FermionicOp#

class FermionicOp(data, num_spin_orbitals=None, *, copy=True, validate=True)[source]#

Bases: SparseLabelOp

N-mode Fermionic operator.

A FermionicOp represents a weighted sum of fermionic creation/annihilation operator terms. These terms are encoded as sparse labels, which are strings consisting of a space-separated list of expressions. Each expression must look like [+-]_<index>, where the <index> is a non-negative integer representing the index of the fermionic mode where the + (creation) or - (annihilation) operation is to be performed. The value of index is bound by the number of spin orbitals (num_spin_orbitals) of the operator (Note: since Python indices are 0-based, the maximum value an index can take is given by num_spin_orbitals-1).

Initialization

A FermionicOp is initialized with a dictionary, mapping terms to their respective coefficients:

from qiskit_nature.second_q.operators import FermionicOp

op = FermionicOp(
    {
        "+_0 -_0": 1.0,
        "+_1 -_1": -1.0,
    },
    num_spin_orbitals=2,
)

By default, this way of initializing will create a full copy of the dictionary of coefficients. If you have very restricted memory resources available, or would like to avoid the additional copy, the dictionary will be stored by reference if you disable copy like so:

some_big_data = {
    "+_0 -_0": 1.0,
    "+_1 -_1": -1.0,
    # ...
}

op = FermionicOp(
    some_big_data,
    num_spin_orbitals=2,
    copy=False,
)

Note

It is the users’ responsibility, that in the above scenario, some_big_data is not changed after initialization of the FermionicOp, since the operator contents are not guaranteed to remain unaffected by such changes.

Algebra

This class supports the following basic arithmetic operations: addition, subtraction, scalar multiplication, operator multiplication, and adjoint. For example,

Addition

FermionicOp({"+_1": 1}, num_spin_orbitals=2) + FermionicOp({"+_0": 1}, num_spin_orbitals=2)

Sum

sum(FermionicOp({label: 1}, num_spin_orbitals=3) for label in ["+_0", "-_1", "+_2 -_2"])

Scalar multiplication

0.5 * FermionicOp({"+_1": 1}, num_spin_orbitals=2)

Operator multiplication

op1 = FermionicOp({"+_0 -_1": 1}, num_spin_orbitals=2)
op2 = FermionicOp({"-_0 +_0 +_1": 1}, num_spin_orbitals=2)
print(op1 @ op2)

Tensor multiplication

op = FermionicOp({"+_0 -_1": 1}, num_spin_orbitals=2)
print(op ^ op)

Adjoint

FermionicOp({"+_0 -_1": 1j}, num_spin_orbitals=2).adjoint()

Iteration

Instances of FermionicOp are iterable. Iterating a FermionicOp yields (term, coefficient) pairs describing the terms contained in the operator.

num_spin_orbitals#

the number of spin orbitals on which this operator acts. This is considered a lower bound, which means that mathematical operations acting on two or more operators will result in a new operator with the maximum number of spin orbitals of any of the involved operators.

Type:: int | None

Note

A FermionicOp can contain qiskit.circuit.ParameterExpression objects as coefficients. However, a FermionicOp containing parameters does not support the following methods:

is_hermitian
to_matrix

Parameters:

data (Mapping[str, _TCoeff]) – the operator data, mapping string-based keys to numerical values.
num_spin_orbitals (int | None) – the number of spin orbitals on which this operator acts.
copy (bool) – when set to False the data will not be copied and the dictionary will be stored by reference rather than by value (which is the default; copy=True). Note, that this requires you to not change the contents of the dictionary after constructing the operator. This also implies validate=False. Use with care!
validate (bool) – when set to False the data keys will not be validated. Note, that the SparseLabelOp base class, makes no assumption about the data keys, so will not perform any validation by itself. Only concrete subclasses are encouraged to implement a key validation method. Disable this setting with care!

Raises:

QiskitNatureError – when an invalid key is encountered during validation.

Attributes

atol = 1e-08#

register_length#

rtol = 1e-05#

Methods

adjoint()#

Return the adjoint of the Operator.

Return type:: Self

argsort(*, weight=False)#

Returns the keys which sort this operator.

Parameters:: weight (bool) – when True, the returned keys will sort this operator according to the coefficient weights of the stored terms; when False, the keys will sort the operator by its keys (i.e. lexicographically).
Returns:: The sequence of keys which sort this operator.
Return type:: Sequence[str]

assign_parameters(parameters)#

Assign parameters to new parameters or values.

Parameters:: parameters (Mapping[ParameterExpression, _TCoeff]) – The mapping from parameters to new parameters or values.
Returns:: A new operator with the parameters assigned.
Return type:: SparseLabelOp

chop(atol=None)#

Chops the real and imaginary parts of the operator coefficients.

This function separately chops the real and imaginary parts of all coefficients to the provided tolerance. Parameters are chopped only if they are exactly zero.

