qp.commutator

commutator(op1, op2, pauli=False)

Compute commutator between two operators in PennyLane

\[[O_1, O_2] = O_1 O_2 - O_2 O_1\]

Parameters:

• op1 (Union[Operator, PauliWord, PauliSentence]) – First operator

• op2 (Union[Operator, PauliWord, PauliSentence]) – Second operator

• pauli (bool) – When True, all results are passed as a PauliSentence instance. Else, results are always returned as Operator instances.

Returns:

The commutator

Return type:

~Operator or ~PauliSentence

Examples

You can compute commutators between operators in PennyLane.

>>> qp.commutator(qp.X(0), qp.Y(0))
2j * Z(0)

>>> op1 = qp.X(0) @ qp.X(1)
>>> op2 = qp.Y(0) @ qp.Y(1)
>>> qp.commutator(op1, op2)
0 * I()

We can return a PauliSentence instance by setting pauli=True.

>>> op1 = qp.X(0) @ qp.X(1)
>>> op2 = qp.Y(0) + qp.Y(1)
>>> res = qp.commutator(op1, op2, pauli=True)
>>> res
2j * Z(0) @ X(1)
+ 2j * X(0) @ Z(1)
>>> isinstance(res, PauliSentence)
True

We can also input PauliWord and PauliSentence instances.

>>> op1 = PauliWord({0:"X", 1:"X"})
>>> op2 = PauliWord({0:"Y"}) + PauliWord({1:"Y"})
>>> res = qp.commutator(op1, op2, pauli=True)
>>> res
2j * Z(0) @ X(1)
+ 2j * X(0) @ Z(1)
>>> isinstance(res, PauliSentence)
True

Note that when pauli=False, even if Pauli operators are used as inputs, qp.commutator returns Operators.

>>> res = qp.commutator(op1, op2, pauli=False)
>>> res
2j * (Z(0) @ X(1)) + 2j * (X(0) @ Z(1))
>>> isinstance(res, qp.operation.Operator)
True

It is worth noting that computing commutators with Paulis is typically faster. Internally, qp.commutator uses the op.pauli_rep whenever that is available for both operators.

Usage Details

The input and result of qp.commutator is not recorded in a tape context (and inside a QNode).

with qp.tape.QuantumTape() as tape:
    a = qp.X(0)      # gets recorded
    b = PauliWord({0:"Y"}) # does not get recorded
    comm = qp.commutator(a, b) # does not get recorded

In this example, we obtain tape.operations = [qp.X(0)]. When desired, we can still record the result of the commutator by using apply(), i.e. qp.apply(comm) inside the recording context.

A peculiarity worth repeating is how in a recording context every created operator is recorded.

with qp.tape.QuantumTape() as tape:
    comm = qp.commutator(qp.X(0), qp.Y(0))

In this example, both PauliX and PauliY get recorded because they were created inside the recording context. To avoid this, create the input to qp.commutator outside the recording context / qnode or insert an extra stop_recording() context (see QueuingManager).

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