---
title: double_commutator (latest version)
description: API reference for qiskit_fermions.operators.library.double_commutator in the latest version of qiskit-fermions
source: https://quantum.cloud.ibm.com/docs/en/api/qiskit-fermions/operators-library-double-commutator
---

# 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   [\[1\]](#id3); the `sign=False` case is the symmetric double commutator $[A, B, C]$ appearing in Equation (2) of   [\[2\]](#id4)):

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`](/docs/api/qiskit-fermions/protocols-supports-commutators "qiskit_fermions.protocols.SupportsCommutators") protocol.

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

```pycon
>>> 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](#id1)]

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

\[[2](#id2)]

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](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*](https://docs.python.org/3/library/functions.html#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*