# This code is part of a Qiskit project.
#
# (C) Copyright IBM 2021, 2023.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""The Fermionic-particle Operator."""
from __future__ import annotations
import re
from collections import defaultdict
from collections.abc import Collection, Mapping
from typing import Iterator, Sequence
import numpy as np
from qiskit_nature.exceptions import QiskitNatureError
from ._bits_container import _BitsContainer
from .polynomial_tensor import PolynomialTensor
from .sparse_label_op import _TCoeff, SparseLabelOp, _to_number
[docs]class FermionicOp(SparseLabelOp):
r"""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 :code:`[+-]_<index>`, where the :code:`<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 :code:`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 :code:`num_spin_orbitals-1`).
**Initialization**
A ``FermionicOp`` is initialized with a dictionary, mapping terms to their respective
coefficients:
.. code-block:: python
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:
.. code-block:: python
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, :code:`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
.. code-block:: python
FermionicOp({"+_1": 1}, num_spin_orbitals=2) + FermionicOp({"+_0": 1}, num_spin_orbitals=2)
Sum
.. code-block:: python
sum(FermionicOp({label: 1}, num_spin_orbitals=3) for label in ["+_0", "-_1", "+_2 -_2"])
Scalar multiplication
.. code-block:: python
0.5 * FermionicOp({"+_1": 1}, num_spin_orbitals=2)
Operator multiplication
.. code-block:: python
op1 = FermionicOp({"+_0 -_1": 1}, num_spin_orbitals=2)
op2 = FermionicOp({"-_0 +_0 +_1": 1}, num_spin_orbitals=2)
print(op1 @ op2)
Tensor multiplication
.. code-block:: python
op = FermionicOp({"+_0 -_1": 1}, num_spin_orbitals=2)
print(op ^ op)
Adjoint
.. code-block:: python
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.
Attributes:
num_spin_orbitals (int | None): 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.
.. note::
A FermionicOp can contain :class:`qiskit.circuit.ParameterExpression` objects as coefficients.
However, a FermionicOp containing parameters does not support the following methods:
- ``is_hermitian``
- ``to_matrix``
"""
_OPERATION_REGEX = re.compile(r"([\+\-]_\d+\s)*[\+\-]_\d+")
def __init__(
self,
data: Mapping[str, _TCoeff],
num_spin_orbitals: int | None = None,
*,
copy: bool = True,
validate: bool = True,
) -> None:
"""
Args:
data: the operator data, mapping string-based keys to numerical values.
num_spin_orbitals: the number of spin orbitals on which this operator acts.
copy: 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: 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.
"""
self.num_spin_orbitals = num_spin_orbitals
super().__init__(data, copy=copy, validate=validate)
@property
def register_length(self) -> int:
if self.num_spin_orbitals is None:
max_index = max(int(term[2:]) for key in self._data for term in key.split())
return max_index + 1
return self.num_spin_orbitals
def _new_instance(
self, data: Mapping[str, _TCoeff], *, other: FermionicOp | None = None
) -> FermionicOp:
num_so = self.num_spin_orbitals
if other is not None:
other_num_so = other.num_spin_orbitals
if num_so is None:
num_so = other_num_so
elif other_num_so is not None:
num_so = max(num_so, other_num_so)
return self.__class__(data, copy=False, num_spin_orbitals=num_so)
def _validate_keys(self, keys: Collection[str]) -> None:
super()._validate_keys(keys)
num_so = self.num_spin_orbitals
max_index = -1
for key in keys:
# 0. explicitly allow the empty key
if key == "":
continue
# 1. validate overall key structure
if not re.fullmatch(FermionicOp._OPERATION_REGEX, key):
raise QiskitNatureError(f"{key} is not a valid FermionicOp label.")
# 2. validate all indices against register length
for term in key.split():
index = int(term[2:])
if num_so is None:
if index > max_index:
max_index = index
elif index >= num_so:
raise QiskitNatureError(
f"The index, {index}, from the label, {key}, exceeds the number of spin "
f"orbitals, {num_so}."
)
self.num_spin_orbitals = max_index + 1 if num_so is None else num_so
@classmethod
def _validate_polynomial_tensor_key(cls, keys: Collection[str]) -> None:
allowed_chars = {"+", "-"}
for key in keys:
if set(key) - allowed_chars:
raise QiskitNatureError(
f"The key {key} is invalid. PolynomialTensor keys may only consists of `+` and "
"`-` characters, for them to be expandable into a FermionicOp."
