def hop(x):
# H*X
nonlocal count
count += 1
assert len(x) == xsize
tda_coeff = reshape_x(x)
res = [np.zeros_like(coeff) if coeff is not None else None for coeff in tda_coeff]
# fix ket and sweep bra and accumulate into res
for ims in range(site_num):
if tda_coeff[ims] is None:
assert tangent_u[ims] is None
continue
# mix-canonical mps
mps_tangent = merge(mps_l_cano, mps_r_cano, ims+1)
mps_tangent[ims] = tensordot(tangent_u[ims], tda_coeff[ims], (-1, 0))
mps_tangent_conj = mps_r_cano.copy()
environ = Environ(mps_tangent, mpo, "R", mps_conj=mps_tangent_conj)
for ims_conj in range(site_num):
ltensor = environ.GetLR(
"L", ims_conj-1, mps_tangent, mpo, itensor=None,
mps_conj=mps_tangent_conj,
method="System"
)
rtensor = environ.GetLR(
"R", ims_conj+1, mps_tangent, mpo, itensor=None,
mps_conj=mps_tangent_conj,
method="Enviro"
)
if tda_coeff[ims_conj] is not None:
# S-a l-S
# d
# O-b-O-f-O
# e
# S-c k-S
path = [
([0, 1], "abc, cek -> abek"),
([2, 0], "abek, bdef -> akdf"),
([1, 0], "akdf, lfk -> adl"),
]
out = multi_tensor_contract(
path, ltensor, asxp(mps_tangent[ims_conj]),
asxp(mpo[ims_conj]), rtensor
)
res[ims_conj] += asnumpy(tensordot(tangent_u[ims_conj], out,
([0,1], [0,1])))
# mps_conj combine
mps_tangent_conj[ims_conj] = mps_l_cano[ims_conj]
res = [mat for mat in res if mat is not None]
return np.concatenate(res, axis=None)
def apply(self, mp: MatrixProduct, canonicalise: bool=False) -> MatrixProduct:
# todo: use meta copy to save time, could be subtle when complex type is involved
# todo: inplace version (saved memory and can be used in `hybrid_exact_propagator`)
new_mps = self.promote_mt_type(mp.copy())
if mp.is_mps:
# mpo x mps
for i, (mt_self, mt_other) in enumerate(zip(self, mp)):
assert mt_self.shape[2] == mt_other.shape[1]
# mt=np.einsum("apqb,cqd->acpbd",mpo[i],mps[i])
mt = xp.moveaxis(
tensordot(mt_self.array, mt_other.array, axes=([2], [1])), 3, 1
)
mt = mt.reshape(
(
mt_self.shape[0] * mt_other.shape[0],
mt_self.shape[1],
mt_self.shape[-1] * mt_other.shape[-1],
)
)
new_mps[i] = mt
elif mp.is_mpo or mp.is_mpdm:
# mpo x mpo
for i, (mt_self, mt_other) in enumerate(zip(self, mp)):
assert mt_self.shape[2] == mt_other.shape[1]
# mt=np.einsum("apqb,cqrd->acprbd",mt_s,mt_o)
mt = xp.moveaxis(
tensordot(mt_self.array, mt_other.array, axes=([2], [1])),
[-3, -2],
[1, 3],
)
mt = mt.reshape(
(
mt_self.shape[0] * mt_other.shape[0],
mt_self.shape[1],
mt_other.shape[2],
mt_self.shape[-1] * mt_other.shape[-1],
)
)
new_mps[i] = mt
else:
assert False
orig_idx = new_mps.qnidx
new_mps.move_qnidx(self.qnidx)
qn = self.qn if not mp.use_dummy_qn else self.dummy_qn
new_mps.qn = [
np.add.outer(np.array(qn_o), np.array(qn_m)).ravel().tolist()
for qn_o, qn_m in zip(qn, new_mps.qn)
]
new_mps.qntot += self.qntot
new_mps.move_qnidx(orig_idx)
# concerns about whether to canonicalise:
# * canonicalise helps to keep mps in a truly canonicalised state
# * canonicalise comes with a cost. Unnecessary canonicalise (for example in P&C evolution and
# expectation calculation) hampers performance.
if canonicalise:
new_mps.canonicalise()
return new_mps
def _update_ms(self,
idx: int,
u: np.ndarray,
vt: np.ndarray,
sigma=None,
qnlset=None,
qnrset=None,
m_trunc=None):
r""" update mps directly after svd
"""
if m_trunc is None:
m_trunc = u.shape[1]
u = u[:, :m_trunc]
vt = vt[:m_trunc, :]
if sigma is None:
# canonicalise, vt is not unitary
if self.is_mpo:
if self.to_right:
norm = np.linalg.norm(vt)
u *= norm
vt /= norm
else:
norm = np.linalg.norm(u)
u /= norm
vt *= norm
else:
sigma = sigma[:m_trunc]
if (not self.is_mpo and self.to_right) or (self.is_mpo
and not self.to_right):
vt = np.einsum("i, ij -> ij", sigma, vt)
else:
u = np.einsum("ji, i -> ji", u, sigma)
if self.to_right:
self[idx + 1] = tensordot(vt, self[idx + 1], axes=1)
ret_mpsi = u.reshape([u.shape[0] // self[idx].pdim_prod] +
list(self[idx].pdim) + [m_trunc])
if qnlset is not None:
self.qn[idx + 1] = qnlset[:m_trunc]
self.qnidx = idx + 1
else:
self[idx - 1] = tensordot(self[idx - 1], u, axes=1)
ret_mpsi = vt.reshape([m_trunc] + list(self[idx].pdim) +
[vt.shape[1] // self[idx].pdim_prod])
if qnrset is not None:
self.qn[idx] = qnrset[:m_trunc]
self.qnidx = idx - 1
if ret_mpsi.nbytes < ret_mpsi.base.nbytes * 0.8:
# do copy here to discard unnecessary data. Note that in NumPy common slicing returns
# a `view` containing the original data. If `ret_mpsi` is used directly the original
# `u` or `vt` is not garbage collected.
ret_mpsi = ret_mpsi.copy()
assert ret_mpsi.any()
self[idx] = ret_mpsi
def renormalization_ddm(cstruct, qnbigl, qnbigr, domain, nexciton, Mmax, percent=0):
"""
get the new mps, mpsdim, mpdqn, complementary mps to get the next guess
with diagonalize reduced density matrix method (> 1 root)
"""
nroots = len(cstruct)
ddm = 0.0
for iroot in range(nroots):
if domain == "R":
ddm += np.tensordot(
cstruct[iroot],
cstruct[iroot],
axes=(range(qnbigl.ndim), range(qnbigl.ndim)),
)
else:
ddm += np.tensordot(
cstruct[iroot],
cstruct[iroot],
axes=(
range(qnbigl.ndim, cstruct[0].ndim),
range(qnbigl.ndim, cstruct[0].ndim),
),
)
ddm /= float(nroots)
if domain == "L":
Uset, Sset, qnnew = svd_qn.Csvd(ddm, qnbigl, qnbigl, nexciton, ddm=True)
else:
Uset, Sset, qnnew = svd_qn.Csvd(ddm, qnbigr, qnbigr, nexciton, ddm=True)
mps, mpsdim, mpsqn, compmps = updatemps(
Uset, Sset, qnnew, None, nexciton, Mmax, percent=percent
)
if domain == "R":
return (
xp.moveaxis(mps.reshape(list(qnbigr.shape) + [mpsdim]), -1, 0),
mpsdim,
mpsqn,
tensordot(
asxp(cstruct[0]),
mps.reshape(list(qnbigr.shape) + [mpsdim]),
axes=(range(qnbigl.ndim, cstruct[0].ndim), range(qnbigr.ndim)),
),
)
else:
return (
mps.reshape(list(qnbigl.shape) + [mpsdim]),
mpsdim,
mpsqn,
tensordot(
mps.reshape(list(qnbigl.shape) + [mpsdim]),
asxp(cstruct[0]),
axes=(range(qnbigl.ndim), range(qnbigl.ndim)),
),
)
def func(t, y):
y0 = y.reshape(shape)
HC = hop(y0)
if not islast:
proj = projector(y0)
if y0.ndim == 3:
HC = tensordot(proj, HC, axes=([2, 3], [0, 1]))
# uncomment this might resolve some numerical problem
# HC = tensordot(proj, HC, axes=([2, 3], [0, 1]))
elif y0.ndim == 4:
HC = tensordot(proj, HC, axes=([3, 4, 5], [0, 1, 2]))
# HC = tensordot(proj, HC, axes=([3, 4, 5], [0, 1, 2]))
return tensordot(HC, S_inv, axes=(-1, 0)).ravel() / coef
def transferMat(mps, mpsconj, domain, siteidx):
"""
calculate the transfer matrix from the left hand or the right hand
"""
val = ones([1, 1])
if domain == "R":
for imps in range(len(mps) - 1, siteidx - 1, -1):
val = tensordot(mpsconj[imps], val, axes=(2, 0))
val = tensordot(val, mps[imps], axes=([1, 2], [1, 2]))
elif domain == "L":
for imps in range(0, siteidx + 1, 1):
val = tensordot(mpsconj[imps], val, axes=(0, 0))
val = tensordot(val, mps[imps], axes=([0, 2], [1, 0]))
return val
def apply(self, mp, canonicalise=False) -> "MpDmBase":
# Note usually mp is an mpo
assert not mp.is_mps
new_mpdm = self.metacopy()
if mp.is_complex:
new_mpdm.to_complex(inplace=True)
# todo: also duplicate with MPO apply. What to do???
for i, (mt_self, mt_other) in enumerate(zip(self, mp)):
assert mt_self.shape[2] == mt_other.shape[1]
# mt=np.einsum("apqb,cqrd->acprbd",mt_s,mt_o)
mt = xp.moveaxis(
tensordot(mt_self.array, mt_other.array, axes=([2], [1])),
[-3, -2],
[1, 3],
)
mt = mt.reshape((
mt_self.shape[0] * mt_other.shape[0],
mt_self.shape[1],
mt_other.shape[2],
mt_self.shape[-1] * mt_other.shape[-1],
))
new_mpdm[i] = mt
qn = mp.dummy_qn
new_mpdm.qn = [
np.add.outer(np.array(qn_o), np.array(qn_m)).ravel().tolist()
for qn_o, qn_m in zip(self.qn, qn)
]
if canonicalise:
new_mpdm.canonicalise()
return new_mpdm
def analysis_1ordm(self):
r""" calculate one-orbital reduced density matrix of each tda root
"""
mps_l_cano, mps_r_cano, tangent_u, tda_coeff_list = self.wfn
for iroot in range(self.nroots):
tda_coeff = tda_coeff_list[iroot]
rdm = None
for ims in range(mps_l_cano.site_num):
if tangent_u[ims] is None:
assert tda_coeff[ims] is None
continue
mps_tangent = merge(mps_l_cano, mps_r_cano, ims+1)
mps_tangent[ims] = tensordot(tangent_u[ims], tda_coeff[ims],[-1,0])
rdm_increment = mps_tangent.calc_1ordm()
if rdm is None:
rdm = rdm_increment
else:
rdm = [mat1+mat2 for mat1, mat2 in zip(rdm, rdm_increment)]
dominant_config = {}
for isite, mat in enumerate(rdm):
quanta = np.argmax(np.diag(mat))
dominant_config[isite] = (quanta, np.diag(mat)[quanta])
logger.info(f"root: {iroot}, config: {dominant_config}")
def _update_ms(
self, idx: int, u: Matrix, vt: Matrix, sigma=None, qnlset=None, qnrset=None, m_trunc=None
):
if m_trunc is None:
m_trunc = u.shape[1]
u = u[:, :m_trunc]
vt = vt[:m_trunc, :]
if sigma is None:
# canonicalise, vt is not unitary
if self.is_mpo:
if self.left:
norm = vt.norm()
u = Matrix(u.array * norm)
vt = Matrix(vt.array / norm)
else:
norm = u.norm()
u = Matrix(u.array / norm)
vt = Matrix(vt.array * norm)
else:
sigma = sigma[:m_trunc]
if (not self.is_mpo and self.left) or (self.is_mpo and not self.left):
vt = einsum("i, ij -> ij", sigma, vt)
else:
u = einsum("ji, i -> ji", u, sigma)
if self.left:
self[idx + 1] = tensordot(vt, self[idx + 1], axes=1)
ret_mpsi = u.reshape(
[u.shape[0] // self[idx].pdim_prod] + list(self[idx].pdim) + [m_trunc]
)
if qnlset is not None:
self.qn[idx + 1] = qnlset[:m_trunc]
else:
self[idx - 1] = tensordot(self[idx - 1], u, axes=1)
ret_mpsi = vt.reshape(
[m_trunc] + list(self[idx].pdim) + [vt.shape[1] // self[idx].pdim_prod]
)
if qnrset is not None:
self.qn[idx] = qnrset[:m_trunc]
if ret_mpsi.nbytes < ret_mpsi.base.nbytes * 0.8:
