mdopt.mps.explicit module#

This module contains the ExplicitMPS class. Hereafter saying the MPS is in the explicit form will mean that the state is stored in the following format: for each three-dimensional tensor at site i denoted by ( ), there exists a singular values diagonal matrix at bond i denoted by <>:

       i    i    i+1  i+1
...---<>---( )---<>---( )---...
            |          |
            |          |

For “ghost” bonds at indices 0, L-1 (i.e., bonds of dimension 1), where L is the length of the MPS, the corresponding singular value tensors at the boundaries would be the identities of the same dimension. We index MPS tensors with i from 0 to L-1, while the singular values matrices are indexed from the left of each tensor, i.e., the singular values matrux with index i is to the left of site i. Essentially, it corresponds to storing each Γ[i] and Λ[i] as shown in fig.4b in reference `[1]`_.

class mdopt.mps.explicit.ExplicitMPS(tensors: List[ndarray], singular_values: List[List], tolerance: float = 1e-12, chi_max: int = 10000)#

Class for finite-size explicit matrix product states (MPS) with open boundary conditions.

tensors#

The “physical” tensors of the MPS, one for each physical site. Each tensor has legs (virtual left, physical, virtual right), in short (vL, i, vR).

Type:

List[np.ndarray]

singular_values#

The singular values at each of the bonds, singular_values[i] is left of tensors[i]. Each singular values list at each bond is normalised to 1.

Type:

List[List]

tolerance#

Absolute tolerance of the normalisation of the singular value spectrum at each bond.

Type:

float

num_sites#

Number of sites.

Type:

int

num_bonds#

Number of non-trivial bonds, which is equal to num_sites - 1.

Type:

int

Raises:

  • ValueError – If the tensors and the singular_values lists do not have corresponding lengths. The number of singular value matrices should be equal to the number of tensors + 1, because there are two trivial singular value matrices at each of the ghost bonds.

  • ValueError – If any of the MPS tensors is not three-dimensional.

  • ValueError – If any of the singular-values tensors is not normalised within the tolerance attribute.

property all_dimensions: List[Tuple[int, ...]]#

Returns the list of all dimensions of the MPS.

property bond_dimensions: List[int]#

Returns the list of all bond dimensions of the MPS.

compress(chi_max: int = 10000, cut: float = 1e-17, renormalise: bool = False, return_truncation_errors: bool = False) → Tuple[ExplicitMPS, List[float | None]]#

Compresses the MPS, i.e., runs the compress_bond method for each bond.

Parameters:

  • chi_max (int) – The maximum bond dimension to keep.

  • cut (float) – Singular values smaller than this will be discarded.

  • renormalise (bool) – Whether to renormalise the singular value spectrum after the cut.

  • return_truncation_errors (bool) – Whether to return the list of truncation errors (for each bond).

Returns:

  • compressed_mps (CanonicalMPS) – The new compressed Matrix Product State.

  • truncation_errors (Optional[List[float]]) – The truncation errors. Returned only if return_truncation_errors is set to True.

compress_bond(bond: int, chi_max: int = 10000, cut: float = 1e-17, renormalise: bool = False, return_truncation_error: bool = False) → Tuple[ExplicitMPS, float | None]#

Compresses the bond at a given site, i.e., reduces its bond dimension. The compression is performed via trimming the singular values at the bond.

Parameters:

  • bond (int) – The index of the bond to compress.

  • chi_max (int) – The maximum bond dimension to keep.

  • cut (float) – Singular values smaller than this will be discarded.

  • renormalise (bool) – Whether to renormalise the singular value spectrum after the cut.

  • return_truncation_error (bool) – Whether to return the truncation error.

Returns:

  • compressed_mps (ExplicitMPS) – The new compressed Matrix Product State.

  • truncation_error (Optional[float]) – The truncation error. Returned only if return_truncation_error is True.

Raises:

ValueError – If the bond index is out of range.

Notes

The bonds are being enumerated from the right side of the tensors. For example, the bond 0 is a bond to the right of tensor 0. Note, the singular value matrices obey a different numbering.

conjugate() → ExplicitMPS#

Returns a complex-conjugated version of the current MPS.

copy() → ExplicitMPS#

Returns a copy of the current MPS.

dense(flatten: bool = True, renormalise: bool = False, norm: None | float | Literal['fro', 'nuc'] = 2) → ndarray#

Returns a dense representation of the MPS.

Warning: this method can cause memory overload for number of sites > ~20!

Parameters:

  • flatten (bool) – Whether to merge all the physical indices to form a vector or not.

  • renormalise (bool) – Whether to renormalise the resulting tensor.

