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_bosons.operators import BosonOperator, ann, cre
>>> from qiskit_bosons.operators.library import double_commutator
>>> op1 = BosonOperator.from_dict({(cre(0), ann(1)): 1.0})
>>> op2 = BosonOperator.from_dict({(cre(1), ann(0)): 1.0})
>>> op3 = BosonOperator.from_dict({(cre(0), ann(0)): 1.0})
>>> result = double_commutator(op1, op2, op3, False)
>>> other = double_commutator(op1, op2, op3, True)
>>> result.equiv(other)
False
>>> scaled = double_commutator(3.0 * op1, op2, op3, False)
>>> scaled.equiv(3.0 * result)
True

[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:

T