Parameters:: atol (float | None) – the tolerance to which to chop. If None, atol will be used.
Returns:: The chopped operator.
Return type:: SparseLabelOp

compose(other, qargs=None, front=False)[source]#

Returns the operator composition with another operator.

Parameters:

other (FermionicOp) – the other operator.
qargs – UNUSED.
front (bool) – If True composition uses right operator multiplication, otherwise left multiplication is used (the default).

Returns:

The operator resulting from the composition.

Return type:

FermionicOp

Note

Composition (&) by default is defined as left matrix multiplication for matrix operators, while @ (equivalent to dot()) is defined as right matrix multiplication. This means that A & B == A.compose(B) is equivalent to B @ A == B.dot(A) when A and B are of the same type.

Setting the front=True keyword argument changes this to right matrix multiplication which is equivalent to the dot() method A.dot(B) == A.compose(B, front=True).

conjugate()#

Returns the conjugate of the SparseLabelOp.

Returns:: The complex conjugate of the starting SparseLabelOp.
Return type:: SparseLabelOp

dot(other, qargs=None)#

Return the right multiplied operator self * other.

Parameters:

other (Operator) – an operator object.
qargs (list or None) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None).

Returns:

The right matrix multiplied Operator.

Return type:

Operator

Note

The dot product can be obtained using the @ binary operator. Hence a.dot(b) is equivalent to a @ b.

equiv(other, *, atol=None, rtol=None)#

Check equivalence of two SparseLabelOp instances up to an accepted tolerance.

Parameters:

other (SparseLabelOp) – the second SparseLabelOp to compare with this instance.
atol (float | None) – Absolute numerical tolerance. The default behavior is to use self.atol.
rtol (float | None) – Relative numerical tolerance. The default behavior is to use self.rtol.

Returns:

True if operators are equivalent, False if not.

Raises:

ValueError – Raised if either operator contains parameters

Return type:

bool

expand(other)[source]#

Returns the reverse-order tensor product with another operator.

Parameters:: other (FermionicOp) – the other operator.
Returns:: The operator resulting from the tensor product, \(othr \otimes self\).
Return type:: FermionicOp

classmethod from_polynomial_tensor(tensor)[source]#

Constructs the operator from a PolynomialTensor.

Parameters:: tensor (PolynomialTensor) – the PolynomialTensor to be expanded.
Returns:: The constructed operator.
Return type:: FermionicOp

classmethod from_terms(terms)[source]#

Constructs a new SparseLabelOp from a sequence returned by terms().

Parameters:: terms (Sequence[tuple[list[tuple[str, int]], _TCoeff]]) – a sequence as returned by terms().
Returns:: The constructed operator.
Return type:: FermionicOp

get(k[, d]) → D[k] if k in D, else d. d defaults to None.#

index_order()[source]#

Convert to the equivalent operator with the terms of each label ordered by index.

Returns a new operator (the original operator is not modified).

Note

You can use this method to achieve the most aggressive simplification of an operator without changing the operation order per index. simplify() does not reorder the terms and, thus, cannot deduce -_0 +_1 and +_1 -_0 +_0 -_0 to be identical labels. Calling this method will reorder the latter label to -_0 +_0 -_0 +_1, after which simplify() will be able to correctly collapse these two labels into one.

Returns:: The index ordered operator.
Return type:: FermionicOp

induced_norm(order=1)#

Returns the p-norm induced by the operator coefficients.

If the operator is represented as a sum of terms

\[\sum_i w_i H_i\]

then the induced \(p\)-norm is

\[\left(\sum_i |w_i|^p \right)^{1/p}\]

This is the standard \(p\)-norm of the operator coefficients considered as a vector (see https://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm). Note that this method does not normal-order or simplify the operator before computing the norm; performing either of those operations can affect the result.

Parameters:: order (int) – Order \(p\) of the norm. The default value is 1.
Returns:: The induced norm.
Return type:: float

Raises:: ValueError – Operator contains parameters.
Return type:: float

is_hermitian(atol=None)[source]#

Checks whether the operator is hermitian.

Parameters:: atol (float | None) – Absolute numerical tolerance. The default behavior is to use self.atol.
Returns:: True if the operator is hermitian up to numerical tolerance, False otherwise.
Raises:: ValueError – Operator contains parameters.
Return type:: bool

is_parameterized()#

Returns whether the operator contains any parameters.

Return type:: bool

is_zero(tol=None)#

Returns true if operator length is zero or all coefficients have value zero.

Parameters:: tol (int | None) – tolerance for checking coefficient values. If this is None, atol will be used instead.
Returns:: If operator length is zero or all coefficients are zero.
Return type:: bool

items() → a set-like object providing a view on D's items#

keys() → a set-like object providing a view on D's keys#

normal_order()[source]#

Convert to the equivalent operator in normal order.

The normal order for fermions is defined [here](https://en.wikipedia.org/wiki/Normal_order#Fermions).

Returns a new operator (the original operator is not modified).