)
[docs] @classmethod
def from_polynomial_tensor(cls, tensor: PolynomialTensor) -> FermionicOp:
cls._validate_polynomial_tensor_key(tensor.keys())
data: dict[str, _TCoeff] = {}
for key in tensor:
if key == "":
data[""] = tensor[key].item()
continue
mat = tensor[key]
label_template = mat.label_template.format(*key)
for value, index in mat.coord_iter():
data[label_template.format(*index)] = value
return cls(data, copy=False, num_spin_orbitals=tensor.register_length).chop()
def __repr__(self) -> str:
data_str = f"{dict(self.items())}"
return "FermionicOp(" f"{data_str}, " f"num_spin_orbitals={self.num_spin_orbitals}, " ")"
def __str__(self) -> str:
pre = (
"Fermionic Operator\n"
f"number spin orbitals={self.num_spin_orbitals}, number terms={len(self)}\n"
)
ret = " " + "\n+ ".join(
[f"{coeff} * ( {label} )" if label else f"{coeff}" for label, coeff in self.items()]
)
return pre + ret
[docs] def terms(self) -> Iterator[tuple[list[tuple[str, int]], _TCoeff]]:
"""Provides an iterator analogous to :meth:`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.
"""
for label in iter(self):
if not label:
yield ([], self[label])
continue
# we hard-code the result of lbl.split("_") as follows:
# lbl[0] is either + or -
# lbl[2:] corresponds to the index
terms = [(lbl[0], int(lbl[2:])) for lbl in label.split()]
yield (terms, self[label])
[docs] @classmethod
def from_terms(cls, terms: Sequence[tuple[list[tuple[str, int]], _TCoeff]]) -> FermionicOp:
data = {
" ".join(f"{action}_{index}" for action, index in label): value
for label, value in terms
}
return cls(data)
def _permute_term(
self, term: list[tuple[str, int]], permutation: Sequence[int]
) -> list[tuple[str, int]]:
return [(action, permutation[index]) for action, index in term]
[docs] def compose(self, other: FermionicOp, qargs=None, front: bool = False) -> FermionicOp:
if not isinstance(other, FermionicOp):
raise TypeError(
f"Unsupported operand type(s) for *: 'FermionicOp' and '{type(other).__name__}'"
)
if front:
return self._tensor(self, other, offset=False)
else:
return self._tensor(other, self, offset=False)
[docs] def tensor(self, other: FermionicOp) -> FermionicOp:
return self._tensor(self, other)
[docs] def expand(self, other: FermionicOp) -> FermionicOp:
return self._tensor(other, self)
@classmethod
def _tensor(cls, a: FermionicOp, b: FermionicOp, *, offset: bool = True) -> FermionicOp:
shift = a.num_spin_orbitals if offset else 0
new_data: dict[str, _TCoeff] = {}
for label1, cf1 in a.items():
for terms2, cf2 in b.terms():
new_label = f"{label1} {' '.join(f'{c}_{i+shift}' for c, i in terms2)}".strip()
if new_label in new_data:
new_data[new_label] += cf1 * cf2
else:
new_data[new_label] = cf1 * cf2
new_op = a._new_instance(new_data, other=b)
if offset:
new_op.num_spin_orbitals = a.num_spin_orbitals + b.num_spin_orbitals
return new_op
[docs] def transpose(self) -> FermionicOp:
data = {}
trans = "".maketrans("+-", "-+")
for label, coeff in self.items():
data[" ".join(lbl.translate(trans) for lbl in reversed(label.split()))] = coeff
return self._new_instance(data)
[docs] def normal_order(self) -> FermionicOp:
"""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 :math:`\\colon c_0 c_0^\\dagger\\colon` where :math:`c_0`
is an annihilation operator, this method returns :math:`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.
"""
ordered_op = FermionicOp.zero()
for terms, coeff in self.terms():
ordered_op += self._normal_order(terms, coeff)
# after successful normal ordering, we remove all zero coefficients
return self._new_instance(
{
label: coeff
for label, coeff in ordered_op.items()
if not np.isclose(_to_number(coeff), 0.0, atol=self.atol)
}
)
def _normal_order(self, terms: list[tuple[str, int]], coeff: _TCoeff) -> FermionicOp:
if not terms:
return self._new_instance({"": coeff})
ordered_op = FermionicOp.zero()
# perform insertion sorting
for i in range(1, len(terms)):
for j in range(i, 0, -1):
right = terms[j]
left = terms[j - 1]
if right[0] == "+" and left[0] == "-":
# swap terms where an annihilation operator is left of a creation operator
terms[j - 1] = right
terms[j] = left
coeff *= -1.0
if right[1] == left[1]:
# if their indices are identical, we incur an additional term because of:
# a_i a_i^\dagger = 1 - a_i^\dagger a_i
new_terms = terms[: (j - 1)] + terms[(j + 1) :]
# we can do so by recursion on this method
ordered_op += self._normal_order(new_terms, -1.0 * coeff)
elif right[0] == left[0]:
# when we have identical neighboring operators, differentiate two cases:
# on identical index, this is an invalid Fermionic operation which evaluates to
# zero: e.g. +_0 +_0 = 0
if right[1] == left[1]:
# thus, we bail on this recursion call
return ordered_op
# otherwise, if the left index is higher than the right one, swap the terms
elif left[1] > right[1]:
terms[j - 1] = right
terms[j] = left
coeff *= -1.0
new_label = " ".join(f"{term[0]}_{term[1]}" for term in terms)
ordered_op += self._new_instance({new_label: coeff})
return ordered_op
[docs] def index_order(self) -> FermionicOp:
"""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. :meth:`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 :meth:`simplify` will be able to correctly collapse
these two labels into one.
