[Op. Arithm.] Improve performance #1

[AlbertMitjans]

1. Use self.hash as a key for storing the eigendecomposition.
2. Use the factors/summands eigvals and eigvecs when there are no overlapping wires.

Note: Check benchmarks below.

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[codecov]

Codecov Report

Merging #3022 (4f0b031) into master (54f76a9) will increase coverage by 0.00%.
The diff coverage is 100.00%.

@@           Coverage Diff           @@
##           master    #3022   +/-   ##
=======================================
  Coverage   99.66%   99.66%           
=======================================
  Files         269      269           
  Lines       22845    22892   +47     
=======================================
+ Hits        22768    22815   +47     
  Misses         77       77           

Impacted Files	Coverage Δ
pennylane/ops/op_math/prod.py	99.60% <100.00%> (+0.02%)	⬆️
pennylane/ops/op_math/sum.py	100.00% <100.00%> (ø)

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[AlbertMitjans]

Comparison of the time taken to compute the expectation value of the sum of 15 tensor products of 9 Pauli operators acting on 9 different wires:

Note: The summands have overlapping wires, thus the full matrix is computed when calling eigvals and diagonalizing_gates.

• Master branch:

• This branch:

[AlbertMitjans]

Comparison of the time taken to compute the eigenvalues of the operator:

(PauliZ(wires=[0]) @ PauliZ(wires=[47]) @ PauliZ(wires=[48]) @ PauliY(wires=[36]) @ PauliZ(wires=[13])) + 
(PauliZ(wires=[27]) @ PauliZ(wires=[30]) @ PauliX(wires=[6]) @ PauliY(wires=[41]) @ PauliY(wires=[1]))

Note: The summands and the factors of each summand have no overlapping wires.

• Master:
  

• Branch:
  

[AlbertMitjans]

Comparison of the time taken to compute the diagonalising gates of the operator:

(PauliZ(wires=[0]) @ PauliZ(wires=[47]) @ PauliZ(wires=[48]) @ PauliY(wires=[36]) @ PauliZ(wires=[13])) + 
(PauliZ(wires=[27]) @ PauliZ(wires=[30]) @ PauliX(wires=[6]) @ PauliY(wires=[41]) @ PauliY(wires=[1]))

Note: The summands and the factors of each summand have no overlapping wires.

• Master:
  

• Branch:
  

[Jaybsoni]

Looks really good! The performance boosts also look really nice 💯! As we discussed I think you can also add this logic into the sparse matrix method, and we should see similar improvements 👍🏼

Happy to approve once that is done and a changelog entry is added

[AlbertMitjans]

Looks really good! The performance boosts also look really nice 💯! As we discussed I think you can also add this logic into the sparse matrix method, and we should see similar improvements 👍🏼

Happy to approve once that is done and a changelog entry is added

@Jaybsoni I believe the changes to the sparse_matrix should be done in PRs #3030 and #3031

Changelog entry added 👍

[albi3ro]

Excellent!