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 restore_LCF_r(A, r, Gi, sanity_checks=False):
if Gi is None:
x = r
else:
x = Gi.dot(r.dot(Gi.conj().T))
M = eps_r_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"
rm1 = mm.simple_diag_matrix(ev, dtype=A.dtype)
Gm1 = EV.conj().T
if Gi is None:
Gi = EV #for left uniform case
r = rm1 #for sanity check
for s in xrange(A.shape[0]):
A[s] = Gm1.dot(A[s].dot(Gi))
if sanity_checks:
rm1_ = eps_r_noop(r, A, A)
if not sp.allclose(rm1_, rm1, atol=1E-12, rtol=1E-12):
log.warning("Sanity Fail in restore_LCF_r!: r is bad!")
log.warning(la.norm(rm1_ - rm1))
Gm1_i = EV
return rm1, Gm1, Gm1_i
def calc_Vsh(A, r_s, sanity_checks=False):
D = A.shape[2]
Dm1 = A.shape[1]
q = A.shape[0]
if q * D - Dm1 <= 0:
return None
R = sp.zeros((D, q, Dm1), dtype=A.dtype, order='C')
for s in xrange(q):
R[:,s,:] = r_s.dot(A[s].conj().T)
R = R.reshape((q * D, Dm1))
Vconj = ns.nullspace_qr(R.conj().T).T
if sanity_checks:
if not sp.allclose(mm.mmul(Vconj.conj(), R), 0):
log.warning("Sanity Fail in calc_Vsh!: VR != 0")
if not sp.allclose(mm.mmul(Vconj, Vconj.conj().T), sp.eye(Vconj.shape[0])):
log.warning("Sanity Fail in calc_Vsh!: V H(V) != eye")
Vconj = Vconj.reshape((q * D - Dm1, D, q))
Vsh = Vconj.T
Vsh = sp.asarray(Vsh, order='C')
if sanity_checks:
Vs = sp.transpose(Vsh, axes=(0, 2, 1)).conj()
M = eps_r_noop(r_s, Vs, A)
if not sp.allclose(M, 0):
log.warning("Sanity Fail in calc_Vsh!: Bad Vsh")
return Vsh
def calc_Vsh(A, r_s, sanity_checks=False):
D = A.shape[2]
Dm1 = A.shape[1]
q = A.shape[0]
if q * D - Dm1 <= 0:
return None
R = sp.zeros((D, q, Dm1), dtype=A.dtype, order='C')
for s in range(q):
R[:, s, :] = r_s.dot(A[s].conj().T)
R = R.reshape((q * D, Dm1))
Vconj = ns.nullspace_qr(R.conj().T).T
if sanity_checks:
if not sp.allclose(mm.mmul(Vconj.conj(), R), 0):
log.warning("Sanity Fail in calc_Vsh!: VR != 0")
if not sp.allclose(mm.mmul(Vconj,
Vconj.conj().T), sp.eye(Vconj.shape[0])):
log.warning("Sanity Fail in calc_Vsh!: V H(V) != eye")
Vconj = Vconj.reshape((q * D - Dm1, D, q))
Vsh = Vconj.T
Vsh = sp.asarray(Vsh, order='C')
if sanity_checks:
Vs = sp.transpose(Vsh, axes=(0, 2, 1)).conj()
M = eps_r_noop(r_s, Vs, A)
if not sp.allclose(M, 0):
log.warning("Sanity Fail in calc_Vsh!: Bad Vsh")
return Vsh
def calc_K_3s(Kp1, C, lm1, rp2, A, AAp1Ap2):
K = eps_r_noop(Kp1, A, A)
Hr = eps_r_op_3s_C123_AAA456(rp2, C, AAp1Ap2)
op_expect = mm.adot(lm1, Hr)
K += Hr
return K, op_expect
def calc_K(Kp1, C, lm1, rp1, A, AAp1):
K = eps_r_noop(Kp1, A, A)
Hr = eps_r_op_2s_C12_AA34(rp1, C, AAp1)
op_expect = mm.adot(lm1, Hr)
K += Hr
return K, op_expect
def calc_K_tp(Kp1, lm1, rp1, A, Ap1, C_tp):
K = eps_r_noop(Kp1, A, A)
Hr = eps_r_op_2s_C12_tp(rp1, C_tp, A, Ap1)
op_expect = mm.adot(lm1, Hr)
K += Hr
return K, op_expect
def calc_BB_Y_2s(C, Vlh, Vrh_p1, l_s_m1, r_s_p1):
Vr_p1 = sp.transpose(Vrh_p1, axes=(0, 2, 1)).conj()
Y = sp.zeros((Vlh.shape[1], Vrh_p1.shape[2]), dtype=C.dtype)
for s in xrange(Vlh.shape[0]):