# do copy here to discard unnecessary data. Note that in NumPy common slicing returns
# a `view` containing the original data. If `ret_mpsi` is used directly the original
# `u` or `vt` is not garbage collected.
ret_mpsi = ret_mpsi.copy()
assert ret_mpsi.any()
self[idx] = ret_mpsi
def test_environ():
mps = Mps.random(holstein_model, 1, 10)
mpo = Mpo(holstein_model)
mps = mps.evolve(mpo, 10)
environ = Environ(mps, mpo)
for i in range(len(mps) - 1):
l = environ.read("L", i)
r = environ.read("R", i + 1)
e = complex(tensordot(l, r, axes=((0, 1, 2), (0, 1, 2)))).real
assert pytest.approx(e) == mps.expectation(mpo)
def full_wfn(self):
dim = np.prod(self.pbond_list)
if 20000 < dim:
raise ValueError("wavefunction too large")
res = ones((1, 1, 1))
for mt in self:
dim1 = res.shape[1] * mt.shape[1]
dim2 = mt.shape[-1]
res = tensordot(res, mt, axes=1).reshape(1, dim1, dim2)
return res[0, :, 0]
def full_operator(self):
dim = np.prod(self.pbond_list)
if 20000 < dim:
raise ValueError("operator too large")
res = ones((1, 1, 1, 1))
for mt in self:
dim1 = res.shape[1] * mt.shape[1]
dim2 = res.shape[2] * mt.shape[2]
dim3 = mt.shape[-1]
res = tensordot(res, mt, axes=1).transpose((0, 1, 3, 2, 4, 5)).reshape(1, dim1, dim2, dim3)
return res[0, :, :, 0]
def calc_reduced_density_matrix(self) -> np.ndarray:
if self.mol_list.scheme < 4:
return self._calc_reduced_density_matrix(self, self.conj_trans())
elif self.mol_list.scheme == 4:
# be careful this method should be read-only
copy = self.copy()
copy.canonicalise(self.mol_list.e_idx())
e_mo = copy[self.mol_list.e_idx()]
return tensordot(e_mo, e_mo.conj(),
axes=((0, 2, 3), (0, 2, 3))).asnumpy()
else:
assert False
def calc_reduced_density_matrix(self) -> np.ndarray:
if self.mol_list.scheme < 4:
mp1 = [mt.reshape(mt.shape[0], mt.shape[1], 1, mt.shape[2]) for mt in self]
mp2 = [mt.reshape(mt.shape[0], 1, mt.shape[1], mt.shape[2]).conj() for mt in self]
return self._calc_reduced_density_matrix(mp1, mp2)
elif self.mol_list.scheme == 4:
# be careful this method should be read-only
copy = self.copy()
copy.canonicalise(self.mol_list.e_idx())
e_mo = copy[self.mol_list.e_idx()]
return tensordot(e_mo.conj(), e_mo, axes=((0, 2), (0, 2))).asnumpy()
else:
assert False
def _calc_reduced_density_matrix(self, mp1, mp2):
# further optimization is difficult. There are totally N^2 intermediate results to remember.
reduced_density_matrix = np.zeros(
(self.mol_list.mol_num, self.mol_list.mol_num), dtype=backend.complex_dtype
)
for i in range(self.mol_list.mol_num):
for j in range(self.mol_list.mol_num):
elem = ones((1, 1))
e_idx = -1
for mt_idx, (mt1, mt2) in enumerate(zip(mp1, mp2)):
if self.ephtable.is_electron(mt_idx):
e_idx += 1
axis_idx1 = int(e_idx == i)
axis_idx2 = int(e_idx == j)
sub_mt1 = mt1[:, axis_idx1, :, :]
sub_mt2 = mt2[:, :, axis_idx2, :]
elem = tensordot(elem, sub_mt1, axes=(0, 0))
elem = tensordot(elem, sub_mt2, axes=[(0, 1), (0, 1)])
else:
elem = tensordot(elem, mt1, axes=(0, 0))
elem = tensordot(elem, mt2, axes=[(0, 1, 2), (0, 2, 1)])
reduced_density_matrix[i][j] = elem.flatten()[0]
return reduced_density_matrix
def dot(self, other):
"""
dot product of two mps / mpo
"""
assert len(self) == len(other)
e0 = eye(1, 1)
# for debugging. It has little computational cost anyway
debug_t = []
for mt1, mt2 in zip(self, other):
# sum_x e0[:,x].m[x,:,:]
debug_t.append(e0)
e0 = tensordot(e0, mt2, 1)
# sum_ij e0[i,p,:] self[i,p,:]
# note, need to flip a (:) index onto top,
# therefore take transpose
if mt1.ndim == 3:
e0 = tensordot(e0, mt1, ([0, 1], [0, 1])).T
elif mt1.ndim == 4:
e0 = tensordot(e0, mt1, ([0, 1, 2], [0, 1, 2])).T
else:
assert False
return e0[0, 0]
def optimize_mps_dmrg(mps, mpo):
"""
1 or 2 site optimization procedure
"""
method = mps.optimize_config.method
procedure = mps.optimize_config.procedure
inverse = mps.optimize_config.inverse
nroots = mps.optimize_config.nroots
assert method in ["2site", "1site"]
# print("optimization method", method)
nexciton = mps.nexciton
# construct the environment matrix
environ = Environ()
environ.construct(mps, mps, mpo, "L")
nMPS = len(mps)
# construct each sweep cycle scheme
if method == "1site":
loop = [["R", i]
for i in range(nMPS - 1, -1, -1)] + [["L", i]
for i in range(0, nMPS)]
else:
loop = [["R", i]
for i in range(nMPS - 1, 0, -1)] + [["L", i]
for i in range(1, nMPS)]
# initial matrix
ltensor = ones((1, 1, 1))
rtensor = ones((1, 1, 1))
energies = []
for isweep, (mmax, percent) in enumerate(procedure):
logger.debug(f"mmax, percent: {mmax}, {percent}")
logger.debug(f"energy: {mps.expectation(mpo)}")
logger.debug(f"{mps}")
for system, imps in loop:
if system == "R":
lmethod, rmethod = "Enviro", "System"
else:
lmethod, rmethod = "System", "Enviro"
if method == "1site":
lsite = imps - 1
addlist = [imps]
else:
lsite = imps - 2
addlist = [imps - 1, imps]
ltensor = environ.GetLR("L",
lsite,
mps,
mps,
mpo,
itensor=ltensor,
method=lmethod)
rtensor = environ.GetLR("R",
imps + 1,
mps,
mps,
mpo,
itensor=rtensor,
method=rmethod)
# get the quantum number pattern
qnmat, qnbigl, qnbigr = construct_qnmat(mps, mpo.ephtable,
mpo.pbond_list, addlist,
method, system)
cshape = qnmat.shape
# hdiag
tmp_ltensor = einsum("aba -> ba", ltensor)
tmp_MPOimps = einsum("abbc -> abc", mpo[imps])
tmp_rtensor = einsum("aba -> ba", rtensor)
if method == "1site":
# S-a c f-S
# O-b-O-g-O
# S-a c f-S
path = [([0, 1], "ba, bcg -> acg"), ([1, 0], "acg, gf -> acf")]
hdiag = multi_tensor_contract(path, tmp_ltensor, tmp_MPOimps,
tmp_rtensor)[(qnmat == nexciton)]
# initial guess b-S-c
# a
cguess = mps[imps][qnmat == nexciton]
else:
# S-a c d f-S
# O-b-O-e-O-g-O
# S-a c d f-S
tmp_MPOimpsm1 = einsum("abbc -> abc", mpo[imps - 1])
path = [
([0, 1], "ba, bce -> ace"),
([0, 1], "edg, gf -> edf"),
([0, 1], "ace, edf -> acdf"),
]
hdiag = multi_tensor_contract(path, tmp_ltensor, tmp_MPOimpsm1,
tmp_MPOimps,
tmp_rtensor)[(qnmat == nexciton)]
# initial guess b-S-c-S-e
# a d
cguess = tensordot(mps[imps - 1], mps[imps],
axes=1)[qnmat == nexciton]
cguess = cguess.asnumpy()
hdiag *= inverse
nonzeros = np.sum(qnmat == nexciton)
# print("Hmat dim", nonzeros)
def hop(c):
# convert c to initial structure according to qn pattern
cstruct = cvec2cmat(cshape, c, qnmat, nexciton)
if method == "1site":
# S-a l-S
# d
# O-b-O-f-O
# e
# S-c k-S
path = [
([0, 1], "abc, adl -> bcdl"),
([2, 0], "bcdl, bdef -> clef"),
([1, 0], "clef, lfk -> cek"),
]
cout = multi_tensor_contract(path, ltensor,
Matrix(cstruct), mpo[imps],
rtensor)
# for small matrices, check hermite:
# a=tensordot(ltensor, mpo[imps], ((1), (0)))
# b=tensordot(a, rtensor, ((4), (1)))
# c=b.transpose((0, 2, 4, 1, 3, 5))
# d=c.reshape(16, 16)
else:
# S-a l-S
# d g
# O-b-O-f-O-j-O
# e h
# S-c k-S
path = [
([0, 1], "abc, adgl -> bcdgl"),
([3, 0], "bcdgl, bdef -> cglef"),
([2, 0], "cglef, fghj -> clehj"),
([1, 0], "clehj, ljk -> cehk"),
]
cout = multi_tensor_contract(
path,
ltensor,
Matrix(cstruct),
mpo[imps - 1],
mpo[imps],
rtensor,
)
# convert structure c to 1d according to qn
return inverse * cout.asnumpy()[qnmat == nexciton]
if nroots != 1:
cguess = [cguess]
for iroot in range(nroots - 1):
cguess.append(np.random.random([nonzeros]) - 0.5)
precond = lambda x, e, *args: x / (hdiag.asnumpy() - e + 1e-4)
e, c = davidson(hop,
cguess,
precond,
max_cycle=100,
nroots=nroots,
max_memory=64000)
# scipy arpack solver : much slower than davidson
# A = spslinalg.LinearOperator((nonzeros,nonzeros), matvec=hop)
# e, c = spslinalg.eigsh(A,k=1, which="SA",v0=cguess)
# print("HC loops:", count[0])
# logger.debug(f"isweep: {isweep}, e: {e}")
energies.append(e)
cstruct = cvec2cmat(cshape, c, qnmat, nexciton, nroots=nroots)
if nroots == 1:
# direct svd the coefficient matrix
mt, mpsdim, mpsqn, compmps = renormalization_svd(
cstruct,
qnbigl,
qnbigr,
system,
nexciton,
Mmax=mmax,
percent=percent,
)
else:
# diagonalize the reduced density matrix
mt, mpsdim, mpsqn, compmps = renormalization_ddm(
cstruct,
qnbigl,
qnbigr,
system,
nexciton,
Mmax=mmax,
percent=percent,
)
if method == "1site":
mps[imps] = mt
if system == "L":
if imps != len(mps) - 1:
mps[imps + 1] = tensordot(compmps,
mps[imps + 1],
axes=1)
mps.qn[imps + 1] = mpsqn
else:
mps[imps] = tensordot(mps[imps], compmps, axes=1)
mps.qn[imps + 1] = [0]
else:
if imps != 0:
mps[imps - 1] = tensordot(mps[imps - 1],
compmps,
axes=1)
mps.qn[imps] = mpsqn
else:
mps[imps] = tensordot(compmps, mps[imps], axes=1)
mps.qn[imps] = [0]
else:
if system == "L":
mps[imps - 1] = mt
mps[imps] = compmps
else:
mps[imps] = mt
mps[imps - 1] = compmps
# mps.dim_list[imps] = mpsdim
mps.qn[imps] = mpsqn
energies = np.array(energies)
if nroots == 1:
logger.debug("Optimization complete, lowest energy = %g",
energies.min())
return energies
def optimize_cv(self, lr_group, direction, isite, num, percent=0.0):
# depending on the spectratype, to restrict the exction
first_LR = lr_group[0]
second_LR = lr_group[1]
