def _restore_CF_diag(self, dbg=False):
nc = self.N_centre
#Want: r[0 <= n < nc] diagonal
Ui = sp.eye(self.D[nc], dtype=self.typ)
for n in xrange(nc, 0, -1):
self.r[n - 1], Um1, Um1_i = tm.restore_LCF_r(self.A[n], self.r[n],
Ui, sanity_checks=self.sanity_checks)
Ui = Um1_i
#Now U is U_0
U = Um1
for s in xrange(self.q[0]):
self.uni_l.A[0][s] = U.dot(self.uni_l.A[0][s])
self.uni_l.A[-1][s] = self.uni_l.A[-1][s].dot(Ui)
self.uni_l.r[-1] = U.dot(self.uni_l.r[-1].dot(U.conj().T))
#And now: l[nc <= n <= N] diagonal
if dbg:
Um1 = sp.eye(self.D[nc - 1], dtype=self.typ)
else:
Um1 = mm.eyemat(self.D[nc - 1], dtype=self.typ) #FIXME: This only works if l[nc - 1] is a special matrix type
for n in xrange(nc, self.N + 1):
self.l[n], U, Ui = tm.restore_RCF_l(self.A[n], self.l[n - 1], Um1,
sanity_checks=self.sanity_checks)
Um1 = U
#Now, Um1 = U_N
Um1_i = Ui
for s in xrange(self.q[0]):
self.uni_r.A[0][s] = Um1.dot(self.uni_r.A[0][s])
self.uni_r.A[-1][s] = self.uni_r.A[-1][s].dot(Um1_i)
self.uni_r.l[-1] = Um1_i.conj().T.dot(self.uni_r.l[-1].dot(Um1_i))
def _restore_CF_diag(self, dbg=False):
nc = self.N_centre
#Want: r[0 <= n < nc] diagonal
Ui = sp.eye(self.D[nc], dtype=self.typ)
for n in xrange(nc, 0, -1):
self.r[n - 1], Um1, Um1_i = tm.restore_LCF_r(
self.A[n], self.r[n], Ui, sanity_checks=self.sanity_checks)
Ui = Um1_i
#Now U is U_0
U = Um1
for s in xrange(self.q[0]):
self.uni_l.A[0][s] = U.dot(self.uni_l.A[0][s])
self.uni_l.A[-1][s] = self.uni_l.A[-1][s].dot(Ui)
self.uni_l.r[-1] = U.dot(self.uni_l.r[-1].dot(U.conj().T))
#And now: l[nc <= n <= N] diagonal
if dbg:
Um1 = sp.eye(self.D[nc - 1], dtype=self.typ)
else:
Um1 = mm.eyemat(
self.D[nc - 1], dtype=self.typ
) #FIXME: This only works if l[nc - 1] is a special matrix type
for n in xrange(nc, self.N + 1):
self.l[n], U, Ui = tm.restore_RCF_l(
self.A[n],
self.l[n - 1],
Um1,
sanity_checks=self.sanity_checks)
Um1 = U
#Now, Um1 = U_N
Um1_i = Ui
for s in xrange(self.q[0]):
self.uni_r.A[0][s] = Um1.dot(self.uni_r.A[0][s])
self.uni_r.A[-1][s] = self.uni_r.A[-1][s].dot(Um1_i)
self.uni_r.l[-1] = Um1_i.conj().T.dot(self.uni_r.l[-1].dot(Um1_i))
def _restore_CF_diag(self):
nc = self.N_centre
self.S_hc = sp.zeros((self.N + 1), dtype=sp.complex128)
#Want: r[0 <= n < nc] diagonal
Ui = sp.eye(self.D[nc], dtype=self.typ)
for n in xrange(nc, 0, -1):
self.r[n - 1], Um1, Um1_i = tm.restore_LCF_r(self.A[n], self.r[n],
Ui, sanity_checks=self.sanity_checks)
self.S_hc[n - 1] = -sp.sum(self.r[n - 1].diag * sp.log2(self.r[n - 1].diag))
Ui = Um1_i
#Now U is U_0
U = Um1
for s in xrange(self.q[0]):
self.A[0][s] = U.dot(self.A[0][s]).dot(Ui)
self.uni_l.r = U.dot(self.uni_l.r.dot(U.conj().T))
#And now: l[nc <= n <= N] diagonal
Um1 = mm.eyemat(self.D[nc - 1], dtype=self.typ)
for n in xrange(nc, self.N + 1):
self.l[n], U, Ui = tm.restore_RCF_l(self.A[n], self.l[n - 1], Um1,
sanity_checks=self.sanity_checks)
self.S_hc[n] = -sp.sum(self.l[n].diag * sp.log2(self.l[n].diag))
Um1 = U
#Now, Um1 = U_N
Um1_i = Ui
for s in xrange(self.q[0]):
self.A[self.N + 1][s] = Um1.dot(self.A[self.N + 1][s]).dot(Um1_i)
self.uni_r.l = Um1_i.conj().T.dot(self.uni_r.l.dot(Um1_i))
def restore_RCF(self, update_l=True, normalize=True, diag_l=True):
"""Use a gauge-transformation to restore right canonical form.