Y += Vlh[s].dot(l_s_m1.dot(eps_r_noop(r_s_p1, C[s], Vr_p1)))
etaBB_sq = mm.adot(Y, Y)
return Y, etaBB_sq
def calc_K(Kp1, C, lm1, rp1, A, AAp1):
K = eps_r_noop(Kp1, A, A)
Hr = eps_r_op_2s_C12_AA34(rp1, C, AAp1)
op_expect = mm.adot(lm1, Hr)
K += Hr
return K, op_expect
def calc_BB_Y_2s(C, Vlh, Vrh_p1, l_s_m1, r_s_p1):
Vr_p1 = sp.transpose(Vrh_p1, axes=(0, 2, 1)).conj()
Y = sp.zeros((Vlh.shape[1], Vrh_p1.shape[2]), dtype=C.dtype)
for s in xrange(Vlh.shape[0]):
Y += Vlh[s].dot(l_s_m1.dot(eps_r_noop(r_s_p1, C[s], Vr_p1)))
etaBB_sq = mm.adot(Y, Y)
return Y, etaBB_sq
def calc_K_3s(Kp1, C, lm1, rp2, A, AAp1Ap2):
K = eps_r_noop(Kp1, A, A)
Hr = eps_r_op_3s_C123_AAA456(rp2, C, AAp1Ap2)
op_expect = mm.adot(lm1, Hr)
K += Hr
return K, op_expect
def calc_K_tp(Kp1, lm1, rp1, A, Ap1, C_tp):
K = eps_r_noop(Kp1, A, A)
Hr = eps_r_op_2s_C12_tp(rp1, C_tp, A, Ap1)
op_expect = mm.adot(lm1, Hr)
K += Hr
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 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 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 restore_RCF_r_seq(A, r, GN=None, sanity_checks=False, sc_data=''):
"""Transforms a sequence of A[n]'s to obtain r[n - 1] = eye(D).
Implements the condition for right-orthonormalization from sub-section
3.1, theorem 1 of arXiv:quant-ph/0608197v2.
Uses a reduced QR 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 r.
r : sequence of ndarray or objects with array interface
The matrices [r0, r1, r2,..., rN], where rN will not be changed, but is
used for sanity checks.
GN : ndarray or scalar
Initial right gauge transformation matrix for site N. 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.
"""
assert len(A) == len(r), 'A and r must have the same length!'
if GN is None:
Gh = mm.eyemat(A[-1].shape[2], dtype=A[-1].dtype)
else:
Gh = GN.conj().T
for n in xrange(len(A) - 1, 0, -1):
q, Dm1, D = A[n].shape
AG = sp.array([Gh.dot(As.conj().T) for As in A[n]]).reshape(
(q * D, Dm1))
Q, R = la.qr(AG, mode='economic')
A[n] = sp.transpose(Q.conj().reshape((q, D, Dm1)), axes=(0, 2, 1))
Gh = R
r[n - 1] = mm.eyemat(Dm1, dtype=A[n].dtype)
if sanity_checks:
r_nm1_ = eps_r_noop(r[n], A[n], A[n])
if not sp.allclose(r_nm1_, r[n - 1].A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_RCF_r!: r is bad! %s %s",
la.norm(r_nm1_ - r[n - 1]), sc_data)
return Gh.conj().T
def restore_RCF_r_seq(A, r, GN=None, sanity_checks=False, sc_data=''):
"""Transforms a sequence of A[n]'s to obtain r[n - 1] = eye(D).
Implements the condition for right-orthonormalization from sub-section
3.1, theorem 1 of arXiv:quant-ph/0608197v2.
Uses a reduced QR 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 r.
r : sequence of ndarray or objects with array interface
The matrices [r0, r1, r2,..., rN], where rN will not be changed, but is
used for sanity checks.
GN : ndarray or scalar
Initial right gauge transformation matrix for site N. 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.