if self.spectratype == "abs":
constrain_qn = 1
else:
constrain_qn = 0
# this function aims at solving the work equation of ZT CV-DMRG
# L = <CV|op_a|CV>+2\eta<op_b|CV>, take a derivative to local CV
# S-a-S-e-S S-a-S-d-S
# | d | | | |
# O-b-O-g-O * CV[isite-1] = -\eta | c |
# | f | | | |
# S-c- -h-S S-b- -e-S
# note to be a_mat * x = vec_b
# the environment matrix
if self.method == "1site":
addlist = [isite - 1]
first_L = asxp(first_LR[isite - 1])
first_R = asxp(first_LR[isite])
second_L = asxp(second_LR[isite - 1])
second_R = asxp(second_LR[isite])
else:
addlist = [isite - 2, isite - 1]
first_L = asxp(first_LR[isite - 2])
first_R = asxp(first_LR[isite])
second_L = asxp(second_LR[isite - 2])
second_R = asxp(second_LR[isite])
if direction == 'left':
system = 'R'
else:
system = 'L'
# this part just be similar with ground state calculation
qnmat, qnbigl, qnbigr = svd_qn.construct_qnmat(
self.cv_mps, self.mpo.pbond_list,
addlist, self.method, system)
xshape = qnmat.shape
nonzeros = np.sum(qnmat == constrain_qn)
if self.method == '1site':
guess = self.cv_mps[isite - 1][qnmat == constrain_qn].reshape(nonzeros, 1)
path_b = [([0, 1], "ab, acd->bcd"),
([1, 0], "bcd, de->bce")]
vec_b = multi_tensor_contract(
path_b, second_L, self.b_oper[isite - 1], second_R
)[qnmat == constrain_qn].reshape(nonzeros, 1)
else:
guess = tensordot(
self.cv_mps[isite - 2], self.cv_mps[isite - 1], axes=(-1, 0)
)
guess = guess[qnmat == constrain_qn].reshape(nonzeros, 1)
path_b = [([0, 1], "ab, acd->bcd"),
([2, 0], "bcd, def->bcef"),
([1, 0], "bcef, fg->bceg")]
vec_b = multi_tensor_contract(
path_b, second_L, self.b_oper[isite - 2],
self.b_oper[isite - 1], second_R
)[qnmat == constrain_qn].reshape(nonzeros, 1)
if self.method == "2site":
a_oper_isite2 = asxp(self.a_oper[isite - 2])
else:
a_oper_isite2 = None
a_oper_isite1 = asxp(self.a_oper[isite - 1])
# use the diagonal part of mat_a to construct the preconditinoner for linear solver
if self.method == "1site":
part_l = xp.einsum('abca->abc', first_L)
part_r = xp.einsum('hfgh->hfg', first_R)
path_pre = [([0, 1], "abc, bdef->acdef"),
([1, 0], "acdef, hfg->acdehg")]
pre_a_mat1 = multi_tensor_contract(path_pre, part_l, a_oper_isite1,
part_r)
path_pre2 = [([0, 1], "acdehg, ceig->adhi")]
pre_a_mat1 = multi_tensor_contract(path_pre2, pre_a_mat1, a_oper_isite1)
pre_a_mat1 = xp.einsum('adhd->adh', pre_a_mat1)[qnmat == constrain_qn]
# pre_a_mat1 = xp.einsum('abca, bdef, cedg, hfgh->adh', first_L, a_oper_isite1,
# a_oper_isite1, first_R)[qnmat == constrain_qn]
cv_shape = self.cv_mps[isite - 1].shape
pre_a_mat2 = xp.ones(cv_shape)[qnmat == constrain_qn]
pre_a_mat = pre_a_mat1 + pre_a_mat2 * self.eta**2
else:
pre_a_mat1 = xp.einsum(
'abca, bdef, cedg, fhij, gihk, ljkl->adhl', first_L, a_oper_isite2, a_oper_isite2,
a_oper_isite1, a_oper_isite1, first_R)[qnmat == constrain_qn]
cv_shape1 = self.cv_mps[isite - 2].shape
cv_shape2 = self.cv_mps[isite - 1].shape
new_shape = [cv_shape1[0], cv_shape1[1], cv_shape2[1], cv_shape2[2]]
pre_a_mat2 = xp.ones(new_shape)[qnmat == constrain_qn]
pre_a_mat = pre_a_mat1 + pre_a_mat2 * self.eta**2
pre_a_mat = np.diag(1./asnumpy(pre_a_mat))
count = 0
def hop(c):
nonlocal count
count += 1
xstruct = asxp(svd_qn.cvec2cmat(xshape, c, qnmat, constrain_qn))
if self.method == "1site":
path_a = [([0, 1], "abcd, aef->bcdef"),
([3, 0], "bcdef, begh->cdfgh"),
([2, 0], "cdfgh, cgij->dfhij"),
([1, 0], "dfhij, fhjk->dik")]
ax1 = multi_tensor_contract(path_a, first_L, xstruct,
a_oper_isite1, a_oper_isite1, first_R)
ax2 = xstruct
ax = ax1 + ax2 * self.eta**2
else:
path_a = [([0, 1], "abcd, aefg->bcdefg"),
([5, 0], "bcdefg, behi->cdfghi"),
([4, 0], "cdfghi, ifjk->cdghjk"),
([3, 0], "cdghjk, chlm->dgjklm"),
([2, 0], "dgjklm, mjno->dgklno"),
([1, 0], "dgklno, gkop->dlnp")]
ax1 = multi_tensor_contract(path_a, first_L, xstruct,
a_oper_isite2, a_oper_isite1,
a_oper_isite2, a_oper_isite1,
first_R)
ax2 = xstruct
ax = ax1 + ax2 * self.eta**2
cout = ax[qnmat == constrain_qn].reshape(nonzeros, 1)
return asnumpy(cout)
mat_a = scipy.sparse.linalg.LinearOperator((nonzeros, nonzeros), matvec=hop)
# for the first two sweep, not use the previous matrix as initial guess
# at the inital stage, they are far from from the optimized one
if num in [1, 2]:
x, info = scipy.sparse.linalg.cg(mat_a, asnumpy(vec_b), atol=0)
else:
x, info = scipy.sparse.linalg.cg(mat_a, asnumpy(vec_b), tol=1.e-5,
x0=guess, M=pre_a_mat, atol=0)
# logger.info(f'hop times:{count}')
self.hop_time.append(count)
if info != 0:
logger.info(f"iteration solver not converged")
# the value of the functional L
l_value = np.inner(hop(x).reshape(1, nonzeros), x.reshape(1, nonzeros)
) - 2 * np.inner(
asnumpy(vec_b).reshape(1, nonzeros), x.reshape(1, nonzeros))
xstruct = svd_qn.cvec2cmat(xshape, x, qnmat, constrain_qn)
x, xdim, xqn, compx = \
solver.renormalization_svd(xstruct, qnbigl, qnbigr, system,
constrain_qn, self.m_max, percent)
if self.method == "1site":
self.cv_mps[isite - 1] = x
if direction == "left":
if isite != 1:
self.cv_mps[isite - 2] = tensordot(
self.cv_mps[isite - 2], compx, axes=(-1, 0))
self.cv_mps.qn[isite - 1] = xqn
else:
self.cv_mps[isite - 1] = tensordot(
compx, self.cv_mps[isite - 1], axes=(-1, 0))
self.cv_mps.qn[isite - 1] = [0]
elif direction == "right":
if isite != len(self.cv_mps):
self.cv_mps[isite] = tensordot(
compx, self.cv_mps[isite], axes=(-1, 0))
self.cv_mps.qn[isite] = xqn
else:
self.cv_mps[isite - 1] = tensordot(
self.cv_mps[isite - 1], compx, axes=(-1, 0))
self.cv_mps.qn[isite] = [0]
else:
if direction == "left":
self.cv_mps[isite - 1] = x
self.cv_mps[isite - 2] = compx
else:
self.cv_mps[isite - 2] = x
self.cv_mps[isite - 1] = compx
self.cv_mps.qn[isite - 1] = xqn
return l_value[0][0]
def optimize_cv(self, lr_group, direction, isite, num, percent=0):
if self.spectratype == "abs":
# quantum number restriction, |1><0|
up_exciton, down_exciton = 1, 0
elif self.spectratype == "emi":
# quantum number restriction, |0><1|
up_exciton, down_exciton = 0, 1
nexciton = 1
first_LR, second_LR, third_LR, forth_LR = lr_group
if self.method == "1site":
add_list = [isite - 1]
first_L = asxp(first_LR[isite - 1])
first_R = asxp(first_LR[isite])
second_L = asxp(second_LR[isite - 1])
second_R = asxp(second_LR[isite])
third_L = asxp(third_LR[isite - 1])
third_R = asxp(third_LR[isite])
forth_L = asxp(forth_LR[isite - 1])
forth_R = asxp(forth_LR[isite])
else:
add_list = [isite - 2, isite - 1]
first_L = asxp(first_LR[isite - 2])
first_R = asxp(first_LR[isite])
second_L = asxp(second_LR[isite - 2])
second_R = asxp(second_LR[isite])
third_L = asxp(third_LR[isite - 2])
third_R = asxp(third_LR[isite])
forth_L = asxp(forth_LR[isite - 2])
forth_R = asxp(forth_LR[isite])
xqnmat, xqnbigl, xqnbigr, xshape = \
self.construct_X_qnmat(add_list, direction)
dag_qnmat, dag_qnbigl, dag_qnbigr = self.swap(xqnmat, xqnbigl, xqnbigr,
direction)
nonzeros = np.sum(self.condition(dag_qnmat,
[down_exciton, up_exciton]))
if self.method == "1site":
guess = moveaxis(self.cv_mpo[isite - 1], (1, 2), (2, 1))
else:
guess = tensordot(moveaxis(self.cv_mpo[isite - 2], (1, 2), (2, 1)),
moveaxis(self.cv_mpo[isite - 1]),
axes=(-1, 0))
guess = guess[self.condition(dag_qnmat,
[down_exciton, up_exciton])].reshape(
nonzeros, 1)
if self.method == "1site":
# define dot path
path_1 = [([0, 1], "abc, adef -> bcdef"),
([2, 0], "bcdef, begh -> cdfgh"),
([1, 0], "cdfgh, fhi -> cdgi")]
path_2 = [([0, 1], "abcd, aefg -> bcdefg"),
([3, 0], "bcdefg, bfhi -> cdeghi"),
([2, 0], "cdeghi, djek -> cghijk"),
([1, 0], "cghijk, gilk -> chjl")]
path_4 = [([0, 1], "ab, acde -> bcde"), ([1,
0], "bcde, ef -> bcdf")]
vecb = multi_tensor_contract(
path_4, forth_L,
moveaxis(self.a_ket_mpo[isite - 1], (1, 2), (2, 1)), forth_R)
vecb = -self.eta * vecb
a_oper_isite = asxp(self.a_oper[isite - 1])
b_oper_isite = asxp(self.b_oper[isite - 1])
h_mpo_isite = asxp(self.h_mpo[isite - 1])
# construct preconditioner
Idt = xp.identity(h_mpo_isite.shape[1])
M1_1 = xp.einsum('aea->ae', first_L)
M1_2 = xp.einsum('eccf->ecf', a_oper_isite)
M1_3 = xp.einsum('dfd->df', first_R)
M1_4 = xp.einsum('bb->b', Idt)
path_m1 = [([0, 1], "ae,b->aeb"), ([2, 0], "aeb,ecf->abcf"),
([1, 0], "abcf, df->abcd")]
pre_M1 = multi_tensor_contract(path_m1, M1_1, M1_4, M1_2, M1_3)
pre_M1 = pre_M1[self.condition(dag_qnmat, [down_exciton, up_exciton])]
M2_1 = xp.einsum('aeag->aeg', second_L)
M2_2 = xp.einsum('eccf->ecf', b_oper_isite)
M2_3 = xp.einsum('gbbh->gbh', h_mpo_isite)
M2_4 = xp.einsum('dfdh->dfh', second_R)
path_m2 = [([0, 1], "aeg,gbh->aebh"), ([2, 0], "aebh,ecf->abchf"),
([1, 0], "abhcf,dfh->abcd")]
pre_M2 = multi_tensor_contract(path_m2, M2_1, M2_3, M2_2, M2_4)
pre_M2 = pre_M2[self.condition(dag_qnmat, [down_exciton, up_exciton])]
M4_1 = xp.einsum('faah->fah', third_L)
M4_4 = xp.einsum('gddi->gdi', third_R)
M4_5 = xp.einsum('cc->c', Idt)
M4_path = [([0, 1], "fah,febg->ahebg"), ([2, 0], "ahebg,hjei->abgji"),
