EdgeVertexOperator¶

class EdgeVertexOperator(coeffs, left_indices, right_indices, boundaries)¶

Bases: object

An edge-vertex operator.

Definition¶

This operator is defined in terms of the edge-vertex (\(E_{jk}\), \(V_j\)) operators:

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

which fulfill the following mixed fermionic-bosonic commutation relations for \(j \neq k \neq l \neq m\): [1]

\[\begin{align} \left\{ E_{jk}, V_k \right\} &= 0 \nonumber \\ \left\{ E_{jk}, E_{kl} \right\} &= 0 \nonumber \\ \left[ V_k, V_l \right] &= 0 \nonumber \\ \left[ E_{jk}, V_l \right] &= 0 \nonumber \\ \left[ E_{jk}, E_{lm} \right] &= 0 \nonumber \, . \end{align}\]

In summary, edge and vertex operators commute, unless they share an index, in which case they anticommute. A simple example can be represented visually like so:

(png, hires.png, pdf)

A visual depication of an edge-vertex operator.

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

\[\text{\texttt{EdgeVertexOperator}} = \sum_i c_i \bigotimes_{lr} E_{lr} \, ,\]

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 \(E_{lr}\) as generalized edge 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 may 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 may 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 EdgeVertexOperator
>>> 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 = EdgeVertexOperator(coeffs, left_indices, right_indices, boundaries)
>>> print(format(op))
  1.000000e0 +0.000000e0j * ()
  2.000000e0 +0.000000e0j * (E(0,1))
  0.000000e0 +4.000000e0j * (E(0,1) V(2))
 -0.000000e0-5.000000e-1j * (V(3) E(0,1))
 -3.000000e0 +0.000000e0j * (E(3,4))

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

>>> op = EdgeVertexOperator.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 * (E(0,1))
  0.000000e0 +4.000000e0j * (E(0,1) V(2))
 -0.000000e0-5.000000e-1j * (V(3) E(0,1))
 -3.000000e0 +0.000000e0j * (E(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 * (E(0,1))
  0.000000e0 +4.000000e0j * (E(0,1) V(2))
 -0.000000e0-5.000000e-1j * (V(3) E(0,1))
 -3.000000e0 +0.000000e0j * (E(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))
EdgeVertexOperator.from_dict({...})

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

>>> print(str(op))
<EdgeVertexOperator 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.edge_vertex_operator.EdgeVertexOperator' 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 = EdgeVertexOperator.one()
>>> (op + op).simplify()
EdgeVertexOperator.from_dict({(): 2+0j})
>>> (op - op).simplify()
EdgeVertexOperator.from_dict({})
>>> op += op
>>> op.simplify()
EdgeVertexOperator.from_dict({(): 2+0j})
>>> op -= op
>>> op.simplify()
EdgeVertexOperator.from_dict({})

Scalar Multiplication/Divison¶

>>> op = EdgeVertexOperator.one()
>>> (2 * op).simplify()
EdgeVertexOperator.from_dict({(): 2+0j})
>>> (op / 2).simplify()
EdgeVertexOperator.from_dict({(): 0.5+0j})
>>> op *= 2
>>> op.simplify()
EdgeVertexOperator.from_dict({(): 2+0j})
>>> op /= 2
>>> op.simplify()
EdgeVertexOperator.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 = EdgeVertexOperator.from_dict({(): 2.0, ((0, 1),): 3.0})
>>> op2 = EdgeVertexOperator.from_dict({(): 1.5, ((2, 2),): 4.0})
>>> comp = (op1 & op2).simplify()
>>> print(format(comp))
  3.000000e0 +0.000000e0j * ()
  4.500000e0 +0.000000e0j * (E(0,1))
  8.000000e0 +0.000000e0j * (V(2))
  1.200000e1 +0.000000e0j * (V(2) E(0,1))
>>> op2 &= op1
>>> print(format(op2.simplify()))
  3.000000e0 +0.000000e0j * ()
  4.500000e0 +0.000000e0j * (E(0,1))
  1.200000e1 +0.000000e0j * (E(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 * (E(0,1))
  9.000000e0 +0.000000e0j * (E(0,1) E(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`()	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.

Attributes

groups¶

An optional vector of group indices for each term.

For more information refer to the grouping module.

Methods

adjoint()¶

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

The generators of this operator (the vertex and edge operators) are individually Hermitian, so the terms themselves are unchanged by the adjoint; only the coefficients are affected:

the coefficients are complex conjugated

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 EdgeVertexOperator
>>> op = EdgeVertexOperator.from_dict({(): -1.0j, ((0, 0), (0, 1)): 1.0})
>>> adj = op.adjoint()
>>> print(format(adj))
 -0.000000e0 +1.000000e0j * ()
  1.000000e0 -0.000000e0j * (V(0) E(0,1))

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 EdgeVertexOperator
>>> op = EdgeVertexOperator.from_dict({(): 1e-7})
>>> zero = EdgeVertexOperator.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.

classmethod from_dict(data)¶

Constructs a new operator from a dictionary.

