# MPO¶

Inheritance Diagram

Methods

 MPO.__init__(sites, Ws[, bc, IdL, IdR, …]) Initialize self. MPO.apply(psi, options) Apply self to an MPS psi and compress psi in place. Applies an MPO to an MPS (in place) naively, without compression. MPO.apply_zipup(psi, options) Applies an MPO to an MPS (in place) with the zip-up method. Return hermition conjugate copy of self. MPO.enlarge_mps_unit_cell([factor]) Repeat the unit cell for infinite MPS boundary conditions; in place. MPO.expectation_value(psi[, tol, max_range]) Calculate /. MPO.extract_segment(first, last) Extract a segment from the MPO. MPO.from_grids(sites, grids[, bc, IdL, IdR, …]) Initialize an MPO from grids. MPO.from_hdf5(hdf5_loader, h5gr, subpath) Load instance from a HDF5 file. Return index of IdL at bond to the left of site i. Return index of IdR at bond to the right of site i. MPO.get_W(i[, copy]) Return W at site i. MPO.get_full_hamiltonian([maxsize]) extract the full Hamiltonian as a d**Lxd**L matrix. MPO.get_grouped_mpo(blocklen) group each blocklen subsequent tensors and return result as a new MPO. MPO.group_sites([n, grouped_sites]) Modify self inplace to group sites. MPO.is_equal(other[, eps, max_range]) Check if self and other represent the same MPO to precision eps. MPO.is_hermitian([eps, max_range]) Check if self is a hermitian MPO. MPO.make_U(dt[, approximation]) Creates the U_I or U_II propagator. Creates the $$U_I$$ propagator with W_I tensors. Creates the $$U_II$$ propagator. MPO.save_hdf5(hdf5_saver, h5gr, subpath) Export self into a HDF5 file. MPO.set_W(i, W) Set W at site i. Sort virtual legs by charges. Sanity check, raises ValueErrors, if something is wrong. MPO.variance(psi[, exp_val]) Calculate  - ^2.

Class Attributes and Properties

 MPO.L Number of physical sites; for an iMPO the len of the MPO unit cell. MPO.chi Dimensions of the virtual bonds. MPO.dim List of local physical dimensions. MPO.finite Distinguish MPO vs iMPO.
class tenpy.networks.mpo.MPO(sites, Ws, bc='finite', IdL=None, IdR=None, max_range=None, explicit_plus_hc=False)[source]

Bases: object

Matrix product operator, finite (MPO) or infinite (iMPO).

Parameters
• sites (list of Site) – Defines the local Hilbert space for each site.

• Ws (list of Array) – The matrices of the MPO. Should have labels wL, wR, p, p*.

• bc ({'finite' | 'segment' | 'infinite'}) – Boundary conditions as described in mps. 'finite' requires Ws[0].get_leg('wL').ind_len = 1.

• IdL ((iterable of) {int | None}) – Indices on the bonds, which correpond to ‘only identities to the left’. A single entry holds for all bonds.

• IdR ((iterable of) {int | None}) – Indices on the bonds, which correpond to ‘only identities to the right’.

• max_range (int | np.inf | None) – Maximum range of hopping/interactions (in unit of sites) of the MPO. None for unknown.

• explicit_plus_hc (bool) – If True, this flag indicates that the hermitian conjugate of the MPO should be computed and added at runtime, i.e., self is not (necessarily) hermitian.

chinfo

The nature of the charge.

Type

ChargeInfo

sites

Defines the local Hilbert space for each site.

Type

list of Site

dtype

The data type of the _W.

Type

type

bc

Boundary conditions as described in mps. 'finite' requires Ws[0].get_leg('wL').ind_len = 1.

Type

{‘finite’ | ‘segment’ | ‘infinite’}

IdL

Indices on the bonds (length L+1), which correpond to ‘only identities to the left’. None for bonds where it is not set. In standard form, this is 0 (except for unset bonds in finite case)

Type

list of {int | None}

IdR

Indices on the bonds (length L+1), which correpond to ‘only identities to the right’. None for bonds where it is not set. In standard form, this is the last index on the bond (except for unset bonds in finite case).