([1, 0], "abgji,gdi->abjd")]
pre_M4 = multi_tensor_contract(M4_path, M4_1, h_mpo_isite, h_mpo_isite,
M4_4)
pre_M4 = xp.einsum('abbd->abd', pre_M4)
pre_M4 = xp.tensordot(pre_M4, M4_5, axes=0)
pre_M4 = xp.moveaxis(pre_M4, [2, 3], [3, 2])[self.condition(
dag_qnmat, [down_exciton, up_exciton])]
pre_M = (pre_M1 + 2 * pre_M2 + pre_M4)
indices = np.array(range(nonzeros))
indptr = np.array(range(nonzeros + 1))
pre_M = scipy.sparse.csc_matrix((asnumpy(pre_M), indices, indptr),
shape=(nonzeros, nonzeros))
M_x = lambda x: scipy.sparse.linalg.spsolve(pre_M, x)
M = scipy.sparse.linalg.LinearOperator((nonzeros, nonzeros), M_x)
count = 0
def hop(x):
nonlocal count
count += 1
dag_struct = asxp(self.dag2mat(xshape, x, dag_qnmat, direction))
if self.method == "1site":
M1 = multi_tensor_contract(path_1, first_L, dag_struct,
a_oper_isite, first_R)
M2 = multi_tensor_contract(path_2, second_L, dag_struct,
b_oper_isite, h_mpo_isite, second_R)
M2 = xp.moveaxis(M2, (1, 2), (2, 1))
M3 = multi_tensor_contract(path_2, third_L, h_mpo_isite,
dag_struct, h_mpo_isite, third_R)
M3 = xp.moveaxis(M3, (1, 2), (2, 1))
cout = M1 + 2 * M2 + M3
cout = cout[self.condition(dag_qnmat,
[down_exciton, up_exciton])].reshape(
nonzeros, 1)
return asnumpy(cout)
# Matrix A and Vector b
vecb = asnumpy(vecb)[self.condition(
dag_qnmat, [down_exciton, up_exciton])].reshape(nonzeros, 1)
mata = scipy.sparse.linalg.LinearOperator((nonzeros, nonzeros),
matvec=hop)
# conjugate gradient method
# x, info = scipy.sparse.linalg.cg(MatA, VecB, atol=0)
if num == 1:
x, info = scipy.sparse.linalg.cg(mata,
vecb,
tol=1.e-5,
maxiter=500,
M=M,
atol=0)
else:
x, info = scipy.sparse.linalg.cg(mata,
vecb,
tol=1.e-5,
x0=guess,
maxiter=500,
M=M,
atol=0)
# logger.info(f"linear eq dim: {nonzeros}")
# logger.info(f'times for hop:{count}')
self.hop_time.append(count)
if info != 0:
logger.warning(
f"cg not converged, vecb.norm:{np.linalg.norm(vecb)}")
l_value = np.inner(
hop(x).reshape(1, nonzeros), x.reshape(1, nonzeros)) - \
2 * np.inner(vecb.reshape(1, nonzeros), x.reshape(1, nonzeros))
x = self.dag2mat(xshape, x, dag_qnmat, direction)
if self.method == "1site":
x = np.moveaxis(x, [1, 2], [2, 1])
x, xdim, xqn, compx = self.x_svd(x,
xqnbigl,
xqnbigr,
nexciton,
direction,
percent=percent)
if self.method == "1site":
self.cv_mpo[isite - 1] = x
if direction == "left":
if isite != 1:
self.cv_mpo[isite - 2] = \
tensordot(self.cv_mpo[isite - 2], compx, axes=(-1, 0))
self.cv_mpo.qn[isite - 1] = xqn
else:
self.cv_mpo[isite - 1] = \
tensordot(compx, self.cv_mpo[isite - 1], axes=(-1, 0))
elif direction == "right":
if isite != len(self.cv_mpo):
self.cv_mpo[isite] = \
tensordot(compx, self.cv_mpo[isite], axes=(-1, 0))
self.cv_mpo.qn[isite] = xqn
else:
self.cv_mpo[isite - 1] = \
tensordot(self.cv_mpo[isite - 1], compx, axes=(-1, 0))
else:
if direction == "left":
self.cv_mpo[isite - 2] = compx
self.cv_mpo[isite - 1] = x
else:
self.cv_mpo[isite - 2] = x
self.cv_mpo[isite - 1] = compx
self.cv_mpo.qn[isite - 1] = xqn
return l_value[0][0]
def optimize_cv(self, lr_group, isite, percent=0.0):
# depending on the spectratype, to restrict the exction
first_LR = lr_group[0]
second_LR = lr_group[1]
constrain_qn = self.cv_mps.qntot
# this function aims at solving the work equation of ZT CV-DMRG
# L = <CV|op_a|CV>+2\eta<op_b|CV>, take a derivative to local CV
# S-a-S-e-S S-a-S-d-S
# | d | | | |
# O-b-O-g-O * CV[isite-1] = -\eta | c |
# | f | | | |
# S-c- -h-S S-b- -e-S
# note to be a_mat * x = vec_b
# the environment matrix
if self.method == "1site":
cidx = [isite - 1]
first_L = asxp(first_LR[isite - 1])
first_R = asxp(first_LR[isite])
second_L = asxp(second_LR[isite - 1])
second_R = asxp(second_LR[isite])
else:
cidx = [isite - 2, isite - 1]
first_L = asxp(first_LR[isite - 2])
first_R = asxp(first_LR[isite])
second_L = asxp(second_LR[isite - 2])
second_R = asxp(second_LR[isite])
# this part just be similar with ground state calculation
qnbigl, qnbigr, qnmat = self.cv_mps._get_big_qn(cidx)
xshape = qnmat.shape
nonzeros = int(np.sum(qnmat == constrain_qn))
if self.method == '1site':
guess = self.cv_mps[isite - 1][qnmat == constrain_qn]
path_b = [([0, 1], "ab, acd->bcd"),
([1, 0], "bcd, de->bce")]
vec_b = multi_tensor_contract(
path_b, second_L, self.b_mps[isite - 1], second_R
)[qnmat == constrain_qn]
else:
guess = tensordot(
self.cv_mps[isite - 2], self.cv_mps[isite - 1], axes=(-1, 0)
)[qnmat == constrain_qn]
path_b = [([0, 1], "ab, acd->bcd"),
([2, 0], "bcd, def->bcef"),
([1, 0], "bcef, fg->bceg")]
vec_b = multi_tensor_contract(
path_b, second_L, self.b_mps[isite - 2],
self.b_mps[isite - 1], second_R
)[qnmat == constrain_qn]
if self.method == "2site":
a_oper_isite2 = asxp(self.a_oper[isite - 2])
else:
a_oper_isite2 = None
a_oper_isite1 = asxp(self.a_oper[isite - 1])
# use the diagonal part of mat_a to construct the preconditinoner
# for linear solver
part_l = xp.einsum('abca->abc', first_L)
part_r = xp.einsum('hfgh->hfg', first_R)
if self.method == "1site":
# S-a d h-S
# O-b -O- f-O
# | e |
# O-c -O- g-O
# S-a i h-S
path_pre = [([0, 1], "abc, bdef -> acdef"),
([1, 0], "acdef, ceig -> adfig")]
a_diag = multi_tensor_contract(path_pre, part_l, a_oper_isite1,
a_oper_isite1)
a_diag = xp.einsum("adfdg -> adfg", a_diag)
a_diag = xp.tensordot(a_diag, part_r,
axes=([2, 3], [1, 2]))[qnmat == constrain_qn]
else:
# S-a d k h-S
# O-b -O- j -O- f-O
# | e l |
# O-c -O- m -O- g-O
# S-a i n h-S
# first left half, second right half, last contraction
path_pre = [([0, 1], "abc, bdej -> acdej"),
([1, 0], "acdej, ceim -> adjim")]
a_diagl = multi_tensor_contract(path_pre, part_l, a_oper_isite2,
a_oper_isite2)
a_diagl = xp.einsum("adjdm -> adjm", a_diagl)
path_pre = [([0, 1], "hfg, jklf -> hgjkl"),
([1, 0], "hgjkl, mlng -> hjkmn")]
a_diagr = multi_tensor_contract(path_pre, part_r, a_oper_isite1,
a_oper_isite1)
a_diagr = xp.einsum("hjkmk -> khjm", a_diagr)
a_diag = xp.tensordot(
a_diagl, a_diagr, axes=([2, 3], [2, 3]))[qnmat == constrain_qn]
a_diag = asnumpy(a_diag + xp.ones(nonzeros) * self.eta**2)
M_x = lambda x: x / a_diag
pre_M = scipy.sparse.linalg.LinearOperator((nonzeros, nonzeros), M_x)
count = 0
# cache oe path
if self.method == "2site":
expr = oe.contract_expression(
"abcd, befh, cfgi, hjkn, iklo, mnop, dglp -> aejm",
first_L, a_oper_isite2, a_oper_isite2, a_oper_isite1,
a_oper_isite1, first_R, xshape,
constants=[0, 1, 2, 3, 4, 5])
def hop(c):
nonlocal count
count += 1
xstruct = asxp(cvec2cmat(xshape, c, qnmat, constrain_qn))
if self.method == "1site":
path_a = [([0, 1], "abcd, aef->bcdef"),
([3, 0], "bcdef, begh->cdfgh"),
([2, 0], "cdfgh, cgij->dfhij"),
([1, 0], "dfhij, fhjk->dik")]
ax1 = multi_tensor_contract(path_a, first_L, xstruct,
a_oper_isite1, a_oper_isite1, first_R)
else:
# opt_einsum v3.2.1 is not bad, ~10% faster than the hand-design
# contraction path for this complicated cases and consumes a little bit less memory
# this is the only place in renormalizer we use opt_einsum now.
# we keep it here just for a demo.
# ax1 = oe.contract("abcd, befh, cfgi, hjkn, iklo, mnop, dglp -> aejm",
# first_L, a_oper_isite2, a_oper_isite2, a_oper_isite1,
# a_oper_isite1, first_R, xstruct)
if USE_GPU:
oe_backend = "cupy"
else:
oe_backend = "numpy"
ax1 = expr(xstruct, backend=oe_backend)
#print(oe.contract_path("abcd, befh, cfgi, hjkn, iklo, mnop, dglp -> aejm",
# first_L, a_oper_isite2, a_oper_isite2, a_oper_isite1,
# a_oper_isite1, first_R, xstruct))
#path_a = [([0, 1], "abcd, aefg->bcdefg"),
# ([5, 0], "bcdefg, behi->cdfghi"),
# ([4, 0], "cdfghi, ifjk->cdghjk"),
# ([3, 0], "cdghjk, chlm->dgjklm"),
# ([2, 0], "dgjklm, mjno->dgklno"),
# ([1, 0], "dgklno, gkop->dlnp")]
#ax1 = multi_tensor_contract(path_a, first_L, xstruct,
# a_oper_isite2, a_oper_isite1,
# a_oper_isite2, a_oper_isite1,
# first_R)
ax = ax1 + xstruct * self.eta**2
cout = ax[qnmat == constrain_qn]
return asnumpy(cout)
mat_a = scipy.sparse.linalg.LinearOperator((nonzeros, nonzeros),
matvec=hop)
x, info = scipy.sparse.linalg.cg(mat_a, asnumpy(vec_b), tol=1.e-5,
x0=asnumpy(guess),
M=pre_M, atol=0)
self.hop_time.append(count)
if info != 0:
logger.info(f"iteration solver not converged")
# the value of the functional L
l_value = xp.dot(asxp(hop(x)), asxp(x)) - 2 * xp.dot(vec_b, asxp(x))
xstruct = cvec2cmat(xshape, x, qnmat, constrain_qn)
self.cv_mps._update_mps(xstruct, cidx, qnbigl, qnbigr, self.m_max, percent)
return float(l_value)
def optimize_mps(mps: Mps, mpo: Mpo, omega: float = None) -> Tuple[List, Mps]:
r""" DMRG ground state algorithm and state-averaged excited states algorithm
Parameters
----------
mps : renormalizer.mps.Mps
initial guess of mps
mpo : renormalizer.mps.Mpo
mpo of Hamiltonian
omega: float, optional
target the eigenpair near omega with special variational function
:math:(\hat{H}-\omega)^2. Default is `None`.
Returns
-------
energy : list
list of energy of each marco sweep.
:math:`[e_0, e_0, \cdots, e_0]` if ``nroots=1``.
