[BUG] qml.Adjoint and qml.Pow do not commute when exponent is fractional

[EmilianoG-byte]

Expected behavior

Taking the qml.adjoint() of a powered-operator, i.e. an operator resulting from qml.pow, returns the same result whether we use lazy=True or lazy=False in qml.adjoint().

Actual behavior

Setting the flag lazy=False when using a fractional power (z) in qml.pow produces a matrix with the imaginary component having the opposite sign as with lazy=True. The correct matrix is the one with lazy=True

Additional information

No response

Source code

import pennylane as qml 

power = 0.5
op = qml.Hadamard(0)
powered_op = qml.pow(op, power, lazy=True)

powered_op_adj_true = qml.adjoint(powered_op, lazy=True)
matrix_true = qml.matrix(powered_op_adj_true)

powered_op_adj_false = qml.adjoint(powered_op, lazy=False)
matrix_false = qml.matrix(powered_op_adj_false)

np.allclose(matrix_true, matrix_false)
>> False

Tracebacks

No response

System information

Name: PennyLane
Version: 0.36.0
Summary: PennyLane is a cross-platform Python library for quantum computing, quantum machine learning, and quantum chemistry. Train a quantum computer the same way as a neural network.
Home-page: https://github.com/PennyLaneAI/pennylane
Author: 
Author-email: 
License: Apache License 2.0
Location: /Users/emiliano.godinez/Documents/software_development/pennylane-iqm/venv/lib/python3.9/site-packages
Requires: appdirs, autograd, autoray, cachetools, networkx, numpy, pennylane-lightning, requests, rustworkx, scipy, semantic-version, toml, typing-extensions
Required-by: pennylane_iqm, PennyLane_Lightning

Platform info:           macOS-14.5-arm64-arm-64bit
Python version:          3.9.19
Numpy version:           1.26.4
Scipy version:           1.13.1
Installed devices:
- iqm.remote (pennylane-iqm-0.0.1)
- default.clifford (PennyLane-0.36.0)
- default.gaussian (PennyLane-0.36.0)
- default.mixed (PennyLane-0.36.0)
- default.qubit (PennyLane-0.36.0)
- default.qubit.autograd (PennyLane-0.36.0)
- default.qubit.jax (PennyLane-0.36.0)
- default.qubit.legacy (PennyLane-0.36.0)
- default.qubit.tf (PennyLane-0.36.0)
- default.qubit.torch (PennyLane-0.36.0)
- default.qutrit (PennyLane-0.36.0)
- default.qutrit.mixed (PennyLane-0.36.0)
- null.qubit (PennyLane-0.36.0)
- lightning.qubit (PennyLane-Lightning-0.36.0)

Existing GitHub issues

• I have searched existing GitHub issues to make sure the issue does not already exist.

[dwierichs]

I'm not sure we have a well-defined behaviour for non-integer powers, because the power function has branch cuts in this case.