>>> from qiskit_fermions.operators import EdgeVertexOperator
>>> op = EdgeVertexOperator.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 * (E(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 edge operator, \(E_{lr}\) (if \(l = r\) then this corresponds to the vertex operator \(V_l\)).
Returns:: A new operator.

classmethod from_terms(terms)¶

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

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

Parameters:: terms – an iterator of terms as produced by iter_terms().
Returns:: A new operator.

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 EdgeVertexOperator
>>> op = EdgeVertexOperator([2.0, 1.0, -1.0j], [0, 0], [0, 1], [0, 0, 1, 2])
>>> op.groups = [0, 1, 1]
>>> reconstructed = EdgeVertexOperator.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()¶

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 EdgeVertexOperator
>>> op = EdgeVertexOperator.one()
>>> op += EdgeVertexOperator.from_dict({((0, 1),): 1.0})
>>> op.get_boundaries()
[0, 0, 1]

Returns:: A list of the operator’s terms boundaries.

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 EdgeVertexOperator
>>> op = EdgeVertexOperator.one()
>>> op += -1j * EdgeVertexOperator.one()
>>> op.get_coeffs()
[(1+0j), -1j]

Returns:: A list of the operator’s coefficients.

get_left_indices()¶

Returns a read-only list of the left indices of all generalized edge 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 EdgeVertexOperator
>>> op = EdgeVertexOperator.from_dict({((0, 0),): 1.0})
>>> op += EdgeVertexOperator.from_dict({((0, 1),): 1.0})
>>> op.get_left_indices()
[0, 0]

Returns:: A list of the left indices of all generalized edge operator terms.

get_right_indices()¶

Returns a read-only list of the right indices of all generalized edge 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 EdgeVertexOperator
>>> op = EdgeVertexOperator.from_dict({((0, 0),): 1.0})
>>> op += EdgeVertexOperator.from_dict({((0, 1),): 1.0})
>>> op.get_right_indices()
[0, 1]

Returns:: A list of the right indices of all generalized edge operator terms.

get_support()¶

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

>>> from qiskit_fermions.operators import EdgeVertexOperator
>>> op = EdgeVertexOperator.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.

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 EdgeVertexOperator
>>> op = EdgeVertexOperator.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 * (E(0,1))
  1.000000e-6 +0.000000e0j * (E(1,0))
>>> op.ichop()
>>> print(format(op))
  1.000000e-4 +0.000000e0j * ()
  1.000000e-6 +0.000000e0j * (E(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(atol=1e-08)¶

Returns whether this operator is Hermitian.

Note

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

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()¶

An iterator over the operator’s terms.

Warning

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

>>> from qiskit_fermions.operators import EdgeVertexOperator
>>> op = EdgeVertexOperator.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()¶

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 EdgeVertexOperator
>>> op = EdgeVertexOperator([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()¶

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 edge 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.

>>> from qiskit_fermions.operators import EdgeVertexOperator
>>> op = EdgeVertexOperator.from_dict({((0, 1), (1, 0), (1, 2), (0, 0), (2, 2)): 1})
>>> print(format(op.normal_ordered().simplify()))
 -1.000000e0 -0.000000e0j * (V(0) V(2) E(0,1) E(1,0) E(1,2))

Returns:: An equivalent but normal-ordered operator.

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 EdgeVertexOperator
>>> op = EdgeVertexOperator(
...     [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.

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 EdgeVertexOperator
>>> op = EdgeVertexOperator.from_dict({(): 2.0})
>>> one = EdgeVertexOperator.one()
>>> op & one == op
True

relabel_modes(permutation)¶

Returns a new operator with relabeled modes.

>>> from qiskit_fermions.operators import EdgeVertexOperator
>>> op = EdgeVertexOperator.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 * (E(2,5) E(3,4))
  1.000000e0 +0.000000e0j * (E(4,2) E(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(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 EdgeVertexOperator
>>> coeffs = [1e-5] * int(1e5)
>>> boundaries = [0] + [0] * int(1e5)
>>> op = EdgeVertexOperator(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()¶

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

If groups is None, this function also returns None. Otherwise, it will return a list of new operators that contain those terms of this operator with the corresponding group index.

>>> from qiskit_fermions.operators import EdgeVertexOperator
>>> op = EdgeVertexOperator(
...     [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))]

Returns:: An optional vector of one new operator for each group index in groups.

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 EdgeVertexOperator
>>> op = EdgeVertexOperator.from_dict({(): 2.0})
>>> zero = EdgeVertexOperator.zero()
>>> op + zero == op
True