:math:`[[e_0, \cdots, e_n], \dots, [e_0, \cdots, e_n]]` if ``nroots=n``.
mps : renormalizer.mps.Mps
optimized ground state mps. The input mps is overwritten and could not
be used anymore.
See Also
--------
renormalizer.utils.configs.OptimizeConfig : The optimization configuration.
"""
algo = mps.optimize_config.algo
method = mps.optimize_config.method
procedure = mps.optimize_config.procedure
inverse = mps.optimize_config.inverse
nroots = mps.optimize_config.nroots
assert method in ["2site", "1site"]
logger.info(f"optimization method: {method}")
logger.info(f"e_rtol: {mps.optimize_config.e_rtol}")
logger.info(f"e_atol: {mps.optimize_config.e_atol}")
if USE_GPU:
oe_backend = "cupy"
else:
oe_backend = "numpy"
# ensure that mps is left or right-canonical
# TODO: start from a mix-canonical MPS
if mps.is_left_canon:
mps.ensure_right_canon()
env = "R"
else:
mps.ensure_left_canon()
env = "L"
# in state-averged calculation, contains C of each state for better initial
# guess
averaged_ms = None
# the index of active site of the returned mps
res_mps_idx = None
# target eigenstate close to omega with (H-omega)^2
# construct the environment matrix
if omega is not None:
identity = Mpo.identity(mpo.model)
mpo = mpo.add(identity.scale(-omega))
environ = Environ(mps, [mpo, mpo], env)
else:
environ = Environ(mps, mpo, env)
macro_iteration_result = []
converged = False
for isweep, (mmax, percent) in enumerate(procedure):
logger.debug(f"isweep: {isweep}")
logger.debug(f"mmax, percent: {mmax}, {percent}")
logger.debug(f"{mps}")
micro_iteration_result = []
for imps in mps.iter_idx_list(full=True):
if method == "2site" and \
((mps.to_right and imps == mps.site_num-1)
or ((not mps.to_right) and imps == 0)):
break
if mps.to_right:
lmethod, rmethod = "System", "Enviro"
else:
lmethod, rmethod = "Enviro", "System"
if method == "1site":
lidx = imps - 1
cidx = [imps]
ridx = imps + 1
elif method == "2site":
if mps.to_right:
lidx = imps - 1
cidx = [imps, imps + 1]
ridx = imps + 2
else:
lidx = imps - 2
cidx = [imps - 1, imps] # center site
ridx = imps + 1
else:
assert False
logger.debug(f"optimize site: {cidx}")
if omega is None:
operator = mpo
else:
operator = [mpo, mpo]
ltensor = environ.GetLR("L",
lidx,
mps,
operator,
itensor=None,
method=lmethod)
rtensor = environ.GetLR("R",
ridx,
mps,
operator,
itensor=None,
method=rmethod)
# get the quantum number pattern
qnbigl, qnbigr, qnmat = mps._get_big_qn(cidx)
cshape = qnmat.shape
nonzeros = np.sum(qnmat == mps.qntot)
logger.debug(f"Hmat dim: {nonzeros}")
# center mo
cmo = [asxp(mpo[idx]) for idx in cidx]
if qnmat.size > 1000 and algo != "direct":
# iterative algorithm
# diagonal elements of H
if omega is None:
tmp_ltensor = xp.einsum("aba -> ba", ltensor)
tmp_cmo0 = xp.einsum("abbc -> abc", cmo[0])
tmp_rtensor = xp.einsum("aba -> ba", rtensor)
if method == "1site":
# S-a c f-S
# O-b-O-g-O
# S-a c f-S
path = [([0, 1], "ba, bcg -> acg"),
([1, 0], "acg, gf -> acf")]
hdiag = multi_tensor_contract(
path, tmp_ltensor, tmp_cmo0,
tmp_rtensor)[(qnmat == mps.qntot)]
else:
# S-a c d f-S
# O-b-O-e-O-g-O
# S-a c d f-S
tmp_cmo1 = xp.einsum("abbc -> abc", cmo[1])
path = [
([0, 1], "ba, bce -> ace"),
([0, 1], "edg, gf -> edf"),
([0, 1], "ace, edf -> acdf"),
]
hdiag = multi_tensor_contract(
path, tmp_ltensor, tmp_cmo0, tmp_cmo1,
tmp_rtensor)[(qnmat == mps.qntot)]
else:
if method == "1site":
# S-a d h-S
# O-b-O-f-O
# | e |
# O-c-O-g-O
# S-a d h-S
hdiag = oe.contract(
"abca, bdef, cedg, hfgh -> adh",
ltensor,
cmo[0],
cmo[0],
rtensor,
backend=oe_backend)[(qnmat == mps.qntot)]
else:
# S-a d h l-S
# O-b-O-f-O-j-O
# | e i |
# O-c-O-g-O-k-O
# S-a d h l-S
hdiag = oe.contract(
"abca, bdef, cedg, fhij, gihk, ljkl -> adhl",
ltensor,
cmo[0],
cmo[0],
cmo[1],
cmo[1],
rtensor,
backend=oe_backend)[(qnmat == mps.qntot)]
hdiag = asnumpy(hdiag * inverse)
# initial guess
if method == "1site":
# initial guess b-S-c
# a
if nroots == 1:
cguess = [asnumpy(mps[cidx[0]])[qnmat == mps.qntot]]
else:
cguess = []
if averaged_ms is not None:
for ms in averaged_ms:
cguess.append(asnumpy(ms)[qnmat == mps.qntot])
else:
# initial guess b-S-c-S-e
# a d
if nroots == 1:
cguess = [
asnumpy(
tensordot(mps[cidx[0]], mps[cidx[1]],
axes=1)[qnmat == mps.qntot])
]
else:
cguess = []
if averaged_ms is not None:
for ms in averaged_ms:
if mps.to_right:
cguess.append(
asnumpy(
tensordot(
ms, mps[cidx[1]],
axes=1)[qnmat == mps.qntot]))
else:
cguess.append(
asnumpy(
tensordot(
mps[cidx[0]], ms,
axes=1)[qnmat == mps.qntot]))
if omega is not None:
if method == "1site":
# S-a e j-S
# O-b-O-g-O
# | f |
# O-c-O-i-O
# S-d h k-S
expr = oe.contract_expression(
"abcd, befg, cfhi, jgik, aej -> dhk",
ltensor,
cmo[0],
cmo[0],
rtensor,
cshape,
constants=[0, 1, 2, 3])
else:
# S-a e j o-S
# O-b-O-g-O-l-O
# | f k |
# O-c-O-i-O-n-O
# S-d h m p-S
expr = oe.contract_expression(
"abcd, befg, cfhi, gjkl, ikmn, olnp, aejo -> dhmp",
ltensor,
cmo[0],
cmo[0],
cmo[1],
cmo[1],
rtensor,
cshape,
constants=[0, 1, 2, 3, 4, 5])
count = 0
def hop(x):
nonlocal count
count += 1
clist = []
if x.ndim == 1:
clist.append(x)
else:
for icol in range(x.shape[1]):
clist.append(x[:, icol])
res = []
for c in clist:
# convert c to initial structure according to qn pattern
cstruct = asxp(cvec2cmat(cshape, c, qnmat, mps.qntot))
if omega is None:
if method == "1site":
# S-a l-S
# d
# O-b-O-f-O
# e
# S-c k-S
path = [
([0, 1], "abc, adl -> bcdl"),
([2, 0], "bcdl, bdef -> clef"),
([1, 0], "clef, lfk -> cek"),
]
cout = multi_tensor_contract(
path, ltensor, cstruct, cmo[0], rtensor)
else:
# S-a l-S
# d g
# O-b-O-f-O-j-O
# e h
# S-c k-S
path = [
([0, 1], "abc, adgl -> bcdgl"),
([3, 0], "bcdgl, bdef -> cglef"),
([2, 0], "cglef, fghj -> clehj"),
([1, 0], "clehj, ljk -> cehk"),
]
cout = multi_tensor_contract(
path,
ltensor,
cstruct,
cmo[0],
cmo[1],
rtensor,
)
else:
cout = expr(cstruct, backend=oe_backend)
# convert structure c to 1d according to qn
res.append(asnumpy(cout)[qnmat == mps.qntot])
if len(res) == 1:
return inverse * res[0]
else:
return inverse * np.stack(res, axis=1)
if len(cguess) < nroots:
cguess.extend([
np.random.random([nonzeros]) - 0.5
for i in range(len(cguess), nroots)
])
if algo == "davidson":
precond = lambda x, e, *args: x / (hdiag - e + 1e-4)
e, c = davidson(hop,
cguess,
precond,
max_cycle=100,
nroots=nroots,
max_memory=64000)
# if one root, davidson return e as np.float
#elif algo == "arpack":
# # scipy arpack solver : much slower than pyscf/davidson
# A = scipy.sparse.linalg.LinearOperator((nonzeros,nonzeros), matvec=hop)
# e, c = scipy.sparse.linalg.eigsh(A, k=nroots, which="SA", v0=cguess)
# # scipy return numpy.array
# if nroots == 1:
# e = e[0]
#elif algo == "lobpcg":
# precond = lambda x: scipy.sparse.diags(1/(hdiag+1e-4)) @ x
# A = scipy.sparse.linalg.LinearOperator((nonzeros,nonzeros),
# matvec=hop, matmat=hop)
# M = scipy.sparse.linalg.LinearOperator((nonzeros,nonzeros),
# matvec=precond, matmat=hop)
# e, c = scipy.sparse.linalg.lobpcg(A, np.array(cguess).T,
# M=M, largest=False)
elif algo == "primme":
precond = lambda x: scipy.sparse.diags(1 /
(hdiag + 1e-4)) @ x
A = scipy.sparse.linalg.LinearOperator(
(nonzeros, nonzeros), matvec=hop, matmat=hop)
M = scipy.sparse.linalg.LinearOperator(
(nonzeros, nonzeros), matvec=precond, matmat=hop)
e, c = primme.eigsh(A,
k=min(nroots, nonzeros),
which="SA",
v0=np.array(cguess).T,
OPinv=M,
method="PRIMME_DYNAMIC",
tol=1e-6)
else:
assert False
logger.debug(f"use {algo}, HC hops: {count}")
else:
logger.debug(f"use direct eigensolver")
# direct algorithm
if omega is None:
if method == "1site":
# S-a l-S
# d
# O-b-O-f-O
# e
# S-c k-S
ham = oe.contract("abc,bdef,lfk->adlcek",
ltensor,
cmo[0],
rtensor,
backend=oe_backend)
ham = ham[:, :, :, qnmat == mps.qntot][
qnmat == mps.qntot, :] * inverse
else:
# S-a l-S
# d g
# O-b-O-f-O-j-O
# e h
# S-c k-S
ham = oe.contract("abc,bdef,fghj,ljk->adglcehk",
ltensor, cmo[0], cmo[1], rtensor)
ham = ham[:, :, :, :, qnmat == mps.qntot][
qnmat == mps.qntot, :] * inverse
else:
if method == "1site":
# S-a e j-S
# O-b-O-g-O
# | f |
# O-c-O-i-O
# S-d h k-S
ham = oe.contract("abcd, befg, cfhi, jgik -> aejdhk",
ltensor, cmo[0], cmo[0], rtensor)
ham = ham[:, :, :, qnmat == mps.qntot][
qnmat == mps.qntot, :] * inverse
else:
# S-a e j o-S
# O-b-O-g-O-l-O
# | f k |
# O-c-O-i-O-n-O
# S-d h m p-S
ham = oe.contract(
"abcd, befg, cfhi, gjkl, ikmn, olnp -> aejodhmp",
ltensor, cmo[0], cmo[0], cmo[1], cmo[1], rtensor)
ham = ham[:, :, :, :, qnmat == mps.qntot][
qnmat == mps.qntot, :] * inverse
w, v = scipy.linalg.eigh(asnumpy(ham))
if nroots == 1:
e = w[0]
c = v[:, 0]
else:
e = w[:nroots]
c = [
v[:, iroot] for iroot in range(min(nroots, v.shape[1]))
]
# if multi roots, both davidson and primme return np.ndarray
if nroots > 1:
e = e.tolist()
logger.debug(f"energy: {e}")
micro_iteration_result.append(e)
cstruct = cvec2cmat(cshape, c, qnmat, mps.qntot, nroots=nroots)
# store the "optimal" mps (usually in the middle of each sweep)
if res_mps_idx is not None and res_mps_idx == imps:
if nroots == 1:
res_mps = mps.copy()
res_mps._update_mps(cstruct, cidx, qnbigl, qnbigr, mmax,
percent)
else:
res_mps = [mps.copy() for i in range(len(cstruct))]
for iroot in range(len(cstruct)):
res_mps[iroot]._update_mps(cstruct[iroot], cidx,
qnbigl, qnbigr, mmax,
percent)
averaged_ms = mps._update_mps(cstruct, cidx, qnbigl, qnbigr, mmax,
percent)
mps._switch_direction()
res_mps_idx = micro_iteration_result.index(min(micro_iteration_result))
macro_iteration_result.append(micro_iteration_result[res_mps_idx])
# check if convergence
if isweep > 0 and percent == 0:
v1, v2 = sorted(macro_iteration_result)[:2]
if np.allclose(v1,
v2,
rtol=mps.optimize_config.e_rtol,
atol=mps.optimize_config.e_atol):
converged = True
break
logger.debug(
f"{isweep+1} sweeps are finished, lowest energy = {sorted(macro_iteration_result)[0]}"
)
if converged:
logger.info("DMRG is converged!")
else:
logger.warning("DMRG is not converged! Please increase the procedure!")
logger.info(
f"The lowest two energies: {sorted(macro_iteration_result)[:2]}.")
