Add functions to support a sparse matrix observable

[soranjh]

Context:

This PR adds a new observable class for supporting sparse matrix objects and adds the functionality to compute the expectation value of the observable via direct matrix-vector multiplication.

Description of the Change:

The observable class SparseHamiltonian is added to the qml.ops.qubit module and the new function expval is created in qml.devices.default_qubit.

Benefits:

The computation of a Hamiltonian expectation value is faster and more memory-efficient with the new functionality.

[codecov]

Codecov Report

Merging #1398 (c97865d) into master (614db57) will increase coverage by 0.00%.
The diff coverage is 100.00%.

@@           Coverage Diff           @@
##           master    #1398   +/-   ##
=======================================
  Coverage   98.23%   98.24%           
=======================================
  Files         160      160           
  Lines       11966    11991   +25     
=======================================
+ Hits        11755    11780   +25     
  Misses        211      211           

Impacted Files	Coverage Δ
pennylane/devices/default_qubit.py	100.00% <100.00%> (ø)
pennylane/ops/qubit.py	98.73% <100.00%> (+0.01%)	⬆️
pennylane/qnode.py	98.01% <100.00%> (+0.01%)	⬆️

---

Continue to review full report at Codecov.

Legend - Click here to learn more
Δ = absolute <relative> (impact), ø = not affected, ? = missing data
Powered by Codecov. Last update 614db57...c97865d. Read the comment docs.

[ixfoduap]

Looks really good! It's crazy that some much work went into understanding how to speed up simulations, and in the end the change is so simple.

My main suggestions are to improve the documentation and to make the tests more compact when specifying the Hamiltonians

[josh146]

Thanks @soranjh! Main comment here is mostly about having qml.SparseHamiltonian match qml.Hermitian in behaviour when it comes to wires.

[josh146]

Nice, thanks @soranjh! I think it's very ready for merging, I just noticed a few very small things to change.

[ixfoduap]

Good to go from my side, but please address the remaining suggestions from @josh146

[josh146]

Thanks @soranjh! I'm happy to now approve this PR, however, I recommend updating the gradient test to test a case where the gradient is not zero. There have been cases in the past where our gradient tests have been using test circuits that were too trivial, leading to bug not being uncovered.

[soranjh]

Thanks @soranjh! I'm happy to now approve this PR, however, I recommend updating the gradient test to test a case where the gradient is not zero. There have been cases in the past where our gradient tests have been using test circuits that were too trivial, leading to bug not being uncovered.

Thanks @josh146. The gradient test is updated for a hydrogen Hamiltonian. Have you observed the codefactor issue "Complex Code" before? I computed the cyclomatic complexity with radon and everything seems normal in pennylane/ops/qubit.py, specially in my additions. Not sure which part of the code raises that issu.

[josh146]

@soranjh I have overridden the complex code warning :)

[soranjh]

@soranjh I have overridden the complex code warning :)

Thanks Josh!