TransferVertexOperator

class TransferVertexOperator(coeffs, left_indices, right_indices, boundaries)

Bases: object

A transfer-vertex operator.

Definition

This operator is defined in terms of the transfer-vertex ( $T_{jk}$ , $V_j$ ) operators:

\begin{align} V_j &= -i \gamma_{2j-1} \gamma_{2j} = -(a_j a_j - a_j a^\dagger_j + a^\dagger_j a_j - a^\dagger_j a^\dagger_j) = 1 - 2 a^\dagger_j a_j \, , \nonumber \\ T_{jk} &= \frac{i}{2} V_j E_{jk} = \frac{i}{2} \gamma_{2j} \gamma_{2k-1} \, , \nonumber \\ T_{kj} &= \frac{i}{2} E_{jk} V_k = -\frac{i}{2} \gamma_{2j-1} \gamma_{2k} \nonumber \end{align}

where $E_{jk}$ is an edge operator of the EdgeVertexOperator and these individual terms fulfill the following mixed fermionic-bosonic commutation relations for $j \lt k \lt l \lt m$ : [1]

\begin{align} \left\{ T_{jk}, V_k \right\} &= 0 \nonumber \\ \left\{ T_{jk}, T_{lk} \right\} &= 0 \nonumber \\ \left[ V_k, V_l \right] &= 0 \nonumber \\ \left[ T_{jk}, V_l \right] &= 0 \nonumber \\ \left[ T_{jk}, T_{lm} \right] &= 0 \nonumber \\ \left[ T_{jk}, T_{kj} \right] &= 0 \nonumber \\ \left[ T_{jk}, T_{km} \right] &= 0 \nonumber \, . \end{align}

A simple example can be represented visually like so:

A visual depication of a transfer-vertex operator.

We can abuse the notation a little bit and define $V_j = T_{jj}$ which reflects how the internal data structure of this operator works. This makes the definition of the entire operator the following:

\text{\texttt{EdgeVertexOperator}} = \sum_i c_i \bigotimes_{lr} T_{lr} \, ,

where $lr$ indexing the involved operator terms and $c_i$ is the (complex) coefficient making up the linear combination of products. The indices $l$ and $r$ can take any value between 0 and the number of fermionic modes acted upon by the operator minus 1.

We will refer to $T_{lr}$ as generalized transfer operators.

Implementation

This class stores the terms and coefficients in multiple sparse vectors, akin to the compressed sparse row format commonly used for sparse matrices. More concretely, a single operator contains 4 arrays:


`coeffs`	A vector of complex coefficients consisting of two 64-bit floating point numbers.
`left_indices`	A vector of 32-bit integers storing the left fermionic mode indices ( $l$ above).
`right_indices`	A vector of 32-bit integers storing the right fermionic mode indices ( $r$ above).
`boundaries`	A vector of integers indicating the boundaries in `actions` and `modes`.

Fermionic modes indexed by left_indices and right_indices are considered spinless.

Note

You can access read-only copies of these internal arrays via their respective methods: get_coeffs(), get_left_indices(), get_right_indices(), and get_boundaries().

This data structure allows for very efficient construction and manipulation of operators. However, it implies that duplicate terms might be contained in an operator at any moment. These must be resolved manually through the use of simplify().

Construction

An operator can be constructed directly by providing the arrays outlined above:

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> coeffs = [1.0, 2.0, -3.0, 4.0j, -0.5j]
>>> left_indices = [0, 3, 0, 2, 3, 0]
>>> right_indices = [1, 4, 1, 2, 3, 1]
>>> boundaries = [0, 0, 1, 2, 4, 6]
>>> op = TransferVertexOperator(coeffs, left_indices, right_indices, boundaries)
>>> print(format(op))
  1.000000e0 +0.000000e0j * ()
  2.000000e0 +0.000000e0j * (T(0,1))
  0.000000e0 +4.000000e0j * (T(0,1) V(2))
 -0.000000e0-5.000000e-1j * (V(3) T(0,1))
 -3.000000e0 +0.000000e0j * (T(3,4))

For convenience, it is possible to construct an operator from a Python dictionary like so:

>>> op = TransferVertexOperator.from_dict(
...     {
...         (): 1.0,
...         ((0, 1),): 2.0,
...         ((3, 4),): -3.0,
...         ((0, 1), (2, 2)): 4.0j,
...         ((3, 3), (0, 1)): -0.5j,
...     }
... )
>>> print(format(op))
  1.000000e0 +0.000000e0j * ()
  2.000000e0 +0.000000e0j * (T(0,1))
  0.000000e0 +4.000000e0j * (T(0,1) V(2))
 -0.000000e0-5.000000e-1j * (V(3) T(0,1))
 -3.000000e0 +0.000000e0j * (T(3,4))

In addition, the following construction and quick helper methods are available:


`zero`()	Constructs the additive identity operator.
`one`()	Constructs the multiplicative identity operator.
`from_terms`(terms)	Constructs a new operator from an iterator of terms (see also `iter_terms()`).
`from_terms_with_groups`(terms)	Constructs a new operator from an iterator of terms with groups (see also `iter_terms_with_groups()`).

Formatting

In the examples above, the constructed operators have been printed using the output from format(), which results in a human-readable form of the operator.

>>> print(format(op))
  1.000000e0 +0.000000e0j * ()
  2.000000e0 +0.000000e0j * (T(0,1))
  0.000000e0 +4.000000e0j * (T(0,1) V(2))
 -0.000000e0-5.000000e-1j * (V(3) T(0,1))
 -3.000000e0 +0.000000e0j * (T(3,4))

Note

The printing order of format(op) gets explicitly sorted before printing. As such, it does not reflect the order of the terms inside the operator.

An alternative form can be obtained from the repr() function, which results in a Python-interpretable representation. In other words, this output can readily be copied and pasted into a Python shell:

>>> print(repr(op))
TransferVertexOperator.from_dict({...})

Finally, for large operators both of these outputs might be very long and undesirable. Then, a very simple form with minimal information can be obtained from the str() function:

>>> print(str(op))
<TransferVertexOperator with 5 terms>

Iteration

Since the underlying data structure is implemented in Rust and has a non-trivial layout, it cannot be iterated over directly:

>>> list(iter(op))
Traceback (most recent call last):
  ...
TypeError: 'qiskit_fermions.operators.transfer_vertex_operator.TransferVertexOperator' object is not iterable

Instead, this class provides custom iterators to fulfill this purpose:

>>> list(sorted(op.iter_terms()))
[([], (1+0j)), ([(0, 1)], (2+0j)), ([(0, 1), (2, 2)], 4j), ([(3, 3), (0, 1)], (-0-0.5j)), ([(3, 4)], (-3+0j))]

See also

iter_terms()

For more relevant implementation details.

The table below lists all available iterators:


`iter_terms`()	An iterator over the operator's terms.
`iter_terms_with_groups`()	An iterator over the operator's terms with their associated group index.

Arithmetics

The following arithmetic operations are supported:

Addition/Subtraction

>>> op = TransferVertexOperator.one()
>>> (op + op).simplify()
TransferVertexOperator.from_dict({(): 2+0j})
>>> (op - op).simplify()
TransferVertexOperator.from_dict({})
>>> op += op
>>> op.simplify()
TransferVertexOperator.from_dict({(): 2+0j})
>>> op -= op
>>> op.simplify()
TransferVertexOperator.from_dict({})

Scalar Multiplication/Divison

>>> op = TransferVertexOperator.one()
>>> (2 * op).simplify()
TransferVertexOperator.from_dict({(): 2+0j})
>>> (op / 2).simplify()
TransferVertexOperator.from_dict({(): 0.5+0j})
>>> op *= 2
>>> op.simplify()
TransferVertexOperator.from_dict({(): 2+0j})
>>> op /= 2
>>> op.simplify()
TransferVertexOperator.from_dict({(): 1+0j})

Operator Composition

Note

Operator composition corresponds to left-multiplication: c = a & b corresponds to $C = B A$ . In other words, the composition of two operators returns a resulting operator that performs “first a and then b”.

>>> op1 = TransferVertexOperator.from_dict({(): 2.0, ((0, 1),): 3.0})
>>> op2 = TransferVertexOperator.from_dict({(): 1.5, ((2, 2),): 4.0})
>>> comp = (op1 & op2).simplify()
>>> print(format(comp))
  3.000000e0 +0.000000e0j * ()
  4.500000e0 +0.000000e0j * (T(0,1))
  8.000000e0 +0.000000e0j * (V(2))
  1.200000e1 +0.000000e0j * (V(2) T(0,1))
>>> op2 &= op1
>>> print(format(op2.simplify()))
  3.000000e0 +0.000000e0j * ()
  4.500000e0 +0.000000e0j * (T(0,1))
  1.200000e1 +0.000000e0j * (T(0,1) V(2))
  8.000000e0 +0.000000e0j * (V(2))
>>> squared = (op1 ** 2).simplify()
>>> print(format(squared))
  4.000000e0 +0.000000e0j * ()
  1.200000e1 +0.000000e0j * (T(0,1))
  9.000000e0 +0.000000e0j * (T(0,1) T(0,1))

Note

For convenience, the right-multiplication is implemented by c = a @ b (resulting in $C = A B$ ).