Type

list of {int | None}

max_range

Maximum range of hopping/interactions (in unit of sites) of the MPO. None for unknown.

Type

int | np.inf | None

grouped

Number of sites grouped together, see group_sites().

Type

int

explicit_plus_hc

If True, this flag indicates that the hermitian conjugate of the MPO should be computed and added at runtime, i.e., self is not (necessarily) hermitian.

Type

bool

_W

The matrices of the MPO. Labels are 'wL', 'wR', 'p', 'p*'.

Type

list of Array

_valid_bc

Class attribute. Valid boundary conditions; the same as for an MPS.

Type

tuple of str

save_hdf5(hdf5_saver, h5gr, subpath)[source]

Export self into a HDF5 file.

This method saves all the data it needs to reconstruct self with from_hdf5().

Specifically, it saves sites, chinfo, max_range (under these names), _W as "tensors", IdL as "index_identity_left", IdR as "index_identity_right", and bc as "boundary_condition". Moreover, it saves L, explicit_plus_hc and grouped as HDF5 attributes, as well as the maximum of chi under the name max_bond_dimension.

Parameters
• hdf5_saver (Hdf5Saver) – Instance of the saving engine.

• h5gr (:classGroup) – HDF5 group which is supposed to represent self.

• subpath (str) – The name of h5gr with a '/' in the end.

classmethod from_hdf5(hdf5_loader, h5gr, subpath)[source]

Load instance from a HDF5 file.

This method reconstructs a class instance from the data saved with save_hdf5().

Parameters
• hdf5_loader (Hdf5Loader) – Instance of the loading engine.

• h5gr (Group) – HDF5 group which is represent the object to be constructed.

• subpath (str) – The name of h5gr with a '/' in the end.

Returns

obj – Newly generated class instance containing the required data.

Return type

cls

classmethod from_grids(sites, grids, bc='finite', IdL=None, IdR=None, Ws_qtotal=None, legs=None, max_range=None, explicit_plus_hc=False)[source]

Initialize an MPO from grids.

Parameters
• sites (list of Site) – Defines the local Hilbert space for each site.

• grids (list of list of list of entries) – For each site (outer-most list) a matrix-grid (corresponding to wL, wR) with entries being or representing (see grid_insert_ops()) onsite-operators.

• bc ({'finite' | 'segment' | 'infinite'}) – Boundary conditions as described in mps.

• IdL ((iterable of) {int | None}) – Indices on the bonds, which correpond to ‘only identities to the left’. A single entry holds for all bonds.

• IdR ((iterable of) {int | None}) – Indices on the bonds, which correpond to ‘only identities to the right’.

• Ws_qtotal ((list of) total charge) – The qtotal to be used for each grid. Defaults to zero charges.

• legs (list of LegCharge) – List of charges for ‘wL’ legs left of each W, L + 1 entries. The last entry should be the conjugate of the ‘wR’ leg, i.e. identical to legs[0] for ‘infinite’ bc. By default, determine the charges automatically. This is limited to cases where there are no “dangling open ends” in the MPO graph. (The MPOGraph can handle those cases, though.)

• max_range (int | np.inf | None) – Maximum range of hopping/interactions (in unit of sites) of the MPO. None for unknown.

• explicit_plus_hc (bool) – If True, the Hermitian conjugate of the MPO is computed at runtime, rather than saved in the MPO.

grid_insert_ops

used to plug in entries of the grid.

tenpy.linalg.np_conserved.grid_outer

used for final conversion.

test_sanity()[source]

Sanity check, raises ValueErrors, if something is wrong.

property L

Number of physical sites; for an iMPO the len of the MPO unit cell.

property dim

List of local physical dimensions.

property finite

Distinguish MPO vs iMPO.

True for an MPO (bc='finite', 'segment'), False for an iMPO (bc='infinite').

property chi

Dimensions of the virtual bonds.

get_W(i, copy=False)[source]

Return W at site i.

set_W(i, W)[source]

Set W at site i.

get_IdL(i)[source]

Return index of IdL at bond to the left of site i.

May be None.

get_IdR(i)[source]

Return index of IdR at bond to the right of site i.