  • norm (Union[str, int]) – Which norm to use for renormalisation of the final tensor.

density_mpo() → List[ndarray]#

Returns the MPO representation (as a list of tensors) of the density matrix defined by a given MPS. This operation is depicted in the following picture:

       i           j
  a    |           |          c              i     j
...---(*)---<>*---(*)---<>*---...        ab  |     |  cd
                                  --> ...---[ ]---[ ]---...
...---( )---<>----( )---<>----...            |     |
  b    |           |          d              k     l
       k           l

In the cartoon, {i,j,k,l} and {a,b,c,d} are single indices, while ab and cd denote multi indices. Here, the ( )’s represent the MPS tensors, the <>’s — the singular values tensors, the [ ]’s — the MPO tensors. The MPS with the physical legs up is complex-conjugated element-wise, this is denoted by the star sign. The empty line between the MPS and its complex-conjugated version stands for the tensor (kronecker) product.

Each tensor in the MPO list has legs (vL, vR, pU, pD), where v stands for “virtual”, p – for “physical”, and L, R, U, D stand for “left”, “right”, “up”, “down”.

entanglement_entropy() → ndarray#

Returns the entanglement entropy for bipartitions at each of the bonds.

left_canonical() → CanonicalMPS#

Returns the current MPS in the left-canonical form given the MPS in the explicit form.

(see eq.19 in https://scipost.org/10.21468/SciPostPhysLectNotes.5 for reference),

marginal(sites_to_marginalise: List[int], renormalise: bool = False) → ExplicitMPS | CanonicalMPS#

Computes a marginal over a subset of sites of an MPS via CanonicalMPS path.

mixed_canonical(orth_centre: int) → CanonicalMPS#

Returns the current MPS in the mixed-canonical form with the orthogonality centre being located at orth_centre.

Parameters:

orth_centre_index (int) – An integer which can take values 0, 1, ..., num_sites-1. Denotes the position of the orthogonality centre – the only non-isometry in the new canonical MPS.

norm() → float#

Computes the norm of the current MPS, that is, the modulus squared of its inner product with itself.

one_site_expectation_value(site: int, operator: ndarray) → float | complex128#

Computes an expectation value of an arbitrary one-site operator (not necessarily unitary) on the given site.

Parameters:

  • site (int) – The site where the operator is applied.

  • operator (np.ndarray) – The one-site operator

Notes

An example of a one-site expectation value is shown in the following diagram:

( )-<>-( )-<>-( )-<>-( )
 |      |      |      |
 |     (o)     |      |
 |      |      |      |
( )-<>-( )-<>-( )-<>-( )
one_site_left_iso(site: int) → ndarray#

Computes a one-site left isometry at a given site: diag(Λ[i]) @ Γ[i] -> scale vL axis by Λ[i].

one_site_left_iso_iter() → Iterable#

Iterator over the left isometries for every site.

one_site_right_iso(site: int) → ndarray#

Computes a one-site right isometry at a given site: Γ[i] @ diag(Λ[i+1]) -> scale vR axis by Λ[i+1].

one_site_right_iso_iter() → Iterable#

Iterator over the right isometries for every site.

property phys_dimensions: List[int]#

Returns the list of all physical dimensions of the MPS.

reverse() → ExplicitMPS#

Returns a reversed version of the current MPS.

right_canonical() → CanonicalMPS#

Returns the current MPS in the right-canonical form given the MPS in the explicit form.

(see eq.19 in https://scipost.org/10.21468/SciPostPhysLectNotes.5 for reference),

two_site_expectation_value(site: int, operator: ndarray) → float | complex128#

Computes an expectation value of an arbitrary two-site operator (not necessarily unitary) on the given site and its next neighbour. The operator has legs (UL, DL, UR, DR), where L, R, U, D stand for “left”, “right”, “up”, “down” accordingly.

Parameters:

  • site (int) – The first site where the operator is applied, the second site to be site + 1.

  • operator (np.ndarray) – The two-site operator.

Notes

An example of a two-site expectation value is shown in the following diagram:

( )-<>-( )-<>-( )-<>-( )
 |      |      |      |
 |     (operator)     |
 |      |      |      |
( )-<>-( )-<>-( )-<>-( )
two_site_left_iso(site: int) → ndarray#

Two-site left isometry from two one-site left isometries.

two_site_left_iso_iter() → Iterable#

Iterator over the two-site left isometries for every site and its right neighbour.

two_site_right_iso(site: int) → ndarray#

Two-site right isometry from two one-site right isometries.

two_site_right_iso_iter() → Iterable#

Iterator over the two-site right isometries for every site and its right neighbour.