Implements the conditions for right canonical form from sub-section
3.1, theorem 1 of arXiv:quant-ph/0608197v2.
This performs two 'almost' gauge transformations, where the 'almost'
means we allow the norm to vary (if "normalize" = True).
The last step (A[1]) is done diffently to the others since G[0],
the gauge-transf. matrix, is just a number, which can be found more
efficiently and accurately without using matrix methods.
The last step (A[1]) is important because, if we have successfully made
r[1] = 1 in the previous steps, it fully determines the normalization
of the state via r[0] ( = l[N]).
Optionally (normalize=False), the function will not attempt to make
A[1] satisfy the orthonorm. condition, and will take G[0] = 1 = G[N],
thus performing a pure gauge-transformation, but not ensuring complete
canonical form.
It is also possible to begin the process from a site n other than N,
in case the sites > n are known to be in the desired form already.
It is also possible to skip the diagonalization of the l's, such that
only the right orthonormalization condition (r_n = eye) is met.
By default, the l's are updated even if diag_l=False.
Parameters
----------
update_l : bool
Whether to call calc_l() after completion (defaults to True)
normalize : bool
Whether to also attempt to normalize the state.
diag_l : bool
Whether to put l in diagonal form (defaults to True)
"""
start = self.N
G_n_i = sp.eye(self.D[start], dtype=self.typ) #This is actually just the number 1
for n in xrange(start, 1, -1):
self.r[n - 1], G_n, G_n_i = tm.restore_RCF_r(self.A[n], self.r[n],
G_n_i, sc_data=('site', n),
zero_tol=self.zero_tol,
sanity_checks=self.sanity_checks)
#Now do A[1]...
#Apply the remaining G[1]^-1 from the previous step.
for s in xrange(self.q[1]):
self.A[1][s] = m.mmul(self.A[1][s], G_n_i)
#Now finish off
tm.eps_r_noop_inplace(self.r[1], self.A[1], self.A[1], out=self.r[0])
if normalize:
G0 = 1. / sp.sqrt(self.r[0].squeeze().real)
self.A[1] *= G0
self.r[0][:] = 1
if self.sanity_checks:
r0 = tm.eps_r_noop(self.r[1], self.A[1], self.A[1])
if not sp.allclose(r0, 1, atol=1E-12, rtol=1E-12):
log.warning("Sanity Fail in restore_RCF!: r_0 is bad / norm failure")
if diag_l:
G_nm1 = sp.eye(self.D[0], dtype=self.typ)
for n in xrange(1, self.N):
self.l[n], G_nm1, G_nm1_i = tm.restore_RCF_l(self.A[n],
self.l[n - 1],
G_nm1,
self.sanity_checks)
#Apply remaining G_Nm1 to A[N]
n = self.N
for s in xrange(self.q[n]):
self.A[n][s] = m.mmul(G_nm1, self.A[n][s])
#Deal with final, scalar l[N]
tm.eps_l_noop_inplace(self.l[n - 1], self.A[n], self.A[n], out=self.l[n])
if self.sanity_checks:
if not sp.allclose(self.l[self.N].real, 1, atol=1E-12, rtol=1E-12):
log.warning("Sanity Fail in restore_RCF!: l_N is bad / norm failure")
log.warning("l_N = %s", self.l[self.N].squeeze().real)
for n in xrange(1, self.N + 1):
r_nm1 = tm.eps_r_noop(self.r[n], self.A[n], self.A[n])
#r_nm1 = tm.eps_r_noop(m.eyemat(self.D[n], self.typ), self.A[n], self.A[n])
if not sp.allclose(r_nm1, self.r[n - 1], atol=1E-11, rtol=1E-11):
log.warning("Sanity Fail in restore_RCF!: r_%u is bad (off by %g)", n, la.norm(r_nm1 - self.r[n - 1]))
elif update_l:
self.calc_l()
def restore_RCF(self, update_l=True, normalize=True, diag_l=True):
"""Use a gauge-transformation to restore right canonical form.
