Low Level Data Structure and Algorithms

Matrix Product (MP)

class renormalizer.mps.mp.MatrixProduct

_array2mt(array, idx, allow_dump=True)

_get_big_qn(cidx: List[int], swap=False)

get the quantum number of L-block and R-block renormalized basis

Parameters:

cidx (list) – a list of center(active) site index. For 1site/2site algorithm, cidx has one/two elements.

Returns:

• qnbigl (np.ndarray) – super-L-block (L-block + active site if necessary) quantum number

• qnbigr (np.ndarray) – super-R-block (active site + R-block if necessary) quantum number

• qnmat (np.ndarray) – L-block + active site + R-block quantum number

_get_sigmaqn(idx)

_push_cano(idx)

_switch_direction()

_update_mps(cstruct, cidx, qnbigl, qnbigr, percent=0)

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.

• 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:

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.

Return type:

averaged_ms

_update_ms(idx: int, u: ndarray, vt: ndarray, sigma=None, qnlset=None, qnrset=None, m_trunc=None)

update mps directly after svd

add(other: MatrixProduct)

angle(other)

append(array)

property bond_dims: List

property bond_dims_exact: ndarray

property bond_dims_mean: int

property bond_list: List

build_empty_mp(num)

build_empty_qn()

build_none_qn()

canonicalise(stop_idx: int = None)

check_left_canonical(rtol: float = None, atol: float = None)

check L-canonical

check_right_canonical(rtol: float = None, atol: float = None)

check R-canonical

compress(temp_m_trunc=None, ret_s=False)

inp: canonicalise MPS (or MPO)

Parameters:

• temp_m_trunc (int or list of int) – Temporary truncation bond dimension. Overwrites the compression configuration in CompressConfig.

• ret_s (bool) – Whether return the singular values at the bonds. The singular values are padded to a rectangular array with zero values.

Return type:

truncated MPS

conj()

complex conjugate

copy()

distance(other) → float

dot(other: MatrixProduct) → complex

dot product of two mps / mpo

dot_ob(other: MatrixProduct) → complex

dot product of two mps / mpo with open boundary, but the boundary of mps/mpo is larger than 1, different from the normal mps/mpo

dump(fname, other_attrs=None)

ensure_left_canonical(rtol: float = None, atol: float = None)

ensure_right_canonical(rtol: float = None, atol: float = None)

classmethod from_mp(model, mplist)

property is_complex

property is_left_canonical

check the qn center in the L-canonical structure

property is_mpdm

property is_mpo

property is_mps

property is_right_canonical

check the qn center in the R-canonical structure

iter_idx_list(full: bool, stop_idx: int = None)

classmethod load(model: Model, fname: str)

metacopy() → MatrixProduct

move_qnidx(dstidx: int)

Quantum number has a boundary site, left hand of the site is L system qn, right hand of the side is R system qn, the sum of quantum number of L system and R system is tot.

property mp_norm: float

if self.is_left_canon:

assert self.check_left_canonical() return np.linalg.norm(np.ravel(self[-1]))

else:

assert self.check_right_canonical() return np.linalg.norm(np.ravel(self[0]))

property pbond_dims

property pbond_list

scale(val, inplace=False)

property site_num

property threshold

to_complex(inplace=False)

property total_bytes

variational_compress(mpo=None, guess=None)

Variational compress an mps/mpdm/mpo

Parameters:

• mpo (renormalizer.mps.Mpo, optional) – Default is None. if mpo is not None, the returned mps is an approximation of mpo @ self

• guess (renormalizer.mps.MatrixProduct, optional) – Initial guess of compressed mps/mpdm/mpo. Default is None.

Note

the variational compress related configurations is defined in self if guess=None, otherwise is defined in guess

Returns:

mp – a new compressed mps/mpdm/mpo, self is not overwritten. guess is overwritten.

Return type:

renormalizer.mps.MatrixProduct

property vbond_dims: List

property vbond_list: List

Matrix Product State (MPS)

class renormalizer.mps.Mps

add(other)

angle(other)

append(array)

property bond_dims: List

property bond_dims_exact: ndarray

property bond_dims_mean: int

property bond_list: List

build_empty_mp(num)

build_empty_qn()

build_none_qn()

calc_1site_rdm(idx=None) → Dict[int, ndarray]

Calculate 1-site reduced density matrix

\(\rho_i = \textrm{Tr}_{j \neq i} | \Psi \rangle \langle \Psi|\)

Parameters:

idx (int, list, tuple, optional) – site index of 1site_rdm. Default is None, which mean all the rdms are calculated.