May be None.

enlarge_mps_unit_cell(factor=2)[source]

Repeat the unit cell for infinite MPS boundary conditions; in place.

Parameters

factor (int) – The new number of sites in the unit cell will be increased from L to factor*L.

group_sites(n=2, grouped_sites=None)[source]

Modify self inplace to group sites.

Group each n sites together using the GroupedSite. This might allow to do TEBD with a Trotter decomposition, or help the convergence of DMRG (in case of too long range interactions).

Parameters
• n (int) – Number of sites to be grouped together.

• grouped_sites (None | list of GroupedSite) – The sites grouped together.

extract_segment(first, last)[source]

Extract a segment from the MPO.

Parameters
• first (int) – The first and last site to include into the segment.

• last (int) – The first and last site to include into the segment.

sort_legcharges()[source]

Sort virtual legs by charges. In place.

The MPO seen as matrix of the wL, wR legs is usually very sparse. This sparsity is captured by the LegCharges for these bonds not being sorted and bunched. This requires a tensordot to do more block-multiplications with smaller blocks. This is in general faster for large blocks, but might lead to a larger overhead for small blocks. Therefore, this function allows to sort the virtual legs by charges.

make_U(dt, approximation='II')[source]

Creates the U_I or U_II propagator.

Approximations of MPO exponentials following .

Parameters
• dt (float|complex) – The time step per application of the propagator. Should be imaginary for real time evolution!

• approximation ('I' | 'II') – Selects the approximation, make_U_I() ('I') or make_U_II() ('II').

Returns

U – The propagator, i.e. approximation $$U ~= exp(H*dt)$$

Return type

MPO

make_U_I(dt)[source]

Creates the $$U_I$$ propagator with W_I tensors.

Parameters

dt (float|complex) – The time step per application of the propagator. Should be imaginary for real time evolution!

Returns

UI – The propagator, i.e. approximation $$U_I ~= exp(H*dt)$$

Return type

MPO

make_U_II(dt)[source]

Creates the $$U_II$$ propagator.

Parameters

dt (float|complex) – The time step per application of the propagator. Should be imaginary for real time evolution!

Returns

U_II – The propagator, i.e. approximation $$UII ~= exp(H*dt)$$

Return type

MPO

expectation_value(psi, tol=1e-10, max_range=100)[source]

Calculate <psi|self|psi>/<psi|psi>.

For a finite MPS, simply contract the network <psi|self|psi>. For an infinite MPS, it assumes that self is the a of terms, with IdL and IdR defined on each site. Under this assumption, it calculates the expectation value of terms with the left-most non-trivial operator inside the MPO unit cell and returns the average value per site.

Parameters
• psi (MPS) – State for which the expectation value should be taken.

• tol (float) – Ignored for finite psi. For infinite MPO containing exponentially decaying long-range terms, stop evaluating further terms if the terms in LP have norm < tol.

• max_range (int) – Ignored for finite psi. Contract at most self.L * max_range sites, even if tol is not reached. In that case, issue a warning.

Returns

exp_val – The expectation value of self with respect to the state psi. For an infinite MPS: the density per site.

Return type

float/complex

variance(psi, exp_val=None)[source]

Calculate <psi|self^2|psi> - <psi|self|psi>^2.

Works only for finite systems. Ignores the norm of psi.

Todo

This is a naive, expensive implementation contracting the full network. Try to follow arXiv:1711.01104 for a better estimate; would that even work in the infinite limit?

Parameters
• psi (MPS) – State for which the variance should be taken.

• exp_val (float/complex | None) – The result of <psi|self|psi> = self.expectation_value(psi) if known; otherwise obtained from expectation_value(). (Set this to 0 to obtain only the part <psi|self^2|psi>.)

dagger()[source]

Return hermition conjugate copy of self.

is_hermitian(eps=1e-10, max_range=None)[source]

Check if self is a hermitian MPO.

Shorthand for self.is_equal(self.dagger(), eps, max_range).

is_equal(other, eps=1e-10, max_range=None)[source]

Check if self and other represent the same MPO to precision eps.

To compare them efficiently we view self and other as MPS and compare the overlaps abs(<self|self> + <other|other> - 2 Re(<self|other>)) < eps*(<self|self>+<other|other>)

Parameters
• other (MPO) – The MPO to compare to.

