def restore_SCF(self):
X = la.cholesky(self.r, lower=True)
Y = la.cholesky(self.l, lower=False)
U, sv, Vh = la.svd(Y.dot(X))
#s contains the Schmidt coefficients,
lam = sv**2
self.S_hc = - np.sum(lam * sp.log2(lam))
S = m.simple_diag_matrix(sv, dtype=self.typ)
Srt = S.sqrt()
g = m.mmul(Srt, Vh, m.invtr(X, lower=True))
g_i = m.mmul(m.invtr(Y, lower=False), U, Srt)
for s in xrange(self.q):
self.A[s] = m.mmul(g, self.A[s], g_i)
if self.sanity_checks:
Sfull = np.asarray(S)
if not np.allclose(g.dot(g_i), np.eye(self.D)):
print "Sanity check failed! Restore_SCF, bad GT!"
l = m.mmul(m.H(g_i), self.l, g_i)
r = m.mmul(g, self.r, m.H(g))
if not np.allclose(Sfull, l):
print "Sanity check failed: Restorce_SCF, left failed!"
if not np.allclose(Sfull, r):
print "Sanity check failed: Restorce_SCF, right failed!"
l = self.eps_l(Sfull)
r = self.eps_r(Sfull)
if not np.allclose(Sfull, l, rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
print "Sanity check failed: Restorce_SCF, left bad!"
if not np.allclose(Sfull, r, rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
print "Sanity check failed: Restorce_SCF, right bad!"
self.l = S
self.r = S
def restore_ONR_n(self, n, G_n_i):
"""Transforms a single A[n] to obtain right orthonormalization.
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 ON_R is fulfilled for n.
Eigenvalues = 0 are a problem here... IOW rank-deficient matrices.
Apparently, they can turn up during a run, but if they do we're screwed.
The fact that M should be positive definite is used to optimize this.
Parameters
----------
n : int
The site number.
G_n_i : ndarray
The inverse gauge transform matrix for site n obtained in the previous step (for n + 1).
Returns
-------
G_n_m1_i : ndarray
The inverse gauge transformation matrix for the site n - 1.
"""
if G_n_i is None:
GGh_n_i = self.r[n]
else:
GGh_n_i = mm.mmul(G_n_i, self.r[n], mm.H(G_n_i))
M = self.eps_r(n, GGh_n_i)
try:
tu = la.cholesky(M) #Assumes M is pos. def.. It should raise LinAlgError if not.
G_nm1 = mm.H(mm.invtr(tu)) #G is now lower-triangular
G_nm1_i = mm.H(tu)
except sp.linalg.LinAlgError:
print "Restore_ON_R_%u: Falling back to eigh()!" % n
e,Gh = la.eigh(M)
G_nm1 = mm.H(mm.mmul(Gh, sp.diag(1/sp.sqrt(e) + 0.j)))
G_nm1_i = la.inv(G_nm1)
if G_n_i is None:
G_n_i = G_nm1_i
if self.sanity_checks:
if not sp.allclose(sp.dot(G_nm1, G_nm1_i), sp.eye(G_nm1.shape[0]), atol=1E-13, rtol=1E-13):
print "Sanity Fail in restore_ONR_%u!: Bad GT at n=%u" % (n, n)
for s in xrange(self.q[n]):
self.A[n][s] = mm.mmul(G_nm1, self.A[n][s], G_n_i)
return G_nm1_i, G_nm1
def restore_ONR_n(self, n, G_n_i):
"""Transforms a single A[n] to obtain right orthonormalization.
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 ON_R is fulfilled for n.
Eigenvalues = 0 are a problem here... IOW rank-deficient matrices.
Apparently, they can turn up during a run, but if they do we're screwed.
The fact that M should be positive definite is used to optimize this.
Parameters
----------
n : int
The site number.
G_n_i : ndarray
The inverse gauge transform matrix for site n obtained in the previous step (for n + 1).
Returns
-------
G_n_m1_i : ndarray
The inverse gauge transformation matrix for the site n - 1.