Returns:

rdm – \(\{0:\rho_0, 1:\rho_1, \cdots\}\). The key is the index of the site.

Return type:

Dict

calc_2site_mutual_entropy() → ndarray

Calculate mutual entropy between two sites. Also known as mutual information

\(m_{ij} = (s_i + s_j - s_{ij})/2\)

See Chemical Physics 323 (2006) 519–531

Returns:

mutual_entropy – mutual entropy with shape (nsite, nsite)

Return type:

2d np.ndarry

calc_2site_rdm()

Calculate 2-site reduced density matrix

\(\rho_{ij} = \textrm{Tr}_{k \neq i, k \neq j} | \Psi \rangle \langle \Psi |\).

Returns:

rdm – \(\{(0,1):\rho_{01}, (0,2):\rho_{02}, \cdots\}\). The key is a tuple of index of the site.

Return type:

Dict

calc_bond_entropy(s_array: array = None) → ndarray

Calculate von Neumann entropy at each bond according to \(S = -\textrm{Tr}(\rho \ln \rho)\) where \(\rho\) is the reduced density matrix of either block.

Parameters:

s_array (np.ndarray, optional) – The singular values array. Calculated by Mps.calc_bond_singular_values().

Returns:

S – a NumPy array containing the entropy values.

Return type:

1D array

calc_bond_singular_values() → ndarray

Calculate the singular values at each bond.

Returns:

Singular values

Return type:

2D array

calc_edof_rdm() → ndarray

Calculate the reduced density matrix of electronic DoF

\(\rho_{ij} = \langle \Psi | a_i^\dagger a_j | \Psi \rangle\)

Note

This function is designed for single-electron system. Fermionic commutation relation is not considered.

calc_entropy(entropy_type)

Calculate 1site, 2site, mutual and bond Von Neumann entropy

\(\textrm{entropy} = -\textrm{Tr}(\rho \ln \rho)\) where \(\ln\) stands for natural logarithm.

1site entropy is the entropy between any site and the other (N-1) sites. 2site entropy is the entropy between any two sites and the other (N-2) sites. mutual entropy characterize the entropy between any two sites. bond entropy is the entropy between L-block and R-block.

Parameters:

entropy_type (str) – “1site”, “2site”, “mutual”, “bond”

Returns:

entropy – if entropy_type = “1site” or “2site”, a dictionary is returned and the key is the index or the tuple of index of mps sites, else an ndarray is returned.

Return type:

dict, ndarray

canonicalise(stop_idx: int = None)

check_left_canonical(rtol: float = None, atol: float = None)

check L-canonical

check_right_canonical(rtol: float = None, atol: float = None)

check R-canonical

compress(temp_m_trunc=None, ret_s=False)

inp: canonicalise MPS (or MPO)

Parameters:

• temp_m_trunc (int or list of int) – Temporary truncation bond dimension. Overwrites the compression configuration in CompressConfig.

• ret_s (bool) – Whether return the singular values at the bonds. The singular values are padded to a rectangular array with zero values.

Return type:

truncated MPS

conj() → Mps

complex conjugate

copy()

property digest

distance(other) → float

dot(other: MatrixProduct) → complex

dot product of two mps / mpo

dot_ob(other: MatrixProduct) → complex

dot product of two mps / mpo with open boundary, but the boundary of mps/mpo is larger than 1, different from the normal mps/mpo

dump(fname)

property e_occupations

Electronic occupations \(a^\dagger_i a_i\) for each electronic DoF. The order is defined by e_dofs.

ensure_left_canonical(rtol: float = None, atol: float = None)

ensure_right_canonical(rtol: float = None, atol: float = None)

evolve(mpo, evolve_dt, normalize=True) → Mps

evolve_exact(h_mpo, evolve_dt, space)

expand_bond_dimension(hint_mpo=None, coef=1e-10, include_ex=True)

expand bond dimension as required in compress_config

expectation(mpo: Mpo | Op | OpSum, self_conj: Mps = None, *, bra: Mps = None) → float | complex

Calculate the expectation value of the operator mpo with respect to the MPS.

For the expectation value \(\langle \psi | \hat O | \psi \rangle\), self is \(|\psi\rangle\) and mpo is \(\hat O\). For the transition amplitude \(\langle \phi | \hat O | \psi \rangle\), bra should be set to \(|\phi\rangle\).

Parameters:

• mpo (Union[MPO, Op, OpSum]) – The operator. Can be provided as: - MPO: Matrix Product Operator - Op: Single symbolic operator - OpSum: Sum of symbolic operators

• self_conj (Mps, optional) – Legacy preconjugated MPS. Unlike bra, its tensors are used as supplied. If neither is provided, uses the conjugate of self.

