Add support for qml.Hamiltonian with lightning.qubit for diff_method='adjoint'

[yitchen-tim]

Issue description

Description of the issue:
When running the following simple code block, it throws the NotImplementedError. There is no issue if I change to other simulators such as default.qubit, or replace the observable to simple observables such as qml.PauliX(0).

import pennylane as qml
import numpy as np

dev = qml.device("lightning.qubit", wires=4)
H, qubits = qml.qchem.molecular_hamiltonian(['H','H'], np.array([0.0, 0.1, 0.0, 0.0, -0.1, 0.0]))

@qml.qnode(dev, diff_method='adjoint')
def circuit(params):
    qml.RX(params, wires=1)
    return qml.expval(H)

circuit(0.123)

• Expected behavior: It returns the expectation value of the Hamiltonian.

• Actual behavior: It throws NotImplementedError.

• Reproduces how often: Every time

• System information:

Name: PennyLane
Version: 0.20.0
Summary: PennyLane is a Python quantum machine learning library by Xanadu Inc.
Home-page: https://github.com/XanaduAI/pennylane
Author:
Author-email:
License: Apache License 2.0
Location: /usr/local/lib/python3.7/site-packages
Requires: appdirs, autograd, autoray, cachetools, networkx, numpy, pennylane-lightning, scipy, semantic-version, toml
Required-by: amazon-braket-pennylane-plugin, PennyLane-Lightning, PennyLane-Qchem
Platform info: Linux-4.14.256-197.484.amzn2.x86_64-x86_64-with-debian-buster-sid
Python version: 3.7.10
Numpy version: 1.19.2
Scipy version: 1.5.2
Installed devices:

• default.gaussian (PennyLane-0.20.0)
• default.mixed (PennyLane-0.20.0)
• default.qubit (PennyLane-0.20.0)
• default.qubit.autograd (PennyLane-0.20.0)
• default.qubit.jax (PennyLane-0.20.0)
• default.qubit.tf (PennyLane-0.20.0)
• default.qubit.torch (PennyLane-0.20.0)
• lightning.qubit (PennyLane-Lightning-0.20.2)
• braket.aws.qubit (amazon-braket-pennylane-plugin-1.3.0)
• braket.local.qubit (amazon-braket-pennylane-plugin-1.3.0)

[antalszava]

Hi @yitchen-tim, thanks for reporting this! 🙂

This is a feature that is not supported with lightning.qubit, the error is meant to refer to that. It's great to know that this would be useful to add in. 👍

[antalszava]

@yitchen-tim a further note here: this is expected not to be working for default.qubit either, see the restrictions to the underlying adjoint_jacobian method in the note box on this page: https://pennylane.readthedocs.io/en/stable/code/api/pennylane.devices.default_qubit.DefaultQubit.html?highlight=adjoint_jacobian#pennylane.devices.default_qubit.DefaultQubit.adjoint_jacobian