double_commutator

double_commutator(op_a, op_b, op_c, sign)

Computes the double-commutator of three operators.

The double-commutator is defined as follows (see also Chapter 13.6, Equation of motion methods, page 479 of [1]; the sign=False case is the symmetric double commutator $[A, B, C]$ appearing in Equation (2) of [2]):

If sign is False, it returns

[[A, B], C]/2 + [A, [B, C]]/2 = (2ABC + 2CBA - BAC - CAB - ACB - BCA)/2.

If sign is True, it returns

\{[A, B], C\}/2 + \{A, [B, C]\}/2 = (2ABC - 2CBA - BAC + CAB - ACB + BCA)/2.

Operators which support this method must implement the SupportsCommutators protocol.

Note

All three inputs must be of the same operator type. This will also determine the output type.

>>> from qiskit_fermions.operators import FermionOperator
>>> from qiskit_fermions.operators.library import double_commutator
>>> op1 = FermionOperator.from_dict({((True, 0), (False, 0)): 1})
>>> op2 = FermionOperator.from_dict({((False, 0), (True, 0)): 2})
>>> op3 = FermionOperator.from_dict(
...     {((True, 0), (False, 0)): 1, ((False, 0), (True, 0)): 2 + 0.5j}
... )
>>> comm = double_commutator(op1, op2, op3, False)
>>> comm = comm.normal_ordered()
>>> canon = comm.simplify()
>>> assert canon == FermionOperator.zero()

[1]

R. McWeeny. Methods of Molecular Quantum Mechanics. 2nd Edition, Academic Press, 1992. ISBN 0-12-486552-6.

[2]

P. J. Ollitrault, A. Miessen, I. Tavernelli. Molecular Quantum Dynamics: A Quantum Computing Perspective. Acc. Chem. Res. 54, 4229-4238 (2021). https://doi.org/10.1021/acs.accounts.1c00514

Parameters

op_a (T) – the operator $A$ above.
op_b (T) – the operator $B$ above.
op_c (T) – the operator $C$ above.
sign (bool) – the nature of the outer (anti-)commutator as per the definition above.

Returns

The double-commutator as per the definition above.

Return type