>>> (op1 @ op2).equiv(op2 & op1)
True

Other Operations

In addition to the magic methods that correspond to the arithmetic operations outlined above, the following methods are available:


`adjoint`()	Returns the Hermitian conjugate (or adjoint) of this operator.
`ichop`([atol])	Removes terms whose coefficient magnitude lies below the provided threshold.
`simplify`([atol])	Returns an equivalent but simplified operator.
`normal_ordered`([reduce])	Returns an equivalent operator with normal ordered terms.
`relabel_modes`(permutation)	Returns a new operator with relabeled modes.

Properties

Finally, various methods exist to check certain properties of an operator:


`is_hermitian`([atol])	Returns whether this operator is Hermitian.

[1]

Gandon et al., arXiv:2512.11418.

Attributes

groups

An optional vector of group indices for each term.

For more information refer to the grouping module.

Methods

adjoint

adjoint()

Returns the Hermitian conjugate (or adjoint) of this operator.

Two things happen to every term:

the coefficients are complex conjugated
the generators within each term are reversed in order

The reversal is required because $(AB)^\dagger = B^\dagger A^\dagger$ . While the individual vertex and transfer generators are Hermitian, they anticommute when they share an index (see the definition above), so the reversed product is not equal to the original one and the order cannot simply be dropped.

Note that this does not make the operator self-adjoint in general: an operator with complex coefficients differs from its adjoint (as the doctest below illustrates).

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict({(): -1.0j, ((0, 0), (0, 1)): 1.0})
>>> adj = op.adjoint()
>>> print(format(adj))
 -0.000000e0 +1.000000e0j * ()
  1.000000e0 -0.000000e0j * (T(0,1) V(0))

equiv

equiv(other, atol=1e-08)

Checks this operator for equivalence with another operator.

Equivalence in this context means approximate equality up to the specified absolute tolerance. To be more precise, this method returns True, when all the absolute values of the coefficients in the difference other - self are below the specified threshold atol.

Note

This is the mathematical comparison you almost always want. It differs from the == operator, which tests exact equality of the stored terms (their coefficients, indices, and internal term boundaries) with no tolerance and no simplification. Two mathematically equal operators can therefore compare unequal under == if they are stored differently – for example an unsimplified a + a versus 2 * a, or terms held in a different order. Use equiv to compare operators up to numerical tolerance.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict({(): 1e-7})
>>> zero = TransferVertexOperator.zero()
>>> op.equiv(zero)
False
>>> op.equiv(zero, 1e-6)
True
>>> op.equiv(zero, 1e-9)
False

Parameters

other – the other operator to compare with.
atol – the absolute tolerance for the comparison. This value defaults to 1e-8.

from_dict

classmethod from_dict(data)

Constructs a new operator from a dictionary.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict(
...     {
...         (): 1.0-1.0j,
...         ((0, 0),): 2.0,
...         ((0, 1),): 2.0j,
...     }
... )
>>> print(format(op))
  1.000000e0 -1.000000e0j * ()
  2.000000e0 +0.000000e0j * (V(0))
  0.000000e0 +2.000000e0j * (T(0,1))

Parameters

data – a dictionary mapping tuples of terms to complex coefficients. Each key is a tuple of (int, int) pairs indicating the indices of the generalized transfer operator, $T_{lr}$ (if $l = r$ then this corresponds to the vertex operator $V_l$ ).

Returns

A new operator.

from_terms

classmethod from_terms(terms)

Constructs a new operator from an iterator of terms (see also iter_terms()).

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict({(): 2.0, ((0, 0),): 1.0, ((0, 1),): -1.0j})
>>> op.equiv(TransferVertexOperator.from_terms(op.iter_terms()))
True

Parameters

terms – an iterator of terms as produced by iter_terms().