• bra (Mps, optional) – The MPS for the bra vector. bra and self_conj cannot both be provided.

Returns:

expectation – The expectation value or transition amplitude. Returns a float if the imaginary part is negligible.

Return type:

Union[float, complex]

Examples

>>> from renormalizer import Mps, Mpo, Model, Op, BasisHalfSpin
>>> model = Model([BasisHalfSpin(0)], [])
>>> mps = Mps.hartree_product_state(model, condition={})
>>> mpo = Mpo(model, Op("X", 0))
>>> mps.expectation(mpo)
0.0
>>> mps.expectation(Op("X", 0))  # direct Op input
0.0

>>> mps2 = Mps.hartree_product_state(model, condition={0: [0,1]})
>>> mps.expectation(mpo, bra=mps2)
1.0

expectations(mpos: List[Mpo | Op | OpSum], self_conj: Mps = None, opt: bool = True, *, bra: Mps = None) → ndarray

classmethod from_dense(model, wfn: ndarray)

classmethod from_mp(model, mplist)

classmethod ground_state(model: Model, max_entangled: bool, normalize: bool = True, condition: Dict = None)

Obtain ground state at \(T = 0\) or \(T = \infty\) (maximum entangled). Electronic DOFs are always at ground state. and vibrational DOFs depend on max_entangled. For Spin-Boson model the electronic DOF also depends on max_entangled.

Parameters:

• model (Model) – system information.

• max_entangled (bool) – temperature of the vibrational DOFs. If set to True, \(T = \infty\) and if set to False, \(T = 0\).

• normalize (bool, optional) – if the returned Mps are normalized when max_entangled=True. Default is True. If normalize=False, the vibrational part is identity.

• condition (dict, optional) – the same as hartree_product_state. only used in ba.BasisMultiElectron cases to define the occupation. Default is None.

Returns:

mps

Return type:

renormalizer.mps.Mps

classmethod hartree_product_state(model, condition: Dict = None, qn_idx: int = None)

Construct a Hartree product state

Parameters:

• model (Model) – Model information.

• condition (Dict) – Dict with format {dof:local_state}. The default local state for dofs not contained in the condition dict is the “0” state. An example is {"e_1":1, "v_0":2, "v_3":[0, 0.707, 0.707]}.

• Note –

  For basis sets containing multiple DoFs (like BasisMultiElectron and BasisMultiElectronVac), the key in the condition dict must be ONE OF THE ACTUAL DOF NAMES or ALL OF THE ACTUAL DOF NAMES from the basis set. In both cases, the value specifies the state for ALL DoFs in that basis.

  For BasisMultiElectron with dof_names = [“S0”, “S1”, “S2”]: - The basis order is [S0, S1, S2] - {"S0": 0} or {"S0": [1, 0, 0]} means S0 is occupied - {"S1": 1} or {"S1": [0, 1, 0]} means S1 is occupied - {"S2": 2} or {"S2": [0, 0, 1]} means S2 is occupied - The key can be ANY of the DoF names: “S0”, “S1”, or “S2” - they all refer to the same basis - Alternatively, {("S0", "S1", "S2"): 1} means S1 is occupied.

  For BasisMultiElectronVac with dof_names = [“S0”, “S1”]: - The basis order is [vacuum, S0, S1] - {"S0": 0} or {"S0": [1, 0, 0]} means vacuum state - {"S1": 1} or {"S1": [0, 1, 0]} means S0 is occupied - {"S0": 2} or {"S0": [0, 0, 1]} means S1 is occupied

• qn_idx (int) – the site index of the quantum number center.

Returns:

Constructed mps (Mps)

property is_complex

property is_left_canonical

check the qn center in the L-canonical structure

property is_mpdm

property is_mpo

property is_mps

property is_right_canonical

check the qn center in the R-canonical structure

iter_idx_list(full: bool, stop_idx: int = None)

classmethod load(model: Model, fname: str)

metacopy() → Mps

move_qnidx(dstidx: int)

Quantum number has a boundary site, left hand of the site is L system qn, right hand of the side is R system qn, the sum of quantum number of L system and R system is tot.

property mp_norm: float

if self.is_left_canon:

assert self.check_left_canonical() return np.linalg.norm(np.ravel(self[-1]))

else:

assert self.check_right_canonical() return np.linalg.norm(np.ravel(self[0]))

property nexciton

property norm

the norm of the total wavefunction

normalize(kind)

normalize the wavefunction

Parameters:

kind (str) – “mps_only”: the mps part is normalized and coeff is not modified; “mps_norm_to_coeff”: the mps part is normalized and the norm is multiplied to coeff; “mps_and_coeff”: both mps and coeff is normalized