Note

This method implements the transformation of an operator to the normal ordered operator. The transformation is calculated by considering all commutation relations between the operators. For example, for the case \(\colon c_0 c_0^\dagger\colon\) where \(c_0\) is an annihilation operator, this method returns \(1 - c_0^\dagger c_0\) due to commutation relations. See the reference: https://en.wikipedia.org/wiki/Normal_order#Multiple_fermions.

Returns:: The normal ordered operator.
Return type:: FermionicOp

classmethod one()#

Constructs a unity-operator.

Returns:: The unity-operator of the given length.
Return type:: SparseLabelOp

parameters()#

Returns a list of the parameters in the operator.

Returns:: A list of the parameters in the operator.
Return type:: list[ParameterExpression]

permute_indices(permutation)#

Permutes the indices of the operator.

This method applies the provided index permutation to all labels of this operator. The provided permutation must be a sequence of integers whose length is equal to the register_length of the operator. The integer at any given index of the sequence indicates the new index which that location will be permuted to. For example:

op = SparseLabelOp({"+_0 -_1 +_2 -_3": 1.0})
permuted_op = op.permute_indices([3, 1, 0, 2])
assert permuted_op == SparseLabelOp({"+_3 -_1 +_0 -_2": 1.0})

Warning

This permutation utility is very powerful. Be mindful of the implications such a permutation might have on other components of the stack. To name an example, the builtin two-qubit reduction of the ParityMapper might not yield the expected results when used on permuted operator.

Parameters:: permutation (Sequence[int]) – a sequence of integers indicating the permutation to be applied. See above for an example.
Returns:: A new operator instance with the permuted indices.
Raises:: ValueError – if the length of the permutation argument does not equal register_length.
Return type:: SparseLabelOp

power(n)#

Return the compose of a operator with itself n times.

Parameters:: n (int) – the number of times to compose with self (n>0).
Returns:: the n-times composed operator.
Return type:: Clifford
Raises:: QiskitError – if the input and output dimensions of the operator are not equal, or the power is not a positive integer.

round(decimals=0)#

Rounds the operator coefficients to a specified number of decimal places.

Parameters:: decimals (int) – the number of decimal places to round coefficients to. By default this will round to the nearest integer value.
Returns:: The rounded operator.
Return type:: SparseLabelOp

simplify(atol=None)[source]#

Simplify the operator.

The simplifications implemented by this method should be: - to eliminate terms whose coefficients are close (w.r.t. atol) to 0. - to combine the coefficients which correspond to equivalent terms (see also the note below)

Note

simplify() should be used to simplify terms whose coefficients are close to zero, up to the specified numerical tolerance. It still differs slightly from chop() because that will chop real and imaginary part components individually.

Note

The meaning of “equivalence” between multiple terms depends on the specific operator subclass. As a restriction this method is required to preserve the order of appearance of the different components within a term. This avoids some possibly unexpected edge cases. However, this also means that some equivalencies cannot be detected. Check for other methods of a specific subclass which may affect the order of terms and can allow for further simplifications to be implemented. For example, check out index_order().

Note

Here is a more specific example: the fermionic term +_0 -_0 +_0 can actually be simplified down to +_0. In other words, these two terms are equivalent. This method will therefore reduce the first term to the second one and combine the associated coefficients. This only works when these sub-terms are not interjected by other ones, because the simplify() method may not re-order terms (see also the previous note and the index_order() method).

This method returns a new operator (the original operator is not modified).

Parameters:: atol (float | None) – Absolute numerical tolerance. The default behavior is to use self.atol.
Returns:: The simplified operator.
Return type:: FermionicOp

sort(*, weight=False)#

Returns a new sorted operator.

Parameters:: weight (bool) – when True, the returned keys will sort this operator according to the coefficient weights of the stored terms; when False, the keys will sort the operator by its keys (i.e. lexicographically).
Returns:: A new operator instance with its contents sorted.
Return type:: SparseLabelOp

tensor(other)[source]#

Returns the tensor product with another SparseLabelOp.

Parameters:: other (FermionicOp) – the other SparseLabelOp.
Returns:: The operator resulting from the tensor product, \(self \otimes other\).
Return type:: FermionicOp

Note

The tensor product can be obtained using the ^ binary operator. Hence a.tensor(b) is equivalent to a ^ b.

terms()[source]#

Provides an iterator analogous to items() but with the labels already split into pairs of operation characters and indices.

Yields:: A tuple with two items; the first one being a list of pairs of the form (char, int) where char is either + or - and the integer corresponds to the fermionic mode index on which the operator gets applied; the second item of the returned tuple is the coefficient of this term.
Return type:: Iterator[tuple[list[tuple[str, int]], _TCoeff]]

transpose()[source]#

Returns the transpose of the operator.

Returns:: The transpose of the operator.
Return type:: FermionicOp

values() → an object providing a view on D's values#

classmethod zero()#

Constructs a zero-operator.

Returns:: The zero-operator of the given length.
Return type:: SparseLabelOp