Returns

A new operator.

from_terms_with_groups

classmethod from_terms_with_groups(terms)

Constructs a new operator from an iterator of terms with groups (see also iter_terms_with_groups()).

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator([2.0, 1.0, -1.0j], [0, 0], [0, 1], [0, 0, 1, 2])
>>> op.groups = [0, 1, 1]
>>> reconstructed = TransferVertexOperator.from_terms_with_groups(op.iter_terms_with_groups())
>>> op.equiv(reconstructed) and op.groups == reconstructed.groups
True

Parameters

terms – an iterator of terms as produced by iter_terms_with_groups().

Returns

A new operator.

get_boundaries

get_boundaries()

Returns a read-only list of the indices indicating the boundaries between operator terms.

Note

This method returns a copy of the internal data.

See also

The explanation of the internal data structure, here.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.one()
>>> op += TransferVertexOperator.from_dict({((0, 1),): 1.0})
>>> op.get_boundaries()
[0, 0, 1]

Returns

A list of the operator’s terms boundaries.

get_coeffs

get_coeffs()

Returns a read-only list of the operator’s coefficients.

Note

This method returns a copy of the internal data.

See also

The explanation of the internal data structure, here.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.one()
>>> op += -1j * TransferVertexOperator.one()
>>> op.get_coeffs()
[(1+0j), -1j]

Returns

A list of the operator’s coefficients.

get_left_indices

get_left_indices()

Returns a read-only list of the left indices of all generalized transfer operator terms.

Note

This method returns a copy of the internal data.

See also

The explanation of the internal data structure, here.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict({((0, 0),): 1.0})
>>> op += TransferVertexOperator.from_dict({((0, 1),): 1.0})
>>> op.get_left_indices()
[0, 0]

Returns

A list of the left indices of all generalized transfer operator terms.

get_right_indices

get_right_indices()

Returns a read-only list of the right indices of all generalized transfer operator terms.

Note

This method returns a copy of the internal data.

See also

The explanation of the internal data structure, here.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict({((0, 0),): 1.0})
>>> op += TransferVertexOperator.from_dict({((0, 1),): 1.0})
>>> op.get_right_indices()
[0, 1]

Returns

A list of the right indices of all generalized transfer operator terms.

get_support

get_support()

Returns the set of mode indices which this operator acts upon.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict(
...     {
...         ((0, 1), (3, 4)): 1,
...         ((7, 7),): 1,
...     }
... )
>>> assert op.get_support() == {0, 1, 3, 4, 7}

Returns

The set of mode indices which this operator acts upon.

group_weights

group_weights()

Returns the mean absolute coefficient magnitude of each group.

The i-th entry is the sum of abs(coeff) over the terms in group i, divided by the number of terms in that group. If groups is None, this function also returns None.

This is the sampling weight of a randomized product formula (e.g. qDRIFT) that draws whole groups rather than individual terms. Computing it natively is considerably cheaper than reducing get_coeffs() and groups in NumPy, because those two accessors each copy one value per ungrouped term out of the operator only for it to be aggregated back down to one value per group, whereas this returns just the num_groups() reduced values.

Note

A group index that no term carries weighs 0.0, which keeps it out of the sample.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator(
...     [1.0, 2.0, -1.0],
...     [0, 1, 2, 3],
...     [1, 0, 3, 2],
...     [0, 1, 3, 4],
... )
>>> print(op.group_weights())
None
>>> op.groups = [0, 1, 0]
>>> op.group_weights()
[1.0, 2.0]

Returns

The mean absolute coefficient magnitude of each group index.

has_groups

has_groups()

Returns whether this operator tracks group indices.

This is equivalent to (but cheaper than) checking op.groups is not None, because it does not copy the group indices out of the operator in order to inspect them.

Note

This returns True even when groups is an empty list, which is the state of a grouped operator that holds no terms.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator(
...     [1.0, 2.0, -1.0],
...     [0, 1, 2, 3],
...     [1, 0, 3, 2],
...     [0, 1, 3, 4],
... )
>>> op.has_groups()
False
>>> op.groups = [0, 1, 0]
>>> op.has_groups()
True

Returns

Whether groups is set on this operator.

ichop

ichop(atol=1e-08)

Removes terms whose coefficient magnitude lies below the provided threshold.

This method modifies the operator in place and returns None.

Caution

This method truncates coefficients greedily! If the acted upon operator may contain separate coefficients for duplicate terms consider calling simplify() instead!

