This commit replaces the use of sympy's multinomial coefficient function
with an inlined version. Sympy is a very large package and very slow to
import (which is why the only other usage is imported at run time instead
of at the module level). In this specific case requiring the entirety of
sympy for simple functions to generate sequences of coefficients is a
bit of overkill this commit adds equivalent private functions to the
polynomial_pauli_rotations module for generating the coefficients. For
performance in general we should only rely on sympy if we need it for
symbolical math and then only if there isn't any other option.

These functions are based heavily on the equivalents in sympy [1],
however they're updated to remove the assumption that non-ints are passed
in, and adjusted for terra's needs and style conventions.

[1] https://github.com/sympy/sympy/blob/sympy-1.5.1/sympy/ntheory/multinomial.py

Co-authored-by: mergify[bot] <37929162+mergify[bot]@users.noreply.github.com>

Just as in Qiskit#4385 and Qiskit#3156 we should not be importing sympy at the
module level. Sympy is a very heavy dependency and is slow to import
(and slow to run too). Adding a sympy import to the module level makes
imports significantly slower. This commit reverts the top level import
of sympy, which was only added for type hints, to fix the import
performance regression.

Just as in #4385 and #3156 we should not be importing sympy at the
module level. Sympy is a very heavy dependency and is slow to import
(and slow to run too). Adding a sympy import to the module level makes
imports significantly slower. This commit reverts the top level import
of sympy, which was only added for type hints, to fix the import
performance regression.

Just as has been demonstrated many times in the past we should not be
importing sympy at the module level. In Qiskit#5259 another import was added
and was immediately flagged as an import speed regression [1]. Sympy is
a very heavy dependency and is slow to import (and slow to run too).
Adding a sympy import to the module level makes and is honestly a
dependency we should be reluctant to add anywhere because it becomes
the primary bottleneck whenever it is used. This commit removes the top
level sympy import and moves to be a runtime import in the single method
that uses sympy. Just as it was done in Qiskit#5149, Qiskit#4385, and Qiskit#3156 before.

[1] https://qiskit.github.io/qiskit/#import.QiskitImport.time_qiskit_import?commits=b5d5a3bb

Just as has been demonstrated many times in the past we should not be
importing sympy at the module level. In #5259 another import was added
and was immediately flagged as an import speed regression [1]. Sympy is
a very heavy dependency and is slow to import (and slow to run too).
Adding a sympy import to the module level makes and is honestly a
dependency we should be reluctant to add anywhere because it becomes
the primary bottleneck whenever it is used. This commit removes the top
level sympy import and moves to be a runtime import in the single method
that uses sympy. Just as it was done in #5149, #4385, and #3156 before.

[1] https://qiskit.github.io/qiskit/#import.QiskitImport.time_qiskit_import?commits=b5d5a3bb

Co-authored-by: Julien Gacon <gaconju@gmail.com>

Just as has been demonstrated many times in the past we should not be
importing sympy at the module level. In Qiskit#5358 another import was added
and was immediately flagged as an import speed regression [1]. Sympy is
a very heavy dependency and is slow to import (and slow to run too).
Adding a sympy import to the module level makes and is honestly a
dependency we should be reluctant to add anywhere because it becomes
the primary bottleneck whenever it is used. This commit removes the top
level sympy import and moves to be a runtime import in the single method
that uses sympy. Just as it was done in Qiskit#5416, Qiskit#5149, Qiskit#4385, and Qiskit#3156
before.

Fixes Qiskit#5575

[1] https://qiskit.github.io/qiskit/#import.QiskitImport.time_qiskit_import?commits=300e9507

Just as has been demonstrated many times in the past we should not be
importing sympy at the module level. In #5358 another import was added
and was immediately flagged as an import speed regression [1]. Sympy is
a very heavy dependency and is slow to import (and slow to run too).
Adding a sympy import to the module level makes and is honestly a
dependency we should be reluctant to add anywhere because it becomes
the primary bottleneck whenever it is used. This commit removes the top
level sympy import and moves to be a runtime import in the single method
that uses sympy. Just as it was done in #5416, #5149, #4385, and #3156
before.

Fixes #5575

[1] https://qiskit.github.io/qiskit/#import.QiskitImport.time_qiskit_import?commits=300e9507