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,
)

নোট

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

নোট

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

প্যারামিটার:

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!

রেইজেস:

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.

রিটার্ন টাইপ:: Self

argsort(*, weight=False)#

Returns the keys which sort this operator.

প্যারামিটার:: 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).
রিটার্নস:: The sequence of keys which sort this operator.
রিটার্ন টাইপ:: Sequence[str]

assign_parameters(parameters)#

Assign parameters to new parameters or values.

প্যারামিটার:: parameters (Mapping[ParameterExpression, complex | ParameterExpression]) -- The mapping from parameters to new parameters or values.
রিটার্নস:: A new operator with the parameters assigned.
রিটার্ন টাইপ:: 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.

প্যারামিটার:: atol (float | None) -- the tolerance to which to chop. If None, atol will be used.
রিটার্নস:: The chopped operator.
রিটার্ন টাইপ:: SparseLabelOp

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

Returns the operator composition with another operator.

প্যারামিটার:

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

রিটার্নস:

The operator resulting from the composition.

রিটার্ন টাইপ:

FermionicOp

নোট

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.

রিটার্নস:: The complex conjugate of the starting SparseLabelOp.
রিটার্ন টাইপ:: SparseLabelOp

dot(other, qargs=None)#

Return the right multiplied operator self * other.

প্যারামিটার:

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).

রিটার্নস:

The right matrix multiplied Operator.

রিটার্ন টাইপ:

Operator

নোট

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.

প্যারামিটার:

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.

রিটার্নস:

True if operators are equivalent, False if not.

রেইজেস:

ValueError -- Raised if either operator contains parameters

রিটার্ন টাইপ:

bool

expand(other)[source]#

Returns the reverse-order tensor product with another operator.

প্যারামিটার:: other (FermionicOp) -- the other operator.
রিটার্নস:: The operator resulting from the tensor product, \(othr \otimes self\).
রিটার্ন টাইপ:: FermionicOp

classmethod from_polynomial_tensor(tensor)[source]#

Constructs the operator from a PolynomialTensor.

প্যারামিটার:: tensor (PolynomialTensor) -- the PolynomialTensor to be expanded.
রিটার্নস:: The constructed operator.
রিটার্ন টাইপ:: FermionicOp

classmethod from_terms(terms)[source]#

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

প্যারামিটার:: terms (Sequence[tuple[list[tuple[str, int]], Union[complex, qiskit.circuit.parameterexpression.ParameterExpression]]]) -- a sequence as returned by terms().
রিটার্নস:: The constructed operator.
রিটার্ন টাইপ:: 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).

নোট

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.

রিটার্নস:: The index ordered operator.
রিটার্ন টাইপ:: 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.

প্যারামিটার:: order (int) -- Order \(p\) of the norm. The default value is 1.
রিটার্নস:: The induced norm.
রিটার্ন টাইপ:: float

রেইজেস:: ValueError -- Operator contains parameters.
রিটার্ন টাইপ:: float

is_hermitian(atol=None)[source]#

Checks whether the operator is hermitian.

প্যারামিটার:: atol (float | None) -- Absolute numerical tolerance. The default behavior is to use self.atol.
রিটার্নস:: True if the operator is hermitian up to numerical tolerance, False otherwise.
রেইজেস:: ValueError -- Operator contains parameters.
রিটার্ন টাইপ:: bool

is_parameterized()#

Returns whether the operator contains any parameters.

রিটার্ন টাইপ:: bool

is_zero(tol=None)#

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

প্যারামিটার:: tol (int | None) -- tolerance for checking coefficient values. If this is None, atol will be used instead.
রিটার্নস:: If operator length is zero or all coefficients are zero.
রিটার্ন টাইপ:: 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).

নোট

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.

রিটার্নস:: The normal ordered operator.
রিটার্ন টাইপ:: FermionicOp

classmethod one()#

Constructs a unity-operator.

রিটার্নস:: The unity-operator of the given length.
রিটার্ন টাইপ:: SparseLabelOp

parameters()#

Returns a list of the parameters in the operator.

রিটার্নস:: A list of the parameters in the operator.
রিটার্ন টাইপ:: list[qiskit.circuit.parameterexpression.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})

সতর্কতা

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.

প্যারামিটার:: permutation (Sequence[int]) -- a sequence of integers indicating the permutation to be applied. See above for an example.
রিটার্নস:: A new operator instance with the permuted indices.
রেইজেস:: ValueError -- if the length of the permutation argument does not equal register_length.
রিটার্ন টাইপ:: SparseLabelOp

power(n)#

Return the compose of a operator with itself n times.

প্যারামিটার:: n (int) -- the number of times to compose with self (n>0).
রিটার্নস:: the n-times composed operator.
রিটার্ন টাইপ:: Clifford
রেইজেস:: 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.

প্যারামিটার:: decimals (int) -- the number of decimal places to round coefficients to. By default this will round to the nearest integer value.
রিটার্নস:: The rounded operator.
রিটার্ন টাইপ:: 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)

নোট

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.

নোট

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().

নোট

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).

প্যারামিটার:: atol (float | None) -- Absolute numerical tolerance. The default behavior is to use self.atol.
রিটার্নস:: The simplified operator.
রিটার্ন টাইপ:: FermionicOp

sort(*, weight=False)#

Returns a new sorted operator.

প্যারামিটার:: 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).
রিটার্নস:: A new operator instance with its contents sorted.
রিটার্ন টাইপ:: SparseLabelOp

tensor(other)[source]#

Returns the tensor product with another SparseLabelOp.

প্যারামিটার:: other (FermionicOp) -- the other SparseLabelOp.
রিটার্নস:: The operator resulting from the tensor product, \(self \otimes other\).
রিটার্ন টাইপ:: FermionicOp

নোট

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.
রিটার্ন টাইপ:: Iterator[tuple[list[tuple[str, int]], Union[complex, qiskit.circuit.parameterexpression.ParameterExpression]]]

transpose()[source]#

Returns the transpose of the operator.

রিটার্নস:: The transpose of the operator.
রিটার্ন টাইপ:: FermionicOp

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

classmethod zero()#

Constructs a zero-operator.

রিটার্নস:: The zero-operator of the given length.
রিটার্ন টাইপ:: SparseLabelOp