for sanity checks.
G0 : ndarray or scalar
Initial left gauge transformation matrix for site 0. Only needed when used
as part of a larger transformation.
sanity_checks : bool (False)
Whether to perform additional sanity checks.
sc_data : string
A string to be appended to sanity check log messages.
"""
if G0 is None:
G = mm.eyemat(A[1].shape[1], dtype=A[1].dtype)
else:
G = G0
for n in xrange(1, len(A)):
q, Dm1, D = A[n].shape
GA = sp.array([G.dot(As) for As in A[n]])
GA = GA.reshape((q * Dm1, D))
Q, G = la.qr(GA, mode='economic')
A[n] = Q.reshape((q, Dm1, D))
l[n] = mm.eyemat(D, dtype=A[n].dtype)
if sanity_checks:
l_ = eps_l_noop(l[n - 1], A[n], A[n])
if not sp.allclose(l_, l[n].A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_LCF_l_seq!: l is bad")
log.warning(la.norm(l_ - l[n].A))
return G
def restore_LCF_l_seq(A, l, G0=None, sanity_checks=False, sc_data=''):
"""Transforms a sequence of A[n]'s to obtain l[n] = eye(D).
Implements the condition for left-orthonormalization.
Uses a reduced QR (RQ) decomposition to avoid inverting anything explicity.
Parameters
----------
A : sequence of ndarray
The parameter tensors for a sequence of sites [None, A1, A2,..., AN].
The first entry is ignored so that the indices match up with l.
l : sequence of ndarray or objects with array interface
The matrices [l0, l1, l2,..., lN], where l0 will not be changed, but is
used for sanity checks.
G0 : ndarray or scalar
Initial left gauge transformation matrix for site 0. Only needed when used
as part of a larger transformation.
sanity_checks : bool (False)
Whether to perform additional sanity checks.
sc_data : string
A string to be appended to sanity check log messages.
"""
if G0 is None:
G = mm.eyemat(A[1].shape[1], dtype=A[1].dtype)
else:
G = G0
for n in xrange(1, len(A)):
q, Dm1, D = A[n].shape
GA = sp.array([G.dot(As) for As in A[n]])
GA = GA.reshape((q * Dm1, D))
Q, G = la.qr(GA, mode='economic')
A[n] = Q.reshape((q, Dm1, D))
l[n] = mm.eyemat(D, dtype=A[n].dtype)
if sanity_checks:
l_ = eps_l_noop(l[n - 1], A[n], A[n])
if not sp.allclose(l_, l[n].A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_LCF_l_seq!: l is bad")
log.warning(la.norm(l_ - l[n].A))
return G
def restore_RCF_l(A, lm1, Gm1, sanity_checks=False):
"""Transforms a single A[n] to obtain diagonal l[n].
Applied after restore_RCF_r(), this completes the full canonical form
of sub-section 3.1, theorem 1 of arXiv:quant-ph/0608197v2.
This function must be called for each n in turn, starting at 1,
passing the gauge transformation matrix from the previous step
as an argument.
Finds a G[n] such that orthonormalization is fulfilled for n.
The diagonal entries of l[n] are sorted in
ascending order (for example l[n] = diag([0, 0, 0.1, 0.2, ...])).
Parameters
----------
A : ndarray
The parameter tensor for the nth site A[n].
lm1 : ndarray or object with array interface
The matrix l[n - 1].
Gm1 : ndarray
The gauge transform matrix for site n obtained in the previous step (for n - 1).
sanity_checks : bool (False)
Whether to perform additional sanity checks.
Returns
-------
l : ndarray or simple_diag_matrix
The new, diagonal matrix l[n].
G : ndarray
The gauge transformation matrix for site n.
G_i : ndarray
Inverse of G.