Return type:

self is overwritten.

property pbond_dims

property pbond_list

property ph_occupations

phonon occupations \(b^\dagger_i b_i\) for each electronic DoF. The order is defined by v_dofs.

classmethod random(model: Model, qntot, m_max, percent=1.0) → Mps

scale(val, inplace=False)

property site_num

property threshold

to_complex(inplace=False) → Mps

todense() → array

property total_bytes

variational_compress(mpo=None, guess=None)

Variational compress an mps/mpdm/mpo

Parameters:

• mpo (renormalizer.mps.Mpo, optional) – Default is None. if mpo is not None, the returned mps is an approximation of mpo @ self

• guess (renormalizer.mps.MatrixProduct, optional) – Initial guess of compressed mps/mpdm/mpo. Default is None.

Note

the variational compress related configurations is defined in self if guess=None, otherwise is defined in guess

Returns:

mp – a new compressed mps/mpdm/mpo, self is not overwritten. guess is overwritten.

Return type:

renormalizer.mps.MatrixProduct

property vbond_dims: List

property vbond_list: List

Matrix Product Operator (MPO)

class renormalizer.mps.Mpo(model: Model = None, terms: Op | List[Op] = None, offset: Quantity = Quantity(0.0, 'a.u.'), algo='qr')

Matrix product operator (MPO)

add(other: MatrixProduct)

angle(other)

append(array)

apply(mp: MatrixProduct, canonicalise: bool = False) → MatrixProduct

property bond_dims: List

property bond_dims_exact: ndarray

property bond_dims_mean: int

property bond_list: List

build_empty_mp(num)

build_empty_qn()

build_none_qn()

canonicalise(stop_idx: int = None)

check_left_canonical(rtol: float = None, atol: float = None)

check L-canonical

check_right_canonical(rtol: float = None, atol: float = None)

check R-canonical

compress(temp_m_trunc=None, ret_s=False)

inp: canonicalise MPS (or MPO)

Parameters:

• temp_m_trunc (int or list of int) – Temporary truncation bond dimension. Overwrites the compression configuration in CompressConfig.

• ret_s (bool) – Whether return the singular values at the bonds. The singular values are padded to a rectangular array with zero values.

Return type:

truncated MPS

conj()

complex conjugate

conj_trans()

contract(mps, algo='svd')

an approximation of mpo @ mps/mpdm/mpo

Parameters:

• mps (Mps, Mpo, MpDm)

• algo (str, optional) – The algorithm to compress mpo @ mps/mpdm/mpo. It could be svd (default) and variational.

Returns:

new_mps – an approximation of mpo @ mps/mpdm/mpo. The input mps is not overwritten.

Return type:

Mps

See also

renormalizer.mps.mp.MatrixProduct.compress

svd compression.

renormalizer.mps.mp.MatrixProduct.variational_compress

variational compression.

copy()

property digest

distance(other) → float

dot(other: MatrixProduct) → complex

dot product of two mps / mpo

dot_ob(other: MatrixProduct) → complex

dot product of two mps / mpo with open boundary, but the boundary of mps/mpo is larger than 1, different from the normal mps/mpo

property dummy_qn

dump(fname, other_attrs=None)

ensure_left_canonical(rtol: float = None, atol: float = None)

ensure_right_canonical(rtol: float = None, atol: float = None)

classmethod exact_propagator(model: HolsteinModel, x, space='GS', shift=0.0)

construct the GS space propagator e^{xH} exact MPO H=sum_{in} omega_{in} b^dagger_{in} b_{in} fortunately, the H is local. so e^{xH} = e^{xh1}e^{xh2}…e^{xhn} the bond dimension is 1 shift is the a constant for H+shift

classmethod finiteT_cv(model, nexciton, m_max, spectratype, percent=1.0)

classmethod from_mp(model, mplist)

classmethod identity(model: Model)

classmethod intersite(model: HolsteinModel, e_opera: dict, ph_opera: dict, scale: Quantity = Quantity(1.0, 'a.u.'))

construct the inter site MPO

Parameters:

• model (HolsteinModel) – the molecular information

• e_opera – the electronic operators. {imol: operator}, such as {1:”a”, 3:r”a^dagger”}

• ph_opera – the vibrational operators. {(imol, iph): operator}, such as {(0,5):”b”}