"""
assert len(A) == len(r), 'A and r must have the same length!'
if GN is None:
Gh = mm.eyemat(A[-1].shape[2], dtype=A[-1].dtype)
else:
Gh = GN.conj().T
for n in xrange(len(A) - 1, 0, -1):
q, Dm1, D = A[n].shape
AG = sp.array([Gh.dot(As.conj().T) for As in A[n]]).reshape((q * D, Dm1))
Q, R = la.qr(AG, mode='economic')
A[n] = sp.transpose(Q.conj().reshape((q, D, Dm1)), axes=(0, 2, 1))
Gh = R
r[n - 1] = mm.eyemat(Dm1, dtype=A[n].dtype)
if sanity_checks:
r_nm1_ = eps_r_noop(r[n], A[n], A[n])
if not sp.allclose(r_nm1_, r[n - 1].A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_RCF_r!: r is bad! %s %s",
la.norm(r_nm1_ - r[n - 1]), sc_data)
return Gh.conj().T
def eps_r_op_3s_C123_AAA456(x, C123, AAA456):
d = C123.shape[0] * C123.shape[1] * C123.shape[2]
S1 = (d, C123.shape[3], C123.shape[4])
S2 = (d, AAA456.shape[3], AAA456.shape[4])
return eps_r_noop(x, C123.reshape(S1), AAA456.reshape(S2))
def eps_r_op_2s_AA12_C34(x, AA12, C34):
d = C34.shape[0] * C34.shape[1]
S1 = (d, AA12.shape[2], AA12.shape[3])
S2 = (d, C34.shape[2], C34.shape[3])
return eps_r_noop(x, AA12.reshape(S1), C34.reshape(S2))
def eps_r_op_2s_C12_tp(x, C12_tp, A1, A2):
res = 0
for al in xrange(len(C12_tp)):
res += eps_r_noop(eps_r_noop(x, C12_tp[al][1], A2), C12_tp[al][0], A1)
return res
def restore_RCF_r(A, r, G_n_i, zero_tol=1E-15, sanity_checks=False, sc_data=''):
"""Transforms a single A[n] to obtain r[n - 1] = eye(D).
Implements the condition for right-orthonormalization from sub-section
3.1, theorem 1 of arXiv:quant-ph/0608197v2.
This function must be called for each n in turn, starting at N + 1,
passing the gauge transformation matrix from the previous step
as an argument.
Finds a G[n-1] such that orthonormalization is fulfilled for n.
If rank-deficiency is encountered, the result fulfills the orthonormality
condition in the occupied subspace with the zeros at the top-left
(for example r = diag([0, 0, 1, 1, 1, 1, 1])).
Parameters
----------
A : ndarray
The parameter tensor for the nth site A[n].
r : ndarray or object with array interface
The matrix r[n].
G_n_i : ndarray
The inverse gauge transform matrix for site n obtained in the previous step (for n + 1).
sanity_checks : bool (False)
Whether to perform additional sanity checks.
zero_tol : float
Tolerance for detecting zeros.
Returns
-------
r_nm1 : ndarray or simple_diag_matrix or eyemat
The new matrix r[n - 1].
G_nm1 : ndarray
The gauge transformation matrix for the site n - 1.
G_n_m1_i : ndarray
The inverse gauge transformation matrix for the site n - 1.
"""
if G_n_i is None:
GGh_n_i = r
else:
GGh_n_i = G_n_i.dot(r.dot(G_n_i.conj().T))
M = eps_r_noop(GGh_n_i, A, A)
X, Xi, new_D = herm_fac_with_inv(M, zero_tol=zero_tol, return_rank=True,
sanity_checks=sanity_checks)
G_nm1 = Xi.conj().T
G_nm1_i = X.conj().T
if G_n_i is None:
G_n_i = G_nm1_i
if sanity_checks:
#GiG may not be equal to eye in the case of rank-deficiency,
#but the rest should lie in the null space of A.
GiG = G_nm1_i.dot(G_nm1)
As = np.sum(A, axis=0)
if not sp.allclose(GiG.dot(As).dot(G_n_i),
As.dot(G_n_i), atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_RCF_r!: Bad GT! %s %s",
la.norm(GiG.dot(As).dot(G_n_i) - As.dot(G_n_i)), sc_data)
for s in xrange(A.shape[0]):
A[s] = G_nm1.dot(A[s]).dot(G_n_i)
if new_D == A.shape[1]:
r_nm1 = mm.eyemat(A.shape[1], A.dtype)
else:
r_nm1 = sp.zeros((A.shape[1]), dtype=A.dtype)
r_nm1[-new_D:] = 1
r_nm1 = mm.simple_diag_matrix(r_nm1, dtype=A.dtype)
if sanity_checks:
r_nm1_ = G_nm1.dot(M).dot(G_nm1.conj().T)
if not sp.allclose(r_nm1_, r_nm1.A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_RCF_r!: r != g old_r gH! %s %s",
la.norm(r_nm1_ - r_nm1), sc_data)
r_nm1_ = eps_r_noop(r, A, A)
if not sp.allclose(r_nm1_, r_nm1.A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_RCF_r!: r is bad! %s %s",
la.norm(r_nm1_ - r_nm1), sc_data)
return r_nm1, G_nm1, G_nm1_i
def eps_r_op_2s_C12_tp(x, C12_tp, A1, A2):
res = 0
for al in xrange(len(C12_tp)):
res += eps_r_noop(eps_r_noop(x, C12_tp[al][1], A2), C12_tp[al][0], A1)
return res
def eps_r_op_2s_AA12_C34(x, AA12, C34):
d = C34.shape[0] * C34.shape[1]
S1 = (d, AA12.shape[2], AA12.shape[3])
S2 = (d, C34.shape[2], C34.shape[3])
return eps_r_noop(x, AA12.reshape(S1), C34.reshape(S2))
def restore_RCF_r(A,
r,
G_n_i,
zero_tol=1E-15,
sanity_checks=False,
sc_data=''):
"""Transforms a single A[n] to obtain r[n - 1] = eye(D).