"""
if Gm1 is None:
x = lm1
else:
x = Gm1.conj().T.dot(lm1.dot(Gm1))
M = eps_l_noop(x, A, A)
ev, EV = la.eigh(
M) #wraps lapack routines, which return eigenvalues in ascending order
if sanity_checks:
assert np.all(ev == np.sort(ev)), "unexpected eigenvalue ordering"
l = mm.simple_diag_matrix(ev, dtype=A.dtype)
G_i = EV
if Gm1 is None:
Gm1 = EV.conj().T #for left uniform case
lm1 = l #for sanity check
for s in xrange(A.shape[0]):
A[s] = Gm1.dot(A[s].dot(G_i))
if sanity_checks:
l_ = eps_l_noop(lm1, A, A)
if not sp.allclose(l_, l, atol=1E-12, rtol=1E-12):
log.warning("Sanity Fail in restore_RCF_l!: l is bad!")
log.warning(la.norm(l_ - l))
G = EV.conj().T
return l, G, G_i
def eps_l_op_2s_AA12_C34(x, AA12, C34):
d = AA12.shape[0] * AA12.shape[1]
S1 = (d, AA12.shape[2], AA12.shape[3])
S2 = (d, C34.shape[2], C34.shape[3])
return eps_l_noop(x, AA12.reshape(S1), C34.reshape(S2))
def eps_l_op_3s_AAA123_C456(x, AAA123, C456):
d = C456.shape[0] * C456.shape[1] * C456.shape[2]
S1 = (d, AAA123.shape[3], AAA123.shape[4])
S2 = (d, C456.shape[3], C456.shape[4])
return eps_l_noop(x, AAA123.reshape(S1), C456.reshape(S2))
def eps_l_op_2s_C34_tp(x, A1, A2, C34_tp):
res = 0
for al in xrange(len(C34_tp)):
res += eps_l_noop(eps_l_noop(x, A1, C34_tp[al][0]), A2, C34_tp[al][1])
return res
def eps_l_op_2s_AA12_C34(x, AA12, C34):
d = AA12.shape[0] * AA12.shape[1]
S1 = (d, AA12.shape[2], AA12.shape[3])
S2 = (d, C34.shape[2], C34.shape[3])
return eps_l_noop(x, AA12.reshape(S1), C34.reshape(S2))
def eps_l_op_3s_AAA123_C456(x, AAA123, C456):
d = C456.shape[0] * C456.shape[1] * C456.shape[2]
S1 = (d, AAA123.shape[3], AAA123.shape[4])
S2 = (d, C456.shape[3], C456.shape[4])
return eps_l_noop(x, AAA123.reshape(S1), C456.reshape(S2))
def restore_RCF_l(A, lm1, Gm1, sanity_checks=False):
"""Transforms a single A[n] to obtain diagonal l[n].
Applied after restore_RCF_r(), this completes the full canonical form
of sub-section 3.1, theorem 1 of arXiv:quant-ph/0608197v2.
This function must be called for each n in turn, starting at 1,
passing the gauge transformation matrix from the previous step
as an argument.
Finds a G[n] such that orthonormalization is fulfilled for n.
The diagonal entries of l[n] are sorted in
ascending order (for example l[n] = diag([0, 0, 0.1, 0.2, ...])).
Parameters
----------
A : ndarray
The parameter tensor for the nth site A[n].
lm1 : ndarray or object with array interface
The matrix l[n - 1].
Gm1 : ndarray
The gauge transform matrix for site n obtained in the previous step (for n - 1).
sanity_checks : bool (False)
Whether to perform additional sanity checks.
Returns
-------
l : ndarray or simple_diag_matrix
The new, diagonal matrix l[n].
G : ndarray
The gauge transformation matrix for site n.
G_i : ndarray
Inverse of G.
"""
if Gm1 is None:
x = lm1
else:
x = Gm1.conj().T.dot(lm1.dot(Gm1))
M = eps_l_noop(x, A, A)
ev, EV = la.eigh(M) #wraps lapack routines, which return eigenvalues in ascending order
if sanity_checks:
assert np.all(ev == np.sort(ev)), "unexpected eigenvalue ordering"
l = mm.simple_diag_matrix(ev, dtype=A.dtype)
G_i = EV
if Gm1 is None:
Gm1 = EV.conj().T #for left uniform case
lm1 = l #for sanity check
for s in xrange(A.shape[0]):
A[s] = Gm1.dot(A[s].dot(G_i))
if sanity_checks:
l_ = eps_l_noop(lm1, A, A)
if not sp.allclose(l_, l, atol=1E-12, rtol=1E-12):
log.warning("Sanity Fail in restore_RCF_l!: l is bad!")
log.warning(la.norm(l_ - l))
G = EV.conj().T
return l, G, G_i
def eps_l_op_2s_C34_tp(x, A1, A2, C34_tp):
res = 0
for al in xrange(len(C34_tp)):
res += eps_l_noop(eps_l_noop(x, A1, C34_tp[al][0]), A2, C34_tp[al][1])
return res