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict({(): 1e-4, ((1, 0),): 1e-6, ((0, 1),): 1e-10})
>>> print(format(op))
  1.000000e-4 +0.000000e0j * ()
 1.000000e-10 +0.000000e0j * (T(0,1))
  1.000000e-6 +0.000000e0j * (T(1,0))
>>> op.ichop()
>>> print(format(op))
  1.000000e-4 +0.000000e0j * ()
  1.000000e-6 +0.000000e0j * (T(1,0))
>>> op.ichop(1e-5)
>>> print(format(op))
  1.000000e-4 +0.000000e0j * ()

Parameters

atol – the absolute tolerance for the cutoff. This value defaults to 1e-8.

is_hermitian

is_hermitian(atol=1e-08)

Returns whether this operator is Hermitian.

Note

This check is implemented using equiv() on the fully reduced normal_ordered() difference of self and its adjoint() and zero().

Note

This check is conservative: it can return False for an operator that is in fact Hermitian. A True result is always reliable.

normal_ordered() contracts every pair of adjacent generators that combines into a shorter term, but two transfer operators sharing a single mode do not: as explained there, $T_{jk} T_{jl}$ can only be rewritten into another length-two product. Since neither form is more canonical than the other, such products are left alone, and a term that would only cancel against its adjoint after rewriting one is not recognized as zero. EdgeVertexOperator has no such gap, because there the analogous product fuses into a single edge operator.

Parameters

atol – The numerical accuracy upto which coefficients are considered equal. This value defaults to 1e-8.

Returns

Whether this operator is Hermitian.

iter_terms

iter_terms()

An iterator over the operator’s terms.

Warning

Mutating the iteration items does not affect the underlying operator data.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict({(): 2.0, ((0, 0),): 1.0, ((0, 1),): -1.0j})
>>> list(sorted(op.iter_terms()))
[([], (2+0j)), ([(0, 0)], (1+0j)), ([(0, 1)], (-0-1j))]

iter_terms_with_groups

iter_terms_with_groups()

An iterator over the operator’s terms with their associated group index.

Warning

Mutating the iteration items does not affect the underlying operator data.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator([2.0, 1.0, -1.0j], [0, 0], [0, 1], [0, 0, 1, 2])
>>> op.groups = [0, 1, 1]
>>> list(op.iter_terms_with_groups())
[([], (2+0j), 0), ([(0, 0)], (1+0j), 1), ([(0, 1)], (-0-1j), 1)]

normal_ordered

normal_ordered(reduce=True)

Returns an equivalent operator with normal ordered terms.

The normal order of an operator term is defined such that all vertex operators appear before all transfer operators. Within each group, the acted-upon modes are ordered lexicographically.

Note

When a term is being reordered, the mixed commutation and anti-commutation relations have to be taken into account. See here for the detailed definitions.

Note

Fewer contractions are available here than for EdgeVertexOperator. Two edge operators sharing a single mode fuse into one, but the analogous transfer product does not: writing $T_{jk} = \frac{i}{2} \gamma_{2j} \gamma_{2k-1}$ , cancelling the shared Majorana always leaves an (even, odd) pair drawn from two different modes, which is not a single generator. Only V(j) V(j), T(j,k) T(j,k) and T(j,k) T(k,j) contract.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict({((0, 1), (1, 0), (1, 2), (0, 0), (2, 2)): 1})
>>> print(format(op.normal_ordered(reduce=False).simplify()))
 -1.000000e0 -0.000000e0j * (V(0) V(2) T(0,1) T(1,0) T(1,2))
>>> print(format(op.normal_ordered().simplify()))
 2.500000e-1 +0.000000e0j * (V(1) V(2) T(1,2))

Parameters

reduce – whether to contract adjacent generators that combine into a scalar or into a pair of vertex operators. See the example above. This value defaults to True.

Returns

An equivalent but normal-ordered operator.

num_groups

num_groups()

Returns the number of groups.

If groups is None, this function also returns None. Otherwise, it will return the number of groups which is defined to be the largest occurring group index plus 1 (which may therefore be used as the index for the next group).

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator(
...     [1.0, 2.0, -1.0],
...     [0, 1, 2, 3],
...     [1, 0, 3, 2],
...     [0, 1, 3, 4],
... )
>>> op.groups = [0, 1, 0]
>>> op.num_groups()
2

Returns

The largest group index in groups plus 1.

one

classmethod one()

Constructs the multiplicative identity operator.