• scale (Quantity) – scalar to scale the mpo

Note

the operator index starts from 0,1,2…

property is_complex

is_hermitian()

property is_left_canonical

check the qn center in the L-canonical structure

property is_mpdm

property is_mpo

property is_mps

property is_right_canonical

check the qn center in the R-canonical structure

iter_idx_list(full: bool, stop_idx: int = None)

classmethod load(model: Model, fname: str)

metacopy()

move_qnidx(dstidx: int)

Quantum number has a boundary site, left hand of the site is L system qn, right hand of the side is R system qn, the sum of quantum number of L system and R system is tot.

property mp_norm: float

if self.is_left_canon:

assert self.check_left_canonical() return np.linalg.norm(np.ravel(self[-1]))

else:

assert self.check_right_canonical() return np.linalg.norm(np.ravel(self[0]))

classmethod onsite(model: Model, opera, dipole=False, dof_set=None)

property pbond_dims

property pbond_list

classmethod ph_onsite(model: HolsteinModel, opera: str, mol_idx: int, ph_idx=0)

promote_mt_type(mp)

scale(val, inplace=False)

property site_num

property threshold

to_complex(inplace=False)

todense()

property total_bytes

try_swap_site(new_model: Model, swap_jw: bool, algo='Hopcroft-Karp')

variational_compress(mpo=None, guess=None)

Variational compress an mps/mpdm/mpo

Parameters:

• mpo (renormalizer.mps.Mpo, optional) – Default is None. if mpo is not None, the returned mps is an approximation of mpo @ self

• guess (renormalizer.mps.MatrixProduct, optional) – Initial guess of compressed mps/mpdm/mpo. Default is None.

Note

the variational compress related configurations is defined in self if guess=None, otherwise is defined in guess

Returns:

mp – a new compressed mps/mpdm/mpo, self is not overwritten. guess is overwritten.

Return type:

renormalizer.mps.MatrixProduct

property vbond_dims: List

property vbond_list: List

Matrix Product Density Matrix (MPDM)

class renormalizer.mps.MpDm

add(other)

angle(other)

append(array)

apply(mp, canonicalise=False) → MpDmBase

property bond_dims: List

property bond_dims_exact: ndarray

property bond_dims_mean: int

property bond_list: List

build_empty_mp(num)

build_empty_qn()

build_none_qn()

calc_1site_rdm(idx=None) → Dict[int, ndarray]

Calculate 1-site reduced density matrix

\(\rho_i = \textrm{Tr}_{j \neq i} | \Psi \rangle \langle \Psi|\)

Parameters:

idx (int, list, tuple, optional) – site index of 1site_rdm. Default is None, which mean all the rdms are calculated.

Returns:

rdm – \(\{0:\rho_0, 1:\rho_1, \cdots\}\). The key is the index of the site.

Return type:

Dict

calc_2site_mutual_entropy() → ndarray

Calculate mutual entropy between two sites. Also known as mutual information

\(m_{ij} = (s_i + s_j - s_{ij})/2\)

See Chemical Physics 323 (2006) 519–531

Returns:

mutual_entropy – mutual entropy with shape (nsite, nsite)

Return type:

2d np.ndarry

calc_2site_rdm()

Calculate 2-site reduced density matrix

\(\rho_{ij} = \textrm{Tr}_{k \neq i, k \neq j} | \Psi \rangle \langle \Psi |\).

Returns:

rdm – \(\{(0,1):\rho_{01}, (0,2):\rho_{02}, \cdots\}\). The key is a tuple of index of the site.

Return type:

Dict

calc_bond_entropy(s_array: array = None) → ndarray

Calculate von Neumann entropy at each bond according to \(S = -\textrm{Tr}(\rho \ln \rho)\) where \(\rho\) is the reduced density matrix of either block.

Parameters:

s_array (np.ndarray, optional) – The singular values array. Calculated by Mps.calc_bond_singular_values().

Returns:

S – a NumPy array containing the entropy values.

Return type:

1D array

calc_bond_singular_values() → ndarray

Calculate the singular values at each bond.

Returns:

Singular values

Return type:

2D array

calc_edof_rdm() → ndarray

Calculate the reduced density matrix of electronic DoF

\(\rho_{ij} = \langle \Psi | a_i^\dagger a_j | \Psi \rangle\)

Note

This function is designed for single-electron system. Fermionic commutation relation is not considered.

calc_entropy(entropy_type)

Calculate 1site, 2site, mutual and bond Von Neumann entropy

\(\textrm{entropy} = -\textrm{Tr}(\rho \ln \rho)\) where \(\ln\) stands for natural logarithm.