# remove the redundant basis near the edge
if nroots == 1:
res_mps = res_mps.normalize().ensure_left_canon().canonicalise()
logger.info(f"{res_mps}")
else:
res_mps = [
mp.normalize().ensure_left_canon().canonicalise() for mp in res_mps
]
logger.info(f"{res_mps[0]}")
return macro_iteration_result, res_mps
def kernel(self, restart=False, include_psi0=False):
r"""calculate the roots
Parameters
----------
restart: bool, optional
if restart from the former converged root. Default is ``False``.
If ``restart = True``, ``include_psi0`` must be the same as the
former calculation.
include_psi0: bool, optional
if the basis of Hamiltonian includes the ground state
:math:`\Psi_0`. Default is ``False``.
Returns
-------
e: np.ndarray
the energy of the states, if ``include_psi0 = True``, the first
element is the ground state energy, otherwise, it is the energy of
the first excited state.
"""
# right canonical mps
mpo = self.hmpo
nroots = self.nroots
algo = self.algo
site_num = mpo.site_num
if not restart:
# make sure that M is not redundant near the edge
mps = self.mps.ensure_right_canon().canonicalise().normalize().canonicalise()
logger.debug(f"reference mps shape, {mps}")
mps_r_cano = mps.copy()
assert mps.to_right
tangent_u = []
for ims, ms in enumerate(mps):
shape = list(ms.shape)
u, s, vt = scipy.linalg.svd(ms.l_combine(), full_matrices=True)
rank = len(s)
if include_psi0 and ims == site_num-1:
tangent_u.append(u.reshape(shape[:-1]+[-1]))
else:
if rank < u.shape[1]:
tangent_u.append(u[:,rank:].reshape(shape[:-1]+[-1]))
else:
tangent_u.append(None) # the tangent space is None
mps[ims] = u[:,:rank].reshape(shape[:-1]+[-1])
vt = xp.einsum("i, ij -> ij", asxp(s), asxp(vt))
if ims == site_num-1:
assert vt.size == 1 and xp.allclose(vt, 1)
else:
mps[ims+1] = asnumpy(tensordot(vt, mps[ims+1], ([-1],[0])))
mps_l_cano = mps.copy()
mps_l_cano.to_right = False
mps_l_cano.qnidx = site_num-1
else:
mps_l_cano, mps_r_cano, tangent_u, tda_coeff_list = self.wfn
cguess = []
for iroot in range(len(tda_coeff_list)):
tda_coeff = tda_coeff_list[iroot]
x = [c.flatten() for c in tda_coeff if c is not None]
x = np.concatenate(x,axis=None)
cguess.append(x)
cguess = np.stack(cguess, axis=1)
xshape = []
xsize = 0
for ims in range(site_num):
if tangent_u[ims] is None:
xshape.append((0,0))
else:
if ims == site_num-1:
xshape.append((tangent_u[ims].shape[-1], 1))
else:
xshape.append((tangent_u[ims].shape[-1], mps_r_cano[ims+1].shape[0]))
xsize += np.prod(xshape[-1])
logger.debug(f"DMRG-TDA H dimension: {xsize}")
if USE_GPU:
oe_backend = "cupy"
else:
oe_backend = "numpy"
mps_tangent = mps_r_cano.copy()
environ = Environ(mps_tangent, mpo, "R")
hdiag = []
for ims in range(site_num):
ltensor = environ.GetLR(
"L", ims-1, mps_tangent, mpo, itensor=None,
method="System"
)
rtensor = environ.GetLR(
"R", ims+1, mps_tangent, mpo, itensor=None,
method="Enviro"
)
if tangent_u[ims] is not None:
u = asxp(tangent_u[ims])
tmp = oe.contract("abc, ded, bghe, agl, chl -> ld", ltensor, rtensor,
asxp(mpo[ims]), u, u, backend=oe_backend)
hdiag.append(asnumpy(tmp))
mps_tangent[ims] = mps_l_cano[ims]
hdiag = np.concatenate(hdiag, axis=None)
count = 0
# recover the vector-like x back to the ndarray tda_coeff
def reshape_x(x):
tda_coeff = []
offset = 0
for shape in xshape:
if shape == (0,0):
tda_coeff.append(None)
else:
size = np.prod(shape)
tda_coeff.append(x[offset:size+offset].reshape(shape))
offset += size
assert offset == xsize
return tda_coeff
def hop(x):
# H*X
nonlocal count
count += 1
assert len(x) == xsize
tda_coeff = reshape_x(x)
res = [np.zeros_like(coeff) if coeff is not None else None for coeff in tda_coeff]
# fix ket and sweep bra and accumulate into res
for ims in range(site_num):
if tda_coeff[ims] is None:
assert tangent_u[ims] is None
continue
# mix-canonical mps
mps_tangent = merge(mps_l_cano, mps_r_cano, ims+1)
mps_tangent[ims] = tensordot(tangent_u[ims], tda_coeff[ims], (-1, 0))
mps_tangent_conj = mps_r_cano.copy()
environ = Environ(mps_tangent, mpo, "R", mps_conj=mps_tangent_conj)
for ims_conj in range(site_num):
ltensor = environ.GetLR(
"L", ims_conj-1, mps_tangent, mpo, itensor=None,
mps_conj=mps_tangent_conj,
method="System"
)
rtensor = environ.GetLR(
"R", ims_conj+1, mps_tangent, mpo, itensor=None,
mps_conj=mps_tangent_conj,
method="Enviro"
)
if tda_coeff[ims_conj] is not None:
# S-a l-S
# d
# O-b-O-f-O
# e
# S-c k-S
path = [
([0, 1], "abc, cek -> abek"),
([2, 0], "abek, bdef -> akdf"),
([1, 0], "akdf, lfk -> adl"),
]
out = multi_tensor_contract(
path, ltensor, asxp(mps_tangent[ims_conj]),
asxp(mpo[ims_conj]), rtensor
)
res[ims_conj] += asnumpy(tensordot(tangent_u[ims_conj], out,
([0,1], [0,1])))
# mps_conj combine
mps_tangent_conj[ims_conj] = mps_l_cano[ims_conj]
res = [mat for mat in res if mat is not None]
return np.concatenate(res, axis=None)
if algo == "davidson":
if restart:
cguess = [cguess[:,i] for i in range(cguess.shape[1])]
else:
cguess = [np.random.random(xsize) - 0.5]
precond = lambda x, e, *args: x / (hdiag - e + 1e-4)
e, c = davidson(
hop, cguess, precond, max_cycle=100,
nroots=nroots, max_memory=64000
)
if nroots == 1:
c = [c]
c = np.stack(c, axis=1)
elif algo == "primme":
if not restart:
cguess = None
def multi_hop(x):
if x.ndim == 1:
return hop(x)
elif x.ndim == 2:
return np.stack([hop(x[:,i]) for i in range(x.shape[1])],axis=1)
else:
assert False
def precond(x):
if x.ndim == 1:
return np.einsum("i, i -> i", 1/(hdiag+1e-4), x)
elif x.ndim ==2:
return np.einsum("i, ij -> ij", 1/(hdiag+1e-4), x)
else:
assert False
A = scipy.sparse.linalg.LinearOperator((xsize,xsize),
matvec=multi_hop, matmat=multi_hop)
M = scipy.sparse.linalg.LinearOperator((xsize,xsize),
matvec=precond, matmat=precond)
e, c = primme.eigsh(A, k=min(nroots,xsize), which="SA",
v0=cguess,
OPinv=M,
method="PRIMME_DYNAMIC",
tol=1e-6)
else:
assert False
logger.debug(f"H*C times: {count}")
tda_coeff_list = []
for iroot in range(nroots):
tda_coeff_list.append(reshape_x(c[:,iroot]))
self.e = np.array(e)
self.wfn = [mps_l_cano, mps_r_cano, tangent_u, tda_coeff_list]
return self.e
def _update_mps(self, cstruct, cidx, qnbigl, qnbigr, Mmax, percent=0):
r"""update mps with basis selection algorithm of J. Chem. Phys. 120,
3172 (2004).
Parameters
---------
cstruct : ndarray, List[ndarray]
The active site coefficient.
cidx : list
The List of active site index.
qnbigl : ndarray
The super-L-block quantum number.
qnbigr : ndarray
The super-R-block quantum number.
Mmax : int
The maximal bond dimension.
percent : float, int
The percentage of renormalized basis which is equally selected from
each quantum number section rather than according to singular
values. ``percent`` is defined in ``procedure`` of
`renormalizer.utils.configs.OptimizeConfig` and ``vprocedure`` of
`renormalizer.utils.configs.CompressConfig`.
Returns
-------
averaged_ms :
if ``cstruct`` is a list, ``averaged_ms`` is a list of rotated ms of
each element in ``cstruct`` as a single site calculation. It is
used for better initial guess in SA-DMRG algorithm. Otherwise,
``None`` is returned.
``self`` is overwritten inplace.
"""
system = "L" if self.to_right else "R"
# step 1: get the selected U, S, V
if type(cstruct) is not list:
# SVD method
# full_matrices = True here to enable increase the bond dimension
Uset, SUset, qnlnew, Vset, SVset, qnrnew = svd_qn.Csvd(
asnumpy(cstruct), qnbigl, qnbigr, self.qntot, system=system)
if self.to_right:
ms, msdim, msqn, compms = select_basis(Uset,
SUset,
qnlnew,
Vset,
Mmax,
percent=percent)
ms = ms.reshape(list(qnbigl.shape) + [msdim])
compms = xp.moveaxis(
compms.reshape(list(qnbigr.shape) + [msdim]), -1, 0)
else:
ms, msdim, msqn, compms = select_basis(Vset,
SVset,
qnrnew,
Uset,
Mmax,
percent=percent)
ms = xp.moveaxis(ms.reshape(list(qnbigr.shape) + [msdim]), -1,
0)
compms = compms.reshape(list(qnbigl.shape) + [msdim])
else:
# state-averaged method
ddm = 0.0
for iroot in range(len(cstruct)):
if self.to_right:
ddm += tensordot(
cstruct[iroot],
cstruct[iroot],
axes=(
range(qnbigl.ndim, cstruct[iroot].ndim),
range(qnbigl.ndim, cstruct[iroot].ndim),
),
)
else:
ddm += tensordot(
cstruct[iroot],
cstruct[iroot],
axes=(range(qnbigl.ndim), range(qnbigl.ndim)),
)
ddm /= len(cstruct)
Uset, Sset, qnnew = svd_qn.Csvd(asnumpy(ddm),
qnbigl,
qnbigr,
self.qntot,
system=system,
ddm=True)
ms, msdim, msqn, compms = select_basis(Uset,
Sset,
qnnew,
None,
Mmax,
percent=percent)
rotated_c = []
averaged_ms = []
if self.to_right:
ms = ms.reshape(list(qnbigl.shape) + [msdim])
for c in cstruct:
compms = tensordot(
ms,
c,
axes=(range(qnbigl.ndim), range(qnbigl.ndim)),
)
rotated_c.append(compms)
compms = rotated_c[0]
else:
ms = ms.reshape(list(qnbigr.shape) + [msdim])
for c in cstruct:
compms = tensordot(
c,
ms,
axes=(range(qnbigl.ndim,
cstruct[0].ndim), range(qnbigr.ndim)),
)
rotated_c.append(compms)
compms = rotated_c[0]
ms = xp.moveaxis(ms, -1, 0)
# step 2, put updated U, S, V back to self
if len(cidx) == 1:
# 1site method
self[cidx[0]] = ms
if self.to_right:
if cidx[0] != self.site_num - 1:
if type(cstruct) is list:
for c in rotated_c:
averaged_ms.append(
tensordot(c, self[cidx[0] + 1], axes=1))
self[cidx[0] + 1] = tensordot(compms,
self[cidx[0] + 1],
axes=1)
self.qn[cidx[0] + 1] = msqn
self.qnidx = cidx[0] + 1
else:
if type(cstruct) is list:
for c in rotated_c:
averaged_ms.append(
tensordot(self[cidx[0]], c, axes=1))
self[cidx[0]] = tensordot(self[cidx[0]], compms, axes=1)
self.qnidx = self.site_num - 1
else:
if cidx[0] != 0:
if type(cstruct) is list:
for c in rotated_c:
averaged_ms.append(
tensordot(self[cidx[0] - 1], c, axes=1))
self[cidx[0] - 1] = tensordot(self[cidx[0] - 1],
compms,
axes=1)
self.qn[cidx[0]] = msqn
self.qnidx = cidx[0] - 1
else:
if type(cstruct) is list:
for c in rotated_c:
averaged_ms.append(
tensordot(c, self[cidx[0]], axes=1))
self[cidx[0]] = tensordot(compms, self[cidx[0]], axes=1)
self.qnidx = 0
else:
if self.to_right:
self[cidx[0]] = ms
self[cidx[1]] = compms
self.qnidx = cidx[1]
else:
self[cidx[1]] = ms
self[cidx[0]] = compms
self.qnidx = cidx[0]
if type(cstruct) is list:
averaged_ms = rotated_c
self.qn[cidx[1]] = msqn
if type(cstruct) is list:
return averaged_ms
else:
return None
def _evolve_dmrg_tdvp_ps(self, mpo, evolve_dt) -> "Mps":
# PhysRevB.94.165116
# TDVP projector splitting
imag_time = np.iscomplex(evolve_dt)
if imag_time:
mps = self.copy()
mps_conj = mps
else:
mps = self.to_complex()
mps_conj = mps.conj() # another copy, so 3x memory is used.