"""
GGh_n_i = m.mmul(
G_n_i, m.H(G_n_i)
) #r[n] does not belong here. The condition is for sum(AA). r[n] = 1 is a consequence.
M = self.eps_r(n, GGh_n_i)
#The following should be more efficient than eigh():
try:
tu = la.cholesky(
M
) #Assumes M is pos. def.. It should raise LinAlgError if not.
G_nm1 = m.H(m.invtr(tu)) #G is now lower-triangular
G_nm1_i = m.H(tu)
except sp.linalg.LinAlgError:
print "restore_ONR_n: Falling back to eigh()!"
e, Gh = la.eigh(M)
G_nm1 = m.H(m.mmul(Gh, sp.diag(1 / sp.sqrt(e) + 0.j)))
G_nm1_i = la.inv(G_nm1)
for s in xrange(self.q[n]):
self.A[n][s] = m.mmul(G_nm1, self.A[n][s], G_n_i)
#It's ok to use the same matrix as out and as an operand here
#since there are > 2 matrices in the chain and it is not the last argument.
return G_nm1_i
def restore_CF(self, ret_g=False):
if self.symm_gauge:
self.restore_SCF()
else:
#First get G such that r = eye
G = la.cholesky(self.r, lower=True)
G_i = m.invtr(G, lower=True)
self.l = m.mmul(m.H(G), self.l, G)
#Now bring l into diagonal form, trace = 1 (guaranteed by r = eye..?)
ev, EV = la.eigh(self.l)
G = G.dot(EV)
G_i = m.H(EV).dot(G_i)
for s in xrange(self.q):
self.A[s] = m.mmul(G_i, self.A[s], G)
#ev contains the squares of the Schmidt coefficients,
self.S_hc = - np.sum(ev * sp.log2(ev))
self.l = m.simple_diag_matrix(ev, dtype=self.typ)
if self.sanity_checks:
M = np.zeros_like(self.r)
for s in xrange(self.q):
M += m.mmul(self.A[s], m.H(self.A[s]))
self.r = m.mmul(G_i, self.r, m.H(G_i))
if not np.allclose(M, self.r,
rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
print "Sanity check failed: RestoreRCF, bad M."
print "Off by: " + str(la.norm(M - self.r))
if not np.allclose(self.r, np.eye(self.D),
rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
print "Sanity check failed: r not identity."
print "Off by: " + str(la.norm(np.eye(self.D) - self.r))
l = self.eps_l(self.l)
r = self.eps_r(self.r)
if not np.allclose(r, self.r,
rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
print "Sanity check failed: Restore_RCF, bad r!"
print "Off by: " + str(la.norm(r - self.r))
if not np.allclose(l, self.l,
rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
print "Sanity check failed: Restore_RCF, bad l!"
print "Off by: " + str(la.norm(l - self.l))
self.r = m.eyemat(self.D, dtype=self.typ)
if ret_g:
return G, G_i
else:
return
def herm_fac_with_inv(A,
lower=False,
zero_tol=1E-15,
return_rank=False,
calc_inv=True,
force_evd=False,
sanity_checks=False,
sc_data=''):
"""Factorizes a Hermitian matrix using either Cholesky or eigenvalue decomposition.
Decomposes a Hermitian A as A = X*X or, if lower == True, A = XX*.
Tries Cholesky first by default, then falls back to EVD if the matrix is
not positive-definite. If Cholesky decomposition is used, X is upper (or lower)
triangular. For the EVD decomposition, the inverse becomes a pseudo-inverse
and all eigenvalues below the zero-tolerance are set to zero.
Parameters
----------
A : ndarray
The Hermitian matrix to be factorized.
lower : bool
Refers to Cholesky factorization. If True, factorize as A = XX*, otherwise as A = X*X
zero_tol : float
Tolerance for detection of zeros in EVD case.
return_rank : bool
Whether to return the rank of A. The detected rank is affected by zero_tol.
calc_inv : bool
Whether to calculate (and return) the inverse of the factor.
force_evd : bool
Whether to force eigenvalue instead of Cholesky decomposition.
sanity_checks : bool
Whether to perform soem basic sanity checks.