1site entropy is the entropy between any site and the other (N-1) sites. 2site entropy is the entropy between any two sites and the other (N-2) sites. mutual entropy characterize the entropy between any two sites. bond entropy is the entropy between L-block and R-block.

Parameters:

entropy_type (str) – “1site”, “2site”, “mutual”, “bond”

Returns:

entropy – if entropy_type = “1site” or “2site”, a dictionary is returned and the key is the index or the tuple of index of mps sites, else an ndarray is returned.

Return type:

dict, ndarray

canonicalise(stop_idx: int = None)

check_left_canonical(rtol: float = None, atol: float = None)

check L-canonical

check_right_canonical(rtol: float = None, atol: float = None)

check R-canonical

compress(temp_m_trunc=None, ret_s=False)

inp: canonicalise MPS (or MPO)

Parameters:

• temp_m_trunc (int or list of int) – Temporary truncation bond dimension. Overwrites the compression configuration in CompressConfig.

• ret_s (bool) – Whether return the singular values at the bonds. The singular values are padded to a rectangular array with zero values.

Return type:

truncated MPS

conj() → Mps

complex conjugate

conj_trans()

contract(mps, algo='svd')

an approximation of mpo @ mps/mpdm/mpo

Parameters:

• mps (Mps, Mpo, MpDm)

• algo (str, optional) – The algorithm to compress mpo @ mps/mpdm/mpo. It could be svd (default) and variational.

Returns:

new_mps – an approximation of mpo @ mps/mpdm/mpo. The input mps is not overwritten.

Return type:

Mps

See also

renormalizer.mps.mp.MatrixProduct.compress

svd compression.

renormalizer.mps.mp.MatrixProduct.variational_compress

variational compression.

copy()

property digest

distance(other) → float

dot(other: MatrixProduct) → complex

dot product of two mps / mpo

dot_ob(other: MatrixProduct) → complex

dot product of two mps / mpo with open boundary, but the boundary of mps/mpo is larger than 1, different from the normal mps/mpo

property dummy_qn

dump(fname)

property e_occupations

Electronic occupations \(a^\dagger_i a_i\) for each electronic DoF. The order is defined by e_dofs.

ensure_left_canonical(rtol: float = None, atol: float = None)

ensure_right_canonical(rtol: float = None, atol: float = None)

evolve(mpo, evolve_dt, normalize=True) → Mps

evolve_exact(h_mpo, evolve_dt, space)

classmethod exact_propagator(model: HolsteinModel, x, space='GS', shift=0.0)

construct the GS space propagator e^{xH} exact MPO H=sum_{in} omega_{in} b^dagger_{in} b_{in} fortunately, the H is local. so e^{xH} = e^{xh1}e^{xh2}…e^{xhn} the bond dimension is 1 shift is the a constant for H+shift

expand_bond_dimension(hint_mpo=None, coef=1e-10, include_ex=True)

expand bond dimension as required in compress_config

expectation(mpo: Mpo | Op | OpSum, self_conj: Mps = None, *, bra: Mps = None) → float | complex

Calculate the expectation value of the operator mpo with respect to the MPS.

For the expectation value \(\langle \psi | \hat O | \psi \rangle\), self is \(|\psi\rangle\) and mpo is \(\hat O\). For the transition amplitude \(\langle \phi | \hat O | \psi \rangle\), bra should be set to \(|\phi\rangle\).

Parameters:

• mpo (Union[MPO, Op, OpSum]) – The operator. Can be provided as: - MPO: Matrix Product Operator - Op: Single symbolic operator - OpSum: Sum of symbolic operators

• self_conj (Mps, optional) – Legacy preconjugated MPS. Unlike bra, its tensors are used as supplied. If neither is provided, uses the conjugate of self.

• bra (Mps, optional) – The MPS for the bra vector. bra and self_conj cannot both be provided.

Returns:

expectation – The expectation value or transition amplitude. Returns a float if the imaginary part is negligible.

