def _calc_lr(self, x, tmp, calc_l=False, A1=None, A2=None, rescale=True,
max_itr=1000, rtol=1E-14, atol=1E-14):
"""Power iteration to obtain eigenvector corresponding to largest
eigenvalue.
x is modified in place.
"""
if A1 is None:
A1 = self.A
if A2 is None:
A2 = self.A
try:
norm = la.get_blas_funcs("nrm2", [x])
except (ValueError, AttributeError):
norm = np.linalg.norm
# try:
# allclose = ac.allclose_mat
# except:
# allclose = np.allclose
# print "Falling back to numpy allclose()!"
n = x.size #we will scale x so that stuff doesn't get too small
x *= n / norm(x.ravel())
tmp[:] = x
for i in xrange(max_itr):
x[:] = tmp
if calc_l:
tm.eps_l_noop_inplace(x, A1, A2, tmp)
else:
tm.eps_r_noop_inplace(x, A1, A2, tmp)
ev_mag = norm(tmp.ravel()) / n
ev = (tmp.mean() / x.mean()).real
tmp *= (1 / ev_mag)
if norm((tmp - x).ravel()) < atol + rtol * n:
# if allclose(tmp, x, rtol, atol):
#print (i, ev, ev_mag, norm((tmp - x).ravel())/n, atol, rtol)
x[:] = tmp
break
# else:
# print (i, ev, ev_mag, norm((tmp - x).ravel())/norm(x.ravel()), atol, rtol)
if rescale and not abs(ev - 1) < atol:
A1 *= 1 / sp.sqrt(ev)
if self.sanity_checks:
if not A1 is A2:
log.warning("Sanity check failed: Re-scaling with A1 <> A2!")
if calc_l:
tm.eps_l_noop_inplace(x, A1, A2, tmp)
else:
tm.eps_r_noop_inplace(x, A1, A2, tmp)
ev = tmp.mean() / x.mean()
if not abs(ev - 1) < atol:
log.warning("Sanity check failed: Largest ev after re-scale = %s", ev)
return x, i < max_itr - 1, i
def restore_LCF(self):
Gm1 = sp.eye(self.D[0], dtype=self.typ) #This is actually just the number 1
for n in xrange(1, self.N):
self.l[n], G, Gi = tm.restore_LCF_l(self.A[n], self.l[n - 1], Gm1,
zero_tol=self.zero_tol,
sanity_checks=self.sanity_checks)
Gm1 = G
#Now do A[N]...
#Apply the remaining G[N - 1] from the previous step.
for s in xrange(self.q[self.N]):
self.A[self.N][s] = Gm1.dot(self.A[self.N][s])
#Now finish off
tm.eps_l_noop_inplace(self.l[self.N - 1], self.A[self.N], self.A[self.N], out=self.l[self.N])
#normalize
GNi = 1. / sp.sqrt(self.l[self.N].squeeze().real)
self.A[self.N] *= GNi
self.l[self.N][:] = 1
if self.sanity_checks:
lN = tm.eps_l_noop(self.l[self.N - 1], self.A[self.N], self.A[self.N])
if not sp.allclose(lN, 1, atol=1E-12, rtol=1E-12):
log.warning("Sanity Fail in restore_LCF!: l_N is bad / norm failure")
#diag r
Gi = sp.eye(self.D[self.N], dtype=self.typ)
for n in xrange(self.N, 1, -1):
self.r[n - 1], Gm1, Gm1_i = tm.restore_LCF_r(self.A[n], self.r[n],
Gi, self.sanity_checks)
Gi = Gm1_i
#Apply remaining G1i to A[1]
for s in xrange(self.q[1]):
self.A[1][s] = self.A[1][s].dot(Gi)
#Deal with final, scalar r[0]
tm.eps_r_noop_inplace(self.r[1], self.A[1], self.A[1], out=self.r[0])
if self.sanity_checks:
if not sp.allclose(self.r[0], 1, atol=1E-12, rtol=1E-12):
log.warning("Sanity Fail in restore_LCF!: r_0 is bad / norm failure")
log.warning("r_0 = %s", self.r[0].squeeze().real)
for n in xrange(1, self.N + 1):
l = tm.eps_l_noop(self.l[n - 1], self.A[n], self.A[n])
if not sp.allclose(l, self.l[n], atol=1E-11, rtol=1E-11):
log.warning("Sanity Fail in restore_LCF!: l_%u is bad (off by %g)", n, la.norm(l - self.l[n]))
log.warning((l - self.l[n]).diagonal().real)