• eps (float) – Precision threshold what counts as zero.

• max_range (None | int) – Ignored for finite MPS; for finite MPS we consider only the terms contained in the sites with indices range(self.L + max_range). None defaults to max_range (or L in case this is infinite or None).

Returns

equal – Whether self equals other to the desired precision.

Return type

bool

apply(psi, options)[source]

Apply self to an MPS psi and compress psi in place.

Options

config ApplyMPO
option summary

chi_list (from Sweep) in Sweep.reset_stats

By default (None) this feature is disabled. [...]

combine (from Sweep) in Sweep

Whether to combine legs into pipes. This combines the virtual and [...]

compression_method

Mandatory. [...]

init_env_data (from Sweep) in DMRGEngine.init_env

Dictionary as returned by self.env.get_initialization_data() from [...]

lanczos_params (from Sweep) in Sweep

Lanczos parameters as described in :cfg:config:Lanczos.

m_temp (from ZipUpApplyMPO) in MPO.apply_zipup

bond dimension will be truncated to m_temp * chi_max

N_sweeps (from VariationalCompression) in VariationalCompression

Number of sweeps to perform.

orthogonal_to (from Sweep) in DMRGEngine.init_env

Deprecated in favor of the orthogonal_to function argument (forwarded fro [...]

start_env (from Sweep) in DMRGEngine.init_env

Number of sweeps to be performed without optimization to update the environment.

start_env_sites (from VariationalCompression) in VariationalCompression

Number of sites to contract for the inital LP/RP environment in case of inf [...]

sweep_0 (from Sweep) in Sweep.reset_stats

Number of sweeps that have already been performed.

trunc_params

Truncation parameters as described in :cfg:config:truncation.

trunc_weight (from ZipUpApplyMPO) in MPO.apply_zipup

reduces cut for Schmidt values to trunc_weight * svd_min

option compression_method: 'SVD' | 'variational' | 'zip_up'

Mandatory. Selects the method to be used for compression. For the SVD compression, trunc_params is the only other option used.

option trunc_params: dict

Truncation parameters as described in truncation.

Parameters
• psi (MPS) – The state to which self should be applied, in place.

• options (dict) – See above.

apply_naively(psi)[source]

Applies an MPO to an MPS (in place) naively, without compression.

This function simply contracts the W tensors of the MPO to the B tensors of the MPS, resulting in an MPS with bond dimension self.chi * psi.chi.

Warning

This function sets only a wild guess for the new singular values. You should either compress the MPS or at least call canonical_form(). If you use apply() instead, this will be done automatically.

Parameters

psi (MPS) – The MPS to which self should be applied. Modified in place!

apply_zipup(psi, options)[source]

Applies an MPO to an MPS (in place) with the zip-up method.

Described in Ref. .

The ‘W’ tensors are contracted to the ‘B’ tensors with intermediate SVD compressions, truncated to bond dimensions chi_max * m_temp.

Warning

The MPS afterwards is only approximately in canonical form (under the assumption that self is close to unity). You should either compress the MPS or at least call canonical_form(). If you use apply() instead, this will be done automatically.

Parameters

Options

config ZipUpApplyMPO
option summary

m_temp

bond dimension will be truncated to m_temp * chi_max

trunc_params

Truncation parameters as described in :cfg:config:truncation.

trunc_weight

reduces cut for Schmidt values to trunc_weight * svd_min

option trunc_params: dict

Truncation parameters as described in truncation.

option m_temp: int

bond dimension will be truncated to m_temp * chi_max

option trunc_weight: float

reduces cut for Schmidt values to trunc_weight * svd_min

get_grouped_mpo(blocklen)[source]

group each blocklen subsequent tensors and return result as a new MPO.

Deprecated since version 0.5.0: Make a copy and use group_sites() instead.

get_full_hamiltonian(maxsize=1000000.0)[source]

extract the full Hamiltonian as a d**Lxd**L matrix.

Deprecated since version 0.5.0: Use tenpy.algorithms.exact_diag.ExactDiag.from_H_mpo() instead.