Return type:

Union[float, complex]

Examples

>>> from renormalizer import Mps, Mpo, Model, Op, BasisHalfSpin
>>> model = Model([BasisHalfSpin(0)], [])
>>> mps = Mps.hartree_product_state(model, condition={})
>>> mpo = Mpo(model, Op("X", 0))
>>> mps.expectation(mpo)
0.0
>>> mps.expectation(Op("X", 0))  # direct Op input
0.0

>>> mps2 = Mps.hartree_product_state(model, condition={0: [0,1]})
>>> mps.expectation(mpo, bra=mps2)
1.0

expectations(mpos: List[Mpo | Op | OpSum], self_conj: Mps = None, opt: bool = True, *, bra: Mps = None) → ndarray

classmethod finiteT_cv(model, nexciton, m_max, spectratype, percent=1.0)

classmethod from_dense(model, wfn: ndarray)

classmethod from_mp(model, mplist)

classmethod from_mps(mps: Mps)

classmethod ground_state(model, max_entangled)

Obtain ground state at \(T = 0\) or \(T = \infty\) (maximum entangled). Electronic DOFs are always at ground state. and vibrational DOFs depend on max_entangled. For Spin-Boson model the electronic DOF also depends on max_entangled.

Parameters:

• model (Model) – system information.

• max_entangled (bool) – temperature of the vibrational DOFs. If set to True, \(T = \infty\) and if set to False, \(T = 0\).

• normalize (bool, optional) – if the returned Mps are normalized when max_entangled=True. Default is True. If normalize=False, the vibrational part is identity.

• condition (dict, optional) – the same as hartree_product_state. only used in ba.BasisMultiElectron cases to define the occupation. Default is None.

Returns:

mps

Return type:

renormalizer.mps.Mps

classmethod hartree_product_state(model, condition: Dict = None, qn_idx: int = None)

Construct a Hartree product state

Parameters:

• model (Model) – Model information.

• condition (Dict) – Dict with format {dof:local_state}. The default local state for dofs not contained in the condition dict is the “0” state. An example is {"e_1":1, "v_0":2, "v_3":[0, 0.707, 0.707]}.

• Note –

  For basis sets containing multiple DoFs (like BasisMultiElectron and BasisMultiElectronVac), the key in the condition dict must be ONE OF THE ACTUAL DOF NAMES or ALL OF THE ACTUAL DOF NAMES from the basis set. In both cases, the value specifies the state for ALL DoFs in that basis.

  For BasisMultiElectron with dof_names = [“S0”, “S1”, “S2”]: - The basis order is [S0, S1, S2] - {"S0": 0} or {"S0": [1, 0, 0]} means S0 is occupied - {"S1": 1} or {"S1": [0, 1, 0]} means S1 is occupied - {"S2": 2} or {"S2": [0, 0, 1]} means S2 is occupied - The key can be ANY of the DoF names: “S0”, “S1”, or “S2” - they all refer to the same basis - Alternatively, {("S0", "S1", "S2"): 1} means S1 is occupied.

  For BasisMultiElectronVac with dof_names = [“S0”, “S1”]: - The basis order is [vacuum, S0, S1] - {"S0": 0} or {"S0": [1, 0, 0]} means vacuum state - {"S1": 1} or {"S1": [0, 1, 0]} means S0 is occupied - {"S0": 2} or {"S0": [0, 0, 1]} means S1 is occupied

• qn_idx (int) – the site index of the quantum number center.

Returns:

Constructed mps (Mps)

classmethod identity(model: Model)

classmethod intersite(model: HolsteinModel, e_opera: dict, ph_opera: dict, scale: Quantity = Quantity(1.0, 'a.u.'))

construct the inter site MPO

Parameters:

• model (HolsteinModel) – the molecular information

• e_opera – the electronic operators. {imol: operator}, such as {1:”a”, 3:r”a^dagger”}

• ph_opera – the vibrational operators. {(imol, iph): operator}, such as {(0,5):”b”}

• scale (Quantity) – scalar to scale the mpo

Note

the operator index starts from 0,1,2…

property is_complex

is_hermitian()

property is_left_canonical

check the qn center in the L-canonical structure

property is_mpdm

property is_mpo

property is_mps

property is_right_canonical

check the qn center in the R-canonical structure

iter_idx_list(full: bool, stop_idx: int = None)

classmethod load(model: Model, fname: str)

classmethod max_entangled_ex(model, normalize=True)

T = infty locally maximal entangled EX state

classmethod max_entangled_gs(model) → MpDm

metacopy() → Mps

move_qnidx(dstidx: int)

Quantum number has a boundary site, left hand of the site is L system qn, right hand of the side is R system qn, the sum of quantum number of L system and R system is tot.

property mp_norm: float

if self.is_left_canon:

assert self.check_left_canonical() return np.linalg.norm(np.ravel(self[-1]))

else:

assert self.check_right_canonical() return np.linalg.norm(np.ravel(self[0]))

property nexciton

property norm

the norm of the total wavefunction

normalize(kind)

normalize the wavefunction

Parameters:

kind (str) – “mps_only”: the mps part is normalized and coeff is not modified; “mps_norm_to_coeff”: the mps part is normalized and the norm is multiplied to coeff; “mps_and_coeff”: both mps and coeff is normalized