# construct the environment matrix
environ = Environ()
# almost half is not used. Not a big deal.
environ.construct(mps, mps_conj, mpo, "L")
environ.construct(mps, mps_conj, mpo, "R")
# a workaround for https://github.com/scipy/scipy/issues/10164
if imag_time:
evolve_dt = -evolve_dt.imag
# used in calculating derivatives
coef = -1
else:
coef = 1j
# statistics for debug output
cmf_rk_steps = []
USE_RK = self.evolve_config.tdvp_ps_rk4
# sweep for 2 rounds
for i in range(2):
for imps in mps.iter_idx_list(full=True):
system = "L" if mps.left else "R"
ltensor = environ.read("L", imps - 1)
rtensor = environ.read("R", imps + 1)
shape = list(mps[imps].shape)
l_array = ltensor.array
r_array = rtensor.array
hop = hop_factory(l_array, r_array, mpo[imps].array, len(shape))
def hop_svt(ms):
# S-a l-S
#
# O-b - b-O
#
# S-c k-S
path = [([0, 1], "abc, ck -> abk"), ([1, 0], "abk, lbk -> al")]
HC = multi_tensor_contract(path, l_array, ms, r_array)
return HC
if USE_RK:
def func(t, y):
return hop(y.reshape(shape)).ravel() / coef
sol = solve_ivp(
func, (0, evolve_dt / 2.0), mps[imps].ravel().array, method="RK45"
)
cmf_rk_steps.append(len(sol.t))
mps_t = sol.y[:, -1]
else:
# Can't use the same func because here H should be Hermitian
def func(y):
return hop(y.reshape(shape)).ravel()
mps_t = expm_krylov(func, (evolve_dt / 2) / coef, mps[imps].ravel().array)
mps_t = mps_t.reshape(shape)
qnbigl, qnbigr = mps._get_big_qn(imps)
u, qnlset, v, qnrset = svd_qn.Csvd(
asnumpy(mps_t),
qnbigl,
qnbigr,
mps.qntot,
QR=True,
system=system,
full_matrices=False,
)
vt = v.T
if mps.is_left_canon and imps != 0:
mps[imps] = vt.reshape([-1] + shape[1:])
mps_conj[imps] = mps[imps].conj()
mps.qn[imps] = qnrset
rtensor = environ.GetLR(
"R", imps, mps, mps_conj, mpo, itensor=rtensor, method="System"
)
r_array = rtensor.array
# reverse update u site
shape_u = u.shape
if USE_RK:
def func_u(t, y):
return hop_svt(y.reshape(shape_u)).ravel() / coef
sol_u = solve_ivp(
func_u, (0, -evolve_dt / 2), u.ravel(), method="RK45"
)
cmf_rk_steps.append(len(sol_u.t))
mps_t = sol_u.y[:, -1]
else:
def func_u(y):
return hop_svt(y.reshape(shape_u)).ravel()
mps_t = expm_krylov(func_u, (-evolve_dt / 2) / coef, u.ravel())
mps_t = mps_t.reshape(shape_u)
mps[imps - 1] = tensordot(
mps[imps - 1].array,
mps_t,
axes=(-1, 0),
)
mps_conj[imps - 1] = mps[imps - 1].conj()
elif mps.is_right_canon and imps != len(mps) - 1:
mps[imps] = u.reshape(shape[:-1] + [-1])
mps_conj[imps] = mps[imps].conj()
mps.qn[imps + 1] = qnlset
ltensor = environ.GetLR(
"L", imps, mps, mps_conj, mpo, itensor=ltensor, method="System"
)
l_array = ltensor.array
# reverse update svt site
shape_svt = vt.shape
if USE_RK:
def func_svt(t, y):
return hop_svt(y.reshape(shape_svt)).ravel() / coef
sol_svt = solve_ivp(
func_svt, (0, -evolve_dt / 2), vt.ravel(), method="RK45"
)
cmf_rk_steps.append(len(sol_svt.t))
mps_t = sol_svt.y[:, -1]
else:
def func_svt(y):
return hop_svt(y.reshape(shape_svt)).ravel()
mps_t = expm_krylov(func_svt, (-evolve_dt / 2) / coef, vt.ravel())
mps_t = mps_t.reshape(shape_svt)
mps[imps + 1] = tensordot(
mps_t,
mps[imps + 1].array,
axes=(1, 0),
)
mps_conj[imps + 1] = mps[imps + 1].conj()
else:
mps[imps] = mps_t
mps_conj[imps] = mps[imps].conj()
mps._switch_direction()
if USE_RK:
steps_stat = stats.describe(cmf_rk_steps)
logger.debug(f"TDVP-PS CMF steps: {steps_stat}")
mps.evolve_config.stat = steps_stat
return mps
def _evolve_dmrg_tdvp_mctdhnew(self, mpo, evolve_dt) -> "Mps":
# new regularization scheme
# JCP 148, 124105 (2018)
# JCP 149, 044119 (2018)
# a workaround for https://github.com/scipy/scipy/issues/10164
imag_time = np.iscomplex(evolve_dt)
if imag_time:
evolve_dt = -evolve_dt.imag
# used in calculating derivatives
coef = -1
else:
coef = 1j
if self.is_left_canon:
assert self.check_left_canonical()
self.canonicalise()
mps = self.to_complex(inplace=True)
# construct the environment matrix
environ = Environ()
environ.construct(mps, mps.conj(), mpo, "R")
# initial matrix
ltensor = ones((1, 1, 1))
rtensor = ones((1, 1, 1))
new_mps = mps.metacopy()
# statistics for debug output
cmf_rk_steps = []
for imps in range(len(mps)):
shape = list(mps[imps].shape)
system = "L" if mps.left else "R"
qnbigl, qnbigr = mps._get_big_qn(imps)
u, s, qnlset, v, s, qnrset = svd_qn.Csvd(
mps[imps].asnumpy(),
qnbigl,
qnbigr,
mps.qntot,
system=system,
full_matrices=False,
)
vt = v.T
mps[imps] = u.reshape(shape[:-1] + [-1])
ltensor = environ.GetLR(
"L", imps - 1, mps, mps.conj(), mpo, itensor=ltensor, method="System"
)
rtensor = environ.GetLR(
"R", imps + 1, mps, mps.conj(), mpo, itensor=rtensor, method="Enviro"
)
epsilon = 1e-10
epsilon = np.sqrt(epsilon)
s = s + epsilon * np.exp(-s / epsilon)
svt = Matrix(np.diag(s).dot(vt))
rtensor = tensordot(rtensor, svt, axes=(2, 1))
rtensor = tensordot(Matrix(vt).conj(), rtensor, axes=(1, 0))
if imps != len(mps) - 1:
mps[imps + 1] = tensordot(svt, mps[imps + 1], axes=(-1, 0))
mps.qn[imps + 1] = qnlset
new_mps.qn[imps + 1] = qnlset.copy()
S_inv = xp.diag(1.0 / s)
hop = hop_factory(ltensor, rtensor, mpo[imps], len(shape))
func = integrand_func_factory(shape, hop, imps == len(mps) - 1, S_inv, coef)
sol = solve_ivp(
func, (0, evolve_dt), mps[imps].ravel().array, method="RK45"
)
cmf_rk_steps.append(len(sol.t))
ms = sol.y[:, -1].reshape(shape)
if imps == len(mps) - 1:
new_mps[imps] = ms * s[0]
else:
new_mps[imps] = ms
mps._switch_direction()
new_mps._switch_direction()
new_mps.canonicalise()
steps_stat = stats.describe(cmf_rk_steps)
logger.debug(f"TDVP-MCTDH CMF steps: {steps_stat}")
# new_mps.evolve_config.stat = steps_stat
return new_mps
def analysis_dominant_config(self, thresh=0.8, alias=None, tda_m_trunc=20,
return_compressed_mps=False):
r""" analyze the dominant configuration of each tda root.
The algorithm is to compress the tda wavefunction to a rank-1 Hartree
state and get the ci coefficient of the largest configuration.
Then, the configuration is subtracted from the tda wavefunction and
redo the first step to get the second largest configuration. The
two steps continue until the thresh is achieved.
Parameters
----------
thresh: float, optional
the threshold to stop the analysis procedure of each root.
:math:`\sum_i |c_i|^2 > thresh`. Default is 0.8.
alias: dict, optional
The alias of each site. For example, ``alias={0:"v_0", 1:"v_2",
2:"v_1"}``. Default is `None`.
tda_m_trunc: int, optional
the ``m`` to compress a tda wavefunction. Default is 20.
return_compressed_mps: bool, optional
If ``True``, return the tda excited state as a single compressed
mps. Default is `False`.
Returns
-------
configs: dict
The dominant configration of each root.
``configs = {0:[(config0, config_name0, ci_coeff0),(config1,
config_name1, ci_coeff1),...], 1:...}``
compressed_mps: List[renormalizer.mps.Mps]
see the description in ``return_compressed_mps``.
Note
----
The compressed_mps is an approximation of the tda wavefunction with
``m=tda_m_trunc``.