Composing the operator that is constructed by this method with another one has no effect.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict({(): 2.0})
>>> one = TransferVertexOperator.one()
>>> op & one == op
True

relabel_modes

relabel_modes(permutation)

Returns a new operator with relabeled modes.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict({
...     ((0, 1), (2, 3)): 1,
...     ((1, 2), (3, 0)): 1,
... })
>>> permutation = [4, 2, 5, 3]
>>> relabeled = op.relabel_modes(permutation)
>>> print(format(relabeled))
  1.000000e0 +0.000000e0j * (T(2,5) T(3,4))
  1.000000e0 +0.000000e0j * (T(4,2) T(5,3))

Parameters

permutation – the index permutation list. Mode i is relabeled to permutation[i], so the list must contain no duplicate entries and must be long enough to index every mode the operator acts upon (its length must exceed the operator’s largest mode index).

Returns

A new operator with its modes relabeled.

Raises

ValueError – if permutation contains duplicate entries, or is too short to relabel some mode the operator acts upon.

simplify

simplify(atol=1e-08)

Returns an equivalent but simplified operator.

The simplification process first sums all coefficients that belong to equal terms and then only retains those whose total coefficient exceeds the specified tolerance (just like ichop()).

When an operator has been arithmetically manipulated or constructed in a way that does not guarantee unique terms, this method should be called before applying any method that filters numerically small coefficients to avoid loss of information. See the example below which showcases how ichop() can truncate terms that sum to a total coefficient magnitude which should not be truncated:

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> coeffs = [1e-5] * int(1e5)
>>> boundaries = [0] + [0] * int(1e5)
>>> op = TransferVertexOperator(coeffs, [], [], boundaries)
>>> canon = op.simplify(1e-4)
>>> assert canon.equiv(op.one(), 1e-6)
>>> op.ichop(1e-4)
>>> assert op.equiv(op.zero(), 1e-6)

Parameters

atol – the absolute tolerance for the cutoff. This value defaults to 1e-8.

Returns

An equivalent but simplified operator.

split_out_groups

split_out_groups(group_indices=None)

Splits this operator into an optional list of new operators based on groups.

If groups is None, this function also returns None. Otherwise, if group_indices is None (the default), it returns a list of one new operator for every group index in groups, in index order. If group_indices is given, only the requested indices are built, in the given order: this avoids the cost of constructing operators for groups that are never used, which is especially beneficial when only a small number of groups out of a much larger total are needed, e.g. when subsampling groups for a randomized product formula. A duplicate index in group_indices is returned once per occurrence.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator(
...     [1.0, 2.0, -1.0],
...     [0, 1, 2, 3],
...     [1, 0, 3, 2],
...     [0, 1, 3, 4],
... )
>>> print(op.split_out_groups())
None
>>> op.groups = [0, 1, 0]
>>> groups = op.split_out_groups()
>>> for g in groups:
...     print(list(sorted(g.iter_terms())))
[([(0, 1)], (1+0j)), ([(3, 2)], (-1+0j))]
[([(1, 0), (2, 3)], (2+0j))]
>>> groups = op.split_out_groups(group_indices=[1])
>>> for g in groups:
...     print(list(sorted(g.iter_terms())))
[([(1, 0), (2, 3)], (2+0j))]

Parameters

group_indices – the group indices for which to build operators, in the desired output order. When omitted, every group is built, in index order.

Returns

An optional vector of one new operator for each requested group index.

zero

classmethod zero()

Constructs the additive identity operator.

Adding the operator that is constructed by this method to another one has no effect.

>>> from qiskit_fermions.operators import TransferVertexOperator
>>> op = TransferVertexOperator.from_dict({(): 2.0})
>>> zero = TransferVertexOperator.zero()
>>> op + zero == op
True

Protocol Methods

_anti_commutator_

static _anti_commutator_(op_a, op_b)

_commutator_

static _commutator_(op_a, op_b)

_double_commutator_

static _double_commutator_(op_a, op_b, op_c, sign)

_fermion_operator_

_fermion_operator_()

Converts this operator into a FermionOperator.

This implements the SupportsFermionOperator protocol by delegating to transfer_vertex_to_fermion().

_majorana_operator_

_majorana_operator_()

Converts this operator into a MajoranaOperator.

This implements the SupportsMajoranaOperator protocol by delegating to transfer_vertex_to_majorana().