Return type:

self is overwritten.

classmethod onsite(model: Model, opera, dipole=False, dof_set=None)

property pbond_dims

property pbond_list

property ph_occupations

phonon occupations \(b^\dagger_i b_i\) for each electronic DoF. The order is defined by v_dofs.

classmethod ph_onsite(model: HolsteinModel, opera: str, mol_idx: int, ph_idx=0)

promote_mt_type(mp)

classmethod random(mpo, qntot, m_max, percent=0)

scale(val, inplace=False)

property site_num

property threshold

to_complex(inplace=False) → Mps

todense()

property total_bytes

try_swap_site(new_model: Model, swap_jw: bool, algo='Hopcroft-Karp')

variational_compress(mpo=None, guess=None)

Variational compress an mps/mpdm/mpo

Parameters:

• mpo (renormalizer.mps.Mpo, optional) – Default is None. if mpo is not None, the returned mps is an approximation of mpo @ self

• guess (renormalizer.mps.MatrixProduct, optional) – Initial guess of compressed mps/mpdm/mpo. Default is None.

Note

the variational compress related configurations is defined in self if guess=None, otherwise is defined in guess

Returns:

mp – a new compressed mps/mpdm/mpo, self is not overwritten. guess is overwritten.

Return type:

renormalizer.mps.MatrixProduct

property vbond_dims: List

property vbond_list: List

Thermal Propagation

class renormalizer.mps.thermalprop.ThermalProp(init_mpdm: MpDm, h_mpo_model: Model = None, exact: bool = False, space: str = 'GS', evolve_config: EvolveConfig = None, dump_mps: bool = None, dump_dir: str = None, job_name: str = None, properties: Property = None, auto_expand: bool = True)

Bases: TdMpsJob

Thermally Propagate initial matrix product density operator (MpDm) in imaginary time.

Parameters:

• init_mpdm (MpDm) – the initial density matrix to be propagated. Usually identity.

• h_mpo_model (Model) – Model for the system Hamiltonian. Default is the same with init_mpdm.model.

• exact (bool) – whether propagate assuming Hamiltonian is local \(\hat H = \sum_i \hat H_i = \sum_{in} \omega_{in} b^\dagger_{in} b_{in}\) and exact propagation is possible through \(e^{xH} = e^{xh_1}e^{xh_2} \cdots e^{xh_n}\). If set to True, properties such as occupations are not calculated during time evolution for better efficiency.

• space (str) – the space of exact propagation. Possible options are "GS" or "EX". If set to "GS", then the exact propagation is performed in zero exciton space. If set to "EX", then the exact propagation is performed in one exciton space, i.e. the vibrations are regarded as displaced oscillators.

• evolve_config (EvolveConfig) – config when evolving the MpDm in imaginary time.

• dump_mps (str) – if dump mps when dumping, “all”, “one”, None; Default to None

• dump_dir (str) – the directory for logging and numerical result output.

• job_name (str) – the name of the calculation job which determines the file name of the logging and numerical result output.

• properties (Property)

dump_dict()

property e_occupations_array

evolve(evolve_dt=None, nsteps=None, evolve_time=None)

Parameters:

• evolve_dt (float) – the time step to run evolve_single_step and process_mps.

• nsteps (int) – the total number of evolution steps

• evolve_time (float) – the total evolution time

Note

evolve_dt math: ` imes` nsteps = evolve_time, otherwise nsteps has a higher priority.

evolve_exact(old_mpdm: MpDm, evolve_dt)

evolve_prop(old_mpdm, evolve_dt)

evolve_single_step(evolve_dt)

Evolve the mps for a single step with step size evolve_dt.

Returns:

new mps after the evolution

property evolve_times_array

get_dump_dict()

Obtain calculated properties to dump in dict type.

Returns:

return a (ordered) dict to dump as npz

init_mps()

Returns:

initial mps of the system

property latest_evolve_time

property ph_occupations_array

process_mps(mps)

Process the newly evolved the mps. Primarily for the calculation of properties. Note that currently self.latest_mps has not been updated.

Parameters:

mps – The evolved new mps of the time step.

stop_evolve_criteria()

property vn_entropy_array

renormalizer.mps.thermalprop.load_thermal_state(model, path: str)

Load thermal propagated state from disk. Return None if the file is not found.

Parameters:

• model (MolList) – system information

• path (str) – the path to load thermal state from. Should be an numpy .npz file.

Returns: Loaded MpDm