Implements the conditions for right canonical form from sub-section
3.1, theorem 1 of arXiv:quant-ph/0608197v2.
This performs two 'almost' gauge transformations, where the 'almost'
means we allow the norm to vary (if "normalize" = True).
The last step (A[1]) is done diffently to the others since G[0],
the gauge-transf. matrix, is just a number, which can be found more
efficiently and accurately without using matrix methods.
The last step (A[1]) is important because, if we have successfully made
r[1] = 1 in the previous steps, it fully determines the normalization
of the state via r[0] ( = l[N]).
Optionally (normalize=False), the function will not attempt to make
A[1] satisfy the orthonorm. condition, and will take G[0] = 1 = G[N],
thus performing a pure gauge-transformation, but not ensuring complete
canonical form.
It is also possible to begin the process from a site n other than N,
in case the sites > n are known to be in the desired form already.
It is also possible to skip the diagonalization of the l's, such that
only the right orthonormalization condition (r_n = eye) is met.
By default, the l's are updated even if diag_l=False.
Parameters
----------
update_l : bool
Whether to call calc_l() after completion (defaults to True)
normalize : bool
Whether to also attempt to normalize the state.
diag_l : bool
Whether to put l in diagonal form (defaults to True)
"""
start = self.N
G_n_i = sp.eye(self.D[start],
dtype=self.typ) #This is actually just the number 1
for n in xrange(start, 1, -1):
self.r[n - 1], G_n, G_n_i = tm.restore_RCF_r(
self.A[n],
self.r[n],
G_n_i,
sc_data=('site', n),
zero_tol=self.zero_tol,
sanity_checks=self.sanity_checks)
#Now do A[1]...
#Apply the remaining G[1]^-1 from the previous step.
for s in xrange(self.q[1]):
self.A[1][s] = m.mmul(self.A[1][s], G_n_i)
#Now finish off
tm.eps_r_noop_inplace(self.r[1], self.A[1], self.A[1], out=self.r[0])
if normalize:
G0 = 1. / sp.sqrt(self.r[0].squeeze().real)
self.A[1] *= G0
self.r[0][:] = 1
if self.sanity_checks:
r0 = tm.eps_r_noop(self.r[1], self.A[1], self.A[1])
if not sp.allclose(r0, 1, atol=1E-12, rtol=1E-12):
log.warning(
"Sanity Fail in restore_RCF!: r_0 is bad / norm failure"
)
if diag_l:
G_nm1 = sp.eye(self.D[0], dtype=self.typ)
for n in xrange(1, self.N):
self.l[n], G_nm1, G_nm1_i = tm.restore_RCF_l(
self.A[n], self.l[n - 1], G_nm1, self.sanity_checks)
#Apply remaining G_Nm1 to A[N]
n = self.N
for s in xrange(self.q[n]):
self.A[n][s] = m.mmul(G_nm1, self.A[n][s])
#Deal with final, scalar l[N]
tm.eps_l_noop_inplace(self.l[n - 1],
self.A[n],
self.A[n],
out=self.l[n])
if self.sanity_checks:
if not sp.allclose(
self.l[self.N].real, 1, atol=1E-12, rtol=1E-12):
log.warning(
"Sanity Fail in restore_RCF!: l_N is bad / norm failure"
)
log.warning("l_N = %s", self.l[self.N].squeeze().real)
for n in xrange(1, self.N + 1):
r_nm1 = tm.eps_r_noop(self.r[n], self.A[n], self.A[n])
#r_nm1 = tm.eps_r_noop(m.eyemat(self.D[n], self.typ), self.A[n], self.A[n])
if not sp.allclose(
r_nm1, self.r[n - 1], atol=1E-11, rtol=1E-11):
log.warning(
"Sanity Fail in restore_RCF!: r_%u is bad (off by %g)",
n, la.norm(r_nm1 - self.r[n - 1]))
elif update_l:
self.calc_l()