"""
mps_l_cano, mps_r_cano, tangent_u, tda_coeff_list = self.wfn
if alias is not None:
assert len(alias) == mps_l_cano.site_num
compressed_mps = []
for iroot in range(self.nroots):
logger.info(f"iroot: {iroot}")
tda_coeff = tda_coeff_list[iroot]
mps_tangent_list = []
weight = []
for ims in range(mps_l_cano.site_num):
if tangent_u[ims] is None:
assert tda_coeff[ims] is None
continue
weight.append(np.sum(tda_coeff[ims]**2))
mps_tangent = merge(mps_l_cano, mps_r_cano, ims+1)
mps_tangent[ims] = asnumpy(tensordot(tangent_u[ims],
tda_coeff[ims],[-1,0]))
mps_tangent_list.append(mps_tangent)
assert np.allclose(np.sum(weight), 1)
# sort the mps_tangent from large weight to small weight
mps_tangent_list = [mps_tangent_list[i] for i in np.argsort(weight,axis=None)[::-1]]
coeff_square_sum = 0
mps_delete = None
config_visited = []
while coeff_square_sum < thresh:
if mps_delete is None:
# first compress it to M=tda_m_trunc
mps_rank1 = compressed_sum(mps_tangent_list, batchsize=5,
temp_m_trunc=tda_m_trunc)
else:
mps_rank1 = compressed_sum([mps_delete] + mps_tangent_list,
batchsize=5, temp_m_trunc=tda_m_trunc)
if coeff_square_sum == 0 and return_compressed_mps:
compressed_mps.append(mps_rank1.copy())
mps_rank1 = mps_rank1.canonicalise().compress(temp_m_trunc=1)
# get config with the largest coeff
config = []
for ims, ms in enumerate(mps_rank1):
ms = ms.array.flatten()**2
quanta = int(np.argmax(ms))
config.append(quanta)
# check if the config has been visited
if config in config_visited:
break
config_visited.append(config)
ci_coeff_list = []
for mps_tangent in mps_tangent_list:
sentinel = xp.ones((1,1))
for ims, ms in enumerate(mps_tangent):
sentinel = sentinel.dot(asxp(ms[:,config[ims],:]))
ci_coeff_list.append(float(sentinel[0,0]))
ci_coeff = np.sum(ci_coeff_list)
coeff_square_sum += ci_coeff**2
if alias is not None:
config_name = [f"{quanta}"+f"{alias[isite]}" for isite, quanta
in enumerate(config) if quanta != 0]
config_name = " ".join(config_name)
self.configs[iroot].append((config, config_name, ci_coeff))
logger.info(f"config: {config}, {config_name}")
else:
self.configs[iroot].append((config, ci_coeff))
logger.info(f"config: {config}")
logger.info(f"ci_coeff: {ci_coeff}, weight:{ci_coeff**2}")
condition = {dof:config[idof] for idof, dof in
enumerate(self.model.dofs)}
mps_delete_increment = Mps.hartree_product_state(self.model, condition).scale(-ci_coeff)
if mps_delete is None:
mps_delete = mps_delete_increment
else:
mps_delete = mps_delete + mps_delete_increment
logger.info(f"coeff_square_sum: {coeff_square_sum}")
return self.configs, compressed_mps
def contract_one_site_multi_mpo(environ, ms, mos, domain, ms_conj=None):
"""
contract one mpos/mps(mpdm) site, mos is a list containing several local mo
_ _
| | | |
S-S- | S-|-S-
O-O- or | O-|-O- (the ancillary bond is traced)
O-O- | O-|-O-
S-S- | S-|-S-
|_| |_|
"""
assert domain in ["L", "R"]
if ms_conj is None:
ms_conj = ms.conj()
if domain == "L":
if ms.ndim == 3:
outtensor = tensordot(environ, ms_conj, ([0], [0]))
for mo in mos:
outtensor = tensordot(outtensor, mo, ([0, -2], [0, 1]))
outtensor = tensordot(outtensor, ms, ([0, -2], [0, 1]))
elif ms.ndim == 4:
outtensor = tensordot(environ, ms_conj.transpose(0, 2, 1, 3),
([0], [0]))
for mo in mos:
outtensor = tensordot(outtensor, mo, ([0, -2], [0, 1]))
outtensor = tensordot(outtensor, ms, ([0, 1, -2], [0, 2, 1]))
else:
raise ValueError(f"MPS ndim is not 3 or 4, got {ms.ndim}")
else:
if ms.ndim == 3:
outtensor = tensordot(environ, ms_conj, ([0], [-1]))
for mo in mos:
outtensor = tensordot(outtensor, mo, ([0, -1], [-1, 1]))
outtensor = tensordot(outtensor, ms, ([0, -1], [-1, 1]))
elif ms.ndim == 4:
outtensor = tensordot(environ, ms_conj.transpose(0, 2, 1, 3),
([0], [-1]))
for mo in mos:
outtensor = tensordot(outtensor, mo, ([0, -1], [-1, 1]))
outtensor = tensordot(outtensor, ms, ([0, 2, -1], [-1, 2, 1]))
else:
raise ValueError(f"MPS ndim is not 3 or 4, got {ms.ndim}")
return outtensor
def optimize_cv(self, lr_group, isite, percent=0):
if self.spectratype == "abs":
# quantum number restriction, |1><0|
up_exciton, down_exciton = 1, 0
elif self.spectratype == "emi":
# quantum number restriction, |0><1|
up_exciton, down_exciton = 0, 1
nexciton = 1
first_LR, second_LR, third_LR, forth_LR = lr_group
if self.method == "1site":
add_list = [isite - 1]
first_L = asxp(first_LR[isite - 1])
first_R = asxp(first_LR[isite])
second_L = asxp(second_LR[isite - 1])
second_R = asxp(second_LR[isite])
third_L = asxp(third_LR[isite - 1])
third_R = asxp(third_LR[isite])
forth_L = asxp(forth_LR[isite - 1])
forth_R = asxp(forth_LR[isite])
else:
add_list = [isite - 2, isite - 1]
first_L = asxp(first_LR[isite - 2])
first_R = asxp(first_LR[isite])
second_L = asxp(second_LR[isite - 2])
second_R = asxp(second_LR[isite])
third_L = asxp(third_LR[isite - 2])
third_R = asxp(third_LR[isite])
forth_L = asxp(forth_LR[isite - 2])
forth_R = asxp(forth_LR[isite])
xqnmat, xqnbigl, xqnbigr, xshape = \
self.construct_X_qnmat(add_list)
dag_qnmat, dag_qnbigl, dag_qnbigr = self.swap(xqnmat, xqnbigl, xqnbigr)
nonzeros = int(
np.sum(self.condition(dag_qnmat, [down_exciton, up_exciton])))
if self.method == "1site":
guess = moveaxis(self.cv_mpo[isite - 1], (1, 2), (2, 1))
else:
guess = tensordot(moveaxis(self.cv_mpo[isite - 2], (1, 2), (2, 1)),
moveaxis(self.cv_mpo[isite - 1]),
axes=(-1, 0))
guess = guess[self.condition(dag_qnmat, [down_exciton, up_exciton])]
if self.method == "1site":
# define dot path
path_1 = [([0, 1], "abcd, aefg -> bcdefg"),
([3, 0], "bcdefg, bfhi -> cdeghi"),
([2, 0], "cdeghi, chjk -> degijk"),
([1, 0], "degijk, gikl -> dejl")]
path_2 = [([0, 1], "abcd, aefg -> bcdefg"),
([3, 0], "bcdefg, bfhi -> cdeghi"),
([2, 0], "cdeghi, djek -> cghijk"),
([1, 0], "cghijk, gilk -> chjl")]
path_3 = [([0, 1], "ab, acde -> bcde"), ([1,
0], "bcde, ef -> bcdf")]
vecb = multi_tensor_contract(
path_3, forth_L, moveaxis(self.b_mpo[isite - 1], (1, 2),
(2, 1)),
forth_R)[self.condition(dag_qnmat, [down_exciton, up_exciton])]
a_oper_isite = asxp(self.a_oper[isite - 1])
h_mpo_isite = asxp(self.h_mpo[isite - 1])
# construct preconditioner
Idt = xp.identity(h_mpo_isite.shape[1])
M1_1 = xp.einsum('abca->abc', first_L)
path_m1 = [([0, 1], "abc, bdef->acdef"), ([1,
0], "acdef, cegh->adfgh")]
M1_2 = multi_tensor_contract(path_m1, M1_1, a_oper_isite, a_oper_isite)
M1_2 = xp.einsum("abcbd->abcd", M1_2)
M1_3 = xp.einsum('ecde->ecd', first_R)
M1_4 = xp.einsum('ff->f', Idt)
path_m1 = [([0, 1], "abcd,ecd->abe"), ([1, 0], "abe,f->abef")]
pre_M1 = multi_tensor_contract(path_m1, M1_2, M1_3, M1_4)
pre_M1 = xp.moveaxis(pre_M1, [-2, -1], [-1, -2])[self.condition(
dag_qnmat, [down_exciton, up_exciton])]
M2_1 = xp.einsum('aeag->aeg', second_L)
M2_2 = xp.einsum('eccf->ecf', a_oper_isite)
M2_3 = xp.einsum('gbbh->gbh', h_mpo_isite)
M2_4 = xp.einsum('dfdh->dfh', second_R)
path_m2 = [([0, 1], "aeg,gbh->aebh"), ([2, 0], "aebh,ecf->abchf"),
([1, 0], "abhcf,dfh->abcd")]
pre_M2 = multi_tensor_contract(path_m2, M2_1, M2_3, M2_2, M2_4)
pre_M2 = pre_M2[self.condition(dag_qnmat, [down_exciton, up_exciton])]
M4_1 = xp.einsum('faah->fah', third_L)
M4_4 = xp.einsum('gddi->gdi', third_R)
M4_5 = xp.einsum('cc->c', Idt)
M4_path = [([0, 1], "fah,febg->ahebg"), ([2, 0], "ahebg,hjei->abgji"),
([1, 0], "abgji,gdi->abjd")]
pre_M4 = multi_tensor_contract(M4_path, M4_1, h_mpo_isite, h_mpo_isite,
M4_4)
pre_M4 = xp.einsum('abbd->abd', pre_M4)
pre_M4 = xp.tensordot(pre_M4, M4_5, axes=0)
pre_M4 = xp.moveaxis(pre_M4, [2, 3], [3, 2])[self.condition(
dag_qnmat, [down_exciton, up_exciton])]
M_x = lambda x: asnumpy(
asxp(x) /
(pre_M1 + 2 * pre_M2 + pre_M4 + xp.ones(nonzeros) * self.eta**2))
pre_M = scipy.sparse.linalg.LinearOperator((nonzeros, nonzeros), M_x)
count = 0
def hop(x):
nonlocal count
count += 1
dag_struct = asxp(self.dag2mat(xshape, x, dag_qnmat))
if self.method == "1site":
M1 = multi_tensor_contract(path_1, first_L, dag_struct,
a_oper_isite, a_oper_isite, first_R)
M2 = multi_tensor_contract(path_2, second_L, dag_struct,
a_oper_isite, h_mpo_isite, second_R)
M2 = xp.moveaxis(M2, (1, 2), (2, 1))
M3 = multi_tensor_contract(path_2, third_L, h_mpo_isite,
dag_struct, h_mpo_isite, third_R)
M3 = xp.moveaxis(M3, (1, 2), (2, 1))
cout = M1 + 2 * M2 + M3 + dag_struct * self.eta**2
cout = cout[self.condition(dag_qnmat, [down_exciton, up_exciton])]
return asnumpy(cout)
# Matrix A
mat_a = scipy.sparse.linalg.LinearOperator((nonzeros, nonzeros),
matvec=hop)
x, info = scipy.sparse.linalg.cg(mat_a,
asnumpy(vecb),
tol=1.e-5,
x0=asnumpy(guess),
maxiter=500,
M=pre_M,
atol=0)
# logger.info(f"linear eq dim: {nonzeros}")
# logger.info(f'times for hop:{count}')
self.hop_time.append(count)
if info != 0:
logger.warning(
f"cg not converged, vecb.norm:{xp.linalg.norm(vecb)}")
l_value = xp.dot(asxp(hop(x)), asxp(x)) - 2 * xp.dot(vecb, asxp(x))
x = self.dag2mat(xshape, x, dag_qnmat)
if self.method == "1site":
x = np.moveaxis(x, [1, 2], [2, 1])
x, xdim, xqn, compx = self.x_svd(x,
xqnbigl,
xqnbigr,
nexciton,
percent=percent)
if self.method == "1site":
self.cv_mpo[isite - 1] = x
if not self.cv_mpo.to_right:
if isite != 1:
self.cv_mpo[isite - 2] = \
tensordot(self.cv_mpo[isite - 2], compx, axes=(-1, 0))
self.cv_mpo.qn[isite - 1] = xqn
self.cv_mpo.qnidx = isite - 2
else:
self.cv_mpo[isite - 1] = \
tensordot(compx, self.cv_mpo[isite - 1], axes=(-1, 0))
self.cv_mpo.qnidx = 0
else:
if isite != len(self.cv_mpo):
self.cv_mpo[isite] = \
tensordot(compx, self.cv_mpo[isite], axes=(-1, 0))
self.cv_mpo.qn[isite] = xqn
self.cv_mpo.qnidx = isite
else:
self.cv_mpo[isite - 1] = \
tensordot(self.cv_mpo[isite - 1], compx, axes=(-1, 0))
self.cv_mpo.qnidx = self.cv_mpo.site_num - 1
else:
if not self.cv_mpo.to_right:
self.cv_mpo[isite - 2] = compx
self.cv_mpo[isite - 1] = x
self.cv_mpo.qnidx = isite - 2
else:
self.cv_mpo[isite - 2] = x
self.cv_mpo[isite - 1] = compx
self.cv_mpo.qnidx = isite - 1
self.cv_mpo.qn[isite - 1] = xqn
return float(l_value)