Returns:
The index ordered operator.
"""
data = defaultdict(complex) # type: dict[str, _TCoeff]
for terms, coeff in self.terms():
label, coeff = self._index_order(terms, coeff)
data[label] += coeff
# after successful index ordering, we remove all zero coefficients
return self._new_instance(
{
label: coeff
for label, coeff in data.items()
if not np.isclose(_to_number(coeff), 0.0, atol=self.atol)
}
)
def _index_order(self, terms: list[tuple[str, int]], coeff: _TCoeff) -> tuple[str, _TCoeff]:
if not terms:
return "", coeff
# perform insertion sorting
for i in range(1, len(terms)):
for j in range(i, 0, -1):
right = terms[j]
left = terms[j - 1]
if left[1] > right[1]:
terms[j - 1] = right
terms[j] = left
coeff *= -1.0
new_label = " ".join(f"{term[0]}_{term[1]}" for term in terms)
return new_label, coeff
[docs] def is_hermitian(self, atol: float | None = None) -> bool:
"""Checks whether the operator is hermitian.
Args:
atol: 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.
"""
if self.is_parameterized():
raise ValueError("is_hermitian is not supported for operators containing parameters.")
atol = self.atol if atol is None else atol
diff = (self - self.adjoint()).normal_order().simplify(atol=atol)
return all(np.isclose(coeff, 0.0, atol=atol) for coeff in diff.values())
[docs] def simplify(self, atol: float | None = None) -> FermionicOp:
"""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::
:meth:`simplify` should be used to simplify terms whose coefficients are close to zero,
up to the specified numerical tolerance. It still differs slightly from :meth:`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 :meth:`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 :meth:`simplify` method may not re-order terms (see also the previous note
and the :meth:`index_order` method).
This method returns a new operator (the original operator is not modified).
Args:
atol: Absolute numerical tolerance. The default behavior is to use ``self.atol``.
Returns:
The simplified operator.
"""
atol = self.atol if atol is None else atol
data = defaultdict(complex) # type: dict[str, _TCoeff]
# TODO: use parallel_map to make this more efficient (?)
for label, coeff in self.items():
label, coeff = self._simplify_label(label, coeff)
data[label] += coeff
simplified_data = {
label: coeff
for label, coeff in data.items()
if not np.isclose(_to_number(coeff), 0.0, atol=atol)
}
return self._new_instance(simplified_data)
def _simplify_label(self, label: str, coeff: _TCoeff) -> tuple[str, _TCoeff]:
bits = _BitsContainer[int]()
# Since Python 3.7, dictionaries are guaranteed to be insert-order preserving. We use this
# to our advantage, to implement an ordered set, which allows us to preserve the label order
# and only remove canceling terms.
new_label: dict[str, None] = {}
for lbl in label.split():
char, index = lbl.split("_")
idx = int(index)
char_b = char == "+"
if idx not in bits:
# we store all relevant information for each register index in 4 bits:
# 1. True if a `+` has been applied on this index
# 2. True if a `-` has been applied on this index
# 3. True if a `+` was applied first, False if a `-` was applied first
# 4. True if the last added operation on this index was `+`, False if `-`
bits[idx] = int(f"{char_b:b}{not char_b:b}{char_b:b}{char_b:b}", base=2)
# and we insert the encountered label into our ordered set
new_label[lbl] = None
elif bits.get_last(idx) == char_b:
# we bail, if we apply the same operator as the last one
return "", 0
elif bits.get_plus(idx) and bits.get_minus(idx):
# If both, `+` and `-`, have already been applied, we cancel the opposite to the
# current one (i.e. `+` will cancel `-` and vice versa).
# 1. we construct the reversed label which is the key we need to pop
pop_lbl = f"{'-' if char_b else '+'}_{idx}"
# 2. we find its index in the insertion order of the new label
pop_idx = list(new_label).index(pop_lbl)
# 3. we use this index plus the current length of the new label to determine the
# number of exchange operations necessary to move the current term next to the
# one being popped
num_exchange = len(new_label) - pop_idx - 1
# 4. we perform the information update by:
# a) updating the coefficient sign
coeff *= -1 if num_exchange % 2 else 1
# b) popping the key we just canceled out
new_label.pop(pop_lbl)
# c) updating the bits container
bits.set_plus_or_minus(idx, not char_b, False)
# d) and updating the last bit to the current char
bits.set_last(idx, char_b)
else:
# else, we simply set the bit of the currently applied char
bits.set_plus_or_minus(idx, char_b, True)
# and track it in our ordered set
new_label[lbl] = None
# we also update the last bit to the current char
bits.set_last(idx, char_b)
return " ".join(new_label), coeff