"""
if not force_evd:
try:
x = la.cholesky(A, lower=lower)
if calc_inv:
xi = mm.invtr(x, lower=lower)
else:
xi = None
nonzeros = A.shape[0]
except sp.linalg.LinAlgError: #this usually means a is not pos. def.
force_evd = True
if force_evd:
ev, EV = la.eigh(
A, turbo=True
) #wraps lapack routines, which return eigenvalues in ascending order
if sanity_checks:
assert np.all(
ev == np.sort(ev)
), "Sanity fail in herm_fac_with_inv(): Unexpected eigenvalue ordering"
if ev.min() < -zero_tol:
log.warning(
"Sanity fail in herm_fac_with_inv(): Discarding negative eigenvalues! %s %s",
ev.min(), sc_data)
nonzeros = np.count_nonzero(ev > zero_tol)
ev_sq = sp.zeros_like(ev, dtype=A.dtype)
ev_sq[-nonzeros:] = sp.sqrt(ev[-nonzeros:])
ev_sq = mm.simple_diag_matrix(ev_sq, dtype=A.dtype)
if calc_inv:
#Replace almost-zero values with zero and perform a pseudo-inverse
ev_sq_i = sp.zeros_like(ev, dtype=A.dtype)
ev_sq_i[-nonzeros:] = 1. / ev_sq[-nonzeros:]
ev_sq_i = mm.simple_diag_matrix(ev_sq_i, dtype=A.dtype)
xi = None
if lower:
x = ev_sq.dot_left(EV)
if calc_inv:
xi = ev_sq_i.dot(EV.conj().T)
else:
x = ev_sq.dot(EV.conj().T)
if calc_inv:
xi = ev_sq_i.dot_left(EV)
if sanity_checks:
if not sp.allclose(A, A.conj().T, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): A is not Hermitian! %s %s",
la.norm(A - A.conj().T), sc_data)
eye = sp.zeros((A.shape[0]), dtype=A.dtype)
eye[-nonzeros:] = 1
eye = mm.simple_diag_matrix(eye)
if lower:
if calc_inv:
if not sp.allclose(xi.dot(x), eye, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): Bad left inverse! %s %s",
la.norm(xi.dot(x) - eye), sc_data)
if not sp.allclose(
xi.dot(A).dot(
xi.conj().T), eye, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): Bad A inverse! %s %s",
la.norm(xi.conj().T.dot(A).dot(xi) - eye), sc_data)
if not sp.allclose(x.dot(x.conj().T), A, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): Bad decomp! %s %s",
la.norm(x.dot(x.conj().T) - A), sc_data)
else:
if calc_inv:
if not sp.allclose(x.dot(xi), eye, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): Bad right inverse! %s %s",
la.norm(x.dot(xi) - eye), sc_data)
if not sp.allclose(
xi.conj().T.dot(A).dot(xi), eye, atol=1E-13,
rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): Bad A inverse! %s %s",
la.norm(xi.conj().T.dot(A).dot(xi) - eye), sc_data)
if not sp.allclose(x.conj().T.dot(x), A, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): Bad decomp! %s %s",
la.norm(x.conj().T.dot(x) - A), sc_data)
if calc_inv:
if return_rank:
return x, xi, nonzeros
else:
return x, xi
else:
if return_rank:
return x, nonzeros
else:
return x
def herm_fac_with_inv(A, lower=False, zero_tol=1E-15, return_rank=False,
calc_inv=True, force_evd=False,
sanity_checks=False, sc_data=''):
"""Factorizes a Hermitian matrix using either Cholesky or eigenvalue decomposition.
Decomposes a Hermitian A as A = X*X or, if lower == True, A = XX*.