Implements the condition for right-orthonormalization from sub-section
3.1, theorem 1 of arXiv:quant-ph/0608197v2.
This function must be called for each n in turn, starting at N + 1,
passing the gauge transformation matrix from the previous step
as an argument.
Finds a G[n-1] such that orthonormalization is fulfilled for n.
If rank-deficiency is encountered, the result fulfills the orthonormality
condition in the occupied subspace with the zeros at the top-left
(for example r = diag([0, 0, 1, 1, 1, 1, 1])).
Parameters
----------
A : ndarray
The parameter tensor for the nth site A[n].
r : ndarray or object with array interface
The matrix r[n].
G_n_i : ndarray
The inverse gauge transform matrix for site n obtained in the previous step (for n + 1).
sanity_checks : bool (False)
Whether to perform additional sanity checks.
zero_tol : float
Tolerance for detecting zeros.
Returns
-------
r_nm1 : ndarray or simple_diag_matrix or eyemat
The new matrix r[n - 1].
G_nm1 : ndarray
The gauge transformation matrix for the site n - 1.
G_n_m1_i : ndarray
The inverse gauge transformation matrix for the site n - 1.
"""
if G_n_i is None:
GGh_n_i = r
else:
GGh_n_i = G_n_i.dot(r.dot(G_n_i.conj().T))
M = eps_r_noop(GGh_n_i, A, A)
X, Xi, new_D = herm_fac_with_inv(M,
zero_tol=zero_tol,
return_rank=True,
sanity_checks=sanity_checks)
G_nm1 = Xi.conj().T
G_nm1_i = X.conj().T
if G_n_i is None:
G_n_i = G_nm1_i
if sanity_checks:
#GiG may not be equal to eye in the case of rank-deficiency,
#but the rest should lie in the null space of A.
GiG = G_nm1_i.dot(G_nm1)
As = np.sum(A, axis=0)
if not sp.allclose(
GiG.dot(As).dot(G_n_i), As.dot(G_n_i), atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_RCF_r!: Bad GT! %s %s",
la.norm(GiG.dot(As).dot(G_n_i) - As.dot(G_n_i)),
sc_data)
for s in xrange(A.shape[0]):
A[s] = G_nm1.dot(A[s]).dot(G_n_i)
if new_D == A.shape[1]:
r_nm1 = mm.eyemat(A.shape[1], A.dtype)
else:
r_nm1 = sp.zeros((A.shape[1]), dtype=A.dtype)
r_nm1[-new_D:] = 1
r_nm1 = mm.simple_diag_matrix(r_nm1, dtype=A.dtype)
if sanity_checks:
r_nm1_ = G_nm1.dot(M).dot(G_nm1.conj().T)
if not sp.allclose(r_nm1_, r_nm1.A, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity Fail in restore_RCF_r!: r != g old_r gH! %s %s",
la.norm(r_nm1_ - r_nm1), sc_data)
r_nm1_ = eps_r_noop(r, A, A)
if not sp.allclose(r_nm1_, r_nm1.A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_RCF_r!: r is bad! %s %s",
la.norm(r_nm1_ - r_nm1), sc_data)
return r_nm1, G_nm1, G_nm1_i
def eps_r_op_3s_C123_AAA456(x, C123, AAA456):
d = C123.shape[0] * C123.shape[1] * C123.shape[2]
S1 = (d, C123.shape[3], C123.shape[4])
S2 = (d, AAA456.shape[3], AAA456.shape[4])
return eps_r_noop(x, C123.reshape(S1), AAA456.reshape(S2))