def matvec(self, v):
x = v.reshape((self.D, self.D))
if self.left: #Multiplying from the left, but x is a col. vector, so use mat_dagger
Ehx = tm.eps_l_noop_inplace(x, self.A1, self.A2, self.out)
if self.pseudo:
QEQhx = Ehx - self.l * m.adot(self.r, x)
res = x - sp.exp(-1.j * self.p) * QEQhx
else:
res = x - sp.exp(-1.j * self.p) * Ehx
else:
Ex = tm.eps_r_noop_inplace(x, self.A1, self.A2, self.out)
if self.pseudo:
QEQx = Ex - self.r * m.adot(self.l, x)
res = x - sp.exp(1.j * self.p) * QEQx
else:
res = x - sp.exp(1.j * self.p) * Ex
return res.ravel()
def matvec(self, v):
x = v.reshape((self.D, self.D))
res = self.tmp
out = self.out
res[:] = x
if self.left: # Multiplying from the left, but x is a col. vector, so use mat_dagger
for k in xrange(len(self.A1)):
out = tm.eps_l_noop_inplace(res, self.A1[k], self.A2[k], out)
tmp = res
res = out
out = tmp
if self.pseudo:
out[:] = self.lL
out *= m.adot(self.rL, x)
res -= out
res *= -sp.exp(-1.0j * self.p)
res += x
else:
res *= -sp.exp(-1.0j * self.p)
res += x
else:
for k in xrange(len(self.A1) - 1, -1, -1):
out = tm.eps_r_noop_inplace(res, self.A1[k], self.A2[k], out)
tmp = res
res = out
out = tmp
if self.pseudo:
out[:] = self.rL
out *= m.adot(self.lL, x)
res -= out
res *= -sp.exp(1.0j * self.p)
res += x
else:
res *= -sp.exp(1.0j * self.p)
res += x
return (
res.copy().ravel()
) # apparently we can't reuse the memory we return (seems to blow up lgmres), so we make a copy
def matvec(self, v):
x = v.reshape((self.D, self.D))
if self.left: #Multiplying from the left, but x is a col. vector, so use mat_dagger
Ehx = tm.eps_l_noop_inplace(x, self.A1, self.A2, self.out)
if self.pseudo:
QEQhx = Ehx - self.l * m.adot(self.r, x)
res = x - sp.exp(-1.j * self.p) * QEQhx
else:
res = x - sp.exp(-1.j * self.p) * Ehx
else:
Ex = tm.eps_r_noop_inplace(x, self.A1, self.A2, self.out)
if self.pseudo:
QEQx = Ex - self.r * m.adot(self.l, x)
res = x - sp.exp(1.j * self.p) * QEQx
else:
res = x - sp.exp(1.j * self.p) * Ex
return res.ravel()
def matvec(self, v):
x = v.reshape((self.D, self.D))
res = self.tmp
out = self.out
res[:] = x
if self.left: #Multiplying from the left, but x is a col. vector, so use mat_dagger
for k in xrange(len(self.A1)):
out = tm.eps_l_noop_inplace(res, self.A1[k], self.A2[k], out)
tmp = res
res = out
out = tmp
if self.pseudo:
out[:] = self.lL
out *= m.adot(self.rL, x)
res -= out
res *= -sp.exp(-1.j * self.p)
res += x
else:
res *= -sp.exp(-1.j * self.p)
res += x
else:
for k in xrange(len(self.A1) - 1, -1, -1):
out = tm.eps_r_noop_inplace(res, self.A1[k], self.A2[k], out)
tmp = res
res = out
out = tmp
if self.pseudo:
out[:] = self.rL
out *= m.adot(self.lL, x)
res -= out
res *= -sp.exp(1.j * self.p)
res += x
else:
res *= -sp.exp(1.j * self.p)
res += x
return res.copy().ravel() #apparently we can't reuse the memory we return (seems to blow up lgmres), so we make a copy
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_LCF(self):
Gm1 = sp.eye(self.D[0],
dtype=self.typ) #This is actually just the number 1
for n in xrange(1, self.N):
self.l[n], G, Gi = tm.restore_LCF_l(
self.A[n],
self.l[n - 1],
Gm1,
zero_tol=self.zero_tol,
sanity_checks=self.sanity_checks)
Gm1 = G
#Now do A[N]...