Tries Cholesky first by default, then falls back to EVD if the matrix is
not positive-definite. If Cholesky decomposition is used, X is upper (or lower)
triangular. For the EVD decomposition, the inverse becomes a pseudo-inverse
and all eigenvalues below the zero-tolerance are set to zero.
Parameters
----------
A : ndarray
The Hermitian matrix to be factorized.
lower : bool
Refers to Cholesky factorization. If True, factorize as A = XX*, otherwise as A = X*X
zero_tol : float
Tolerance for detection of zeros in EVD case.
return_rank : bool
Whether to return the rank of A. The detected rank is affected by zero_tol.
calc_inv : bool
Whether to calculate (and return) the inverse of the factor.
force_evd : bool
Whether to force eigenvalue instead of Cholesky decomposition.
sanity_checks : bool
Whether to perform soem basic sanity checks.
"""
if not force_evd:
try:
x = la.cholesky(A, lower=lower)
if calc_inv:
xi = mm.invtr(x, lower=lower)
else:
xi = None
nonzeros = A.shape[0]
except sp.linalg.LinAlgError: #this usually means a is not pos. def.
force_evd = True
if force_evd:
ev, EV = la.eigh(A, turbo=True) #wraps lapack routines, which return eigenvalues in ascending order
if sanity_checks:
assert np.all(ev == np.sort(ev)), "Sanity fail in herm_fac_with_inv(): Unexpected eigenvalue ordering"
if ev.min() < -zero_tol:
log.warning("Sanity fail in herm_fac_with_inv(): Discarding negative eigenvalues! %s %s",
ev.min(), sc_data)
nonzeros = np.count_nonzero(ev > zero_tol)
ev_sq = sp.zeros_like(ev, dtype=A.dtype)
ev_sq[-nonzeros:] = sp.sqrt(ev[-nonzeros:])
ev_sq = mm.simple_diag_matrix(ev_sq, dtype=A.dtype)
if calc_inv:
#Replace almost-zero values with zero and perform a pseudo-inverse
ev_sq_i = sp.zeros_like(ev, dtype=A.dtype)
ev_sq_i[-nonzeros:] = 1. / ev_sq[-nonzeros:]
ev_sq_i = mm.simple_diag_matrix(ev_sq_i, dtype=A.dtype)
xi = None
if lower:
x = ev_sq.dot_left(EV)
if calc_inv:
xi = ev_sq_i.dot(EV.conj().T)
else:
x = ev_sq.dot(EV.conj().T)
if calc_inv:
xi = ev_sq_i.dot_left(EV)
if sanity_checks:
if not sp.allclose(A, A.conj().T, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): A is not Hermitian! %s %s",
la.norm(A - A.conj().T), sc_data)
eye = sp.zeros((A.shape[0]), dtype=A.dtype)
eye[-nonzeros:] = 1
eye = mm.simple_diag_matrix(eye)
if lower:
if calc_inv:
if not sp.allclose(xi.dot(x), eye, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): Bad left inverse! %s %s",
la.norm(xi.dot(x) - eye), sc_data)
if not sp.allclose(xi.dot(A).dot(xi.conj().T), eye, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): Bad A inverse! %s %s",
la.norm(xi.conj().T.dot(A).dot(xi) - eye), sc_data)
if not sp.allclose(x.dot(x.conj().T), A, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): Bad decomp! %s %s",
la.norm(x.dot(x.conj().T) - A), sc_data)
else:
if calc_inv:
if not sp.allclose(x.dot(xi), eye, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): Bad right inverse! %s %s",
la.norm(x.dot(xi) - eye), sc_data)
if not sp.allclose(xi.conj().T.dot(A).dot(xi), eye, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): Bad A inverse! %s %s",
la.norm(xi.conj().T.dot(A).dot(xi) - eye), sc_data)
if not sp.allclose(x.conj().T.dot(x), A, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): Bad decomp! %s %s",
la.norm(x.conj().T.dot(x) - A), sc_data)
if calc_inv:
if return_rank:
return x, xi, nonzeros
else:
return x, xi
else:
if return_rank:
return x, nonzeros
else:
return x