#Apply the remaining G[N - 1] from the previous step.
for s in xrange(self.q[self.N]):
self.A[self.N][s] = Gm1.dot(self.A[self.N][s])
#Now finish off
tm.eps_l_noop_inplace(self.l[self.N - 1],
self.A[self.N],
self.A[self.N],
out=self.l[self.N])
#normalize
GNi = 1. / sp.sqrt(self.l[self.N].squeeze().real)
self.A[self.N] *= GNi
self.l[self.N][:] = 1
if self.sanity_checks:
lN = tm.eps_l_noop(self.l[self.N - 1], self.A[self.N],
self.A[self.N])
if not sp.allclose(lN, 1, atol=1E-12, rtol=1E-12):
log.warning(
"Sanity Fail in restore_LCF!: l_N is bad / norm failure")
#diag r
Gi = sp.eye(self.D[self.N], dtype=self.typ)
for n in xrange(self.N, 1, -1):
self.r[n - 1], Gm1, Gm1_i = tm.restore_LCF_r(
self.A[n], self.r[n], Gi, self.sanity_checks)
Gi = Gm1_i
#Apply remaining G1i to A[1]
for s in xrange(self.q[1]):
self.A[1][s] = self.A[1][s].dot(Gi)
#Deal with final, scalar r[0]
tm.eps_r_noop_inplace(self.r[1], self.A[1], self.A[1], out=self.r[0])
if self.sanity_checks:
if not sp.allclose(self.r[0], 1, atol=1E-12, rtol=1E-12):
log.warning(
"Sanity Fail in restore_LCF!: r_0 is bad / norm failure")
log.warning("r_0 = %s", self.r[0].squeeze().real)
for n in xrange(1, self.N + 1):
l = tm.eps_l_noop(self.l[n - 1], self.A[n], self.A[n])
if not sp.allclose(l, self.l[n], atol=1E-11, rtol=1E-11):
log.warning(
"Sanity Fail in restore_LCF!: l_%u is bad (off by %g)",
n, la.norm(l - self.l[n]))
log.warning((l - self.l[n]).diagonal().real)
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 _calc_lr_ARPACK(self, x, tmp, calc_l=False, A1=None, A2=None, rescale=True,
tol=1E-14, ncv=None, k=1):
if A1 is None:
A1 = self.A
if A2 is None:
A2 = self.A
if self.D == 1:
x.fill(1)
if calc_l:
ev = tm.eps_l_noop(x, A1, A2)[0, 0]
else:
ev = tm.eps_r_noop(x, A1, A2)[0, 0]
if rescale and not abs(ev - 1) < tol:
A1 *= 1 / sp.sqrt(ev)
return x, True, 1
try:
norm = la.get_blas_funcs("nrm2", [x])
except (ValueError, AttributeError):
norm = np.linalg.norm
n = x.size #we will scale x so that stuff doesn't get too small
#start = time.clock()
opE = EOp(A1, A2, calc_l)
x *= n / norm(x.ravel())
try:
ev, eV = las.eigs(opE, which='LM', k=k, v0=x.ravel(), tol=tol, ncv=ncv)
conv = True
except las.ArpackNoConvergence:
log.warning("Reset! (l? %s)", calc_l)
ev, eV = las.eigs(opE, which='LM', k=k, tol=tol, ncv=ncv)
conv = True
#print ev2
#print ev2 * ev2.conj()
ind = ev.argmax()
ev = np.real_if_close(ev[ind])
ev = np.asscalar(ev)
eV = eV[:, ind]
#remove any additional phase factor
eVmean = eV.mean()
eV *= sp.sqrt(sp.conj(eVmean) / eVmean)
if eV.mean() < 0:
eV *= -1
eV = eV.reshape(self.D, self.D)
eV *= n / norm(eV.ravel())
x[:] = eV
#print "splinalg: %g" % (time.clock() - start)
#print "Herm? %g" % norm((eV - m.H(eV)).ravel())
#print "Norm of diff: %g" % norm((eV - x).ravel())
#print "Norms: (%g, %g)" % (norm(eV.ravel()), norm(x.ravel()))
if rescale and not abs(ev - 1) < tol:
A1 *= 1 / sp.sqrt(ev)
if self.sanity_checks:
if not A1 is A2:
log.warning("Sanity check failed: Re-scaling with A1 <> A2!")
if calc_l:
tm.eps_l_noop_inplace(x, A1, A2, tmp)
else:
tm.eps_r_noop_inplace(x, A1, A2, tmp)
ev = tmp.mean() / x.mean()
if not abs(ev - 1) < tol:
log.warning("Sanity check failed: Largest ev after re-scale = %s", ev)
return x, conv, opE.calls