add private helper functions for squaring Hamiltonian terms and colle…

[anthayes92]

In order to evaluate flow constraints of the cost Hamiltonian in the max weighted cycle QAOA problem, we need to reliably expand quadratic terms and simplify the resulting expressions.

Add a private function to find the coefficients and operators of the term of the Hamiltonian to be squared.
Add a private function to collect duplicate terms of a Hamiltonian.

Example of current and desired behaviour:

Using the tensor product operation @ as currently implemented we find:

>>> coeffs = [0, -1, -1, 1, 1]
>>> ops = [qml.Identity(wires=[0]), qml.PauliZ(wires=[0]), qml.PauliZ(wires=[1]), qml.PauliZ(wires=[2]), qml.PauliZ(wires=[4])]
>>> H = qml.Hamiltonian(coeffs, ops)
>>> square_H = H @ H
>>> print(square_H)
  (0) [I0]
+ (1) [Z0 Z0]
+ (1) [Z1 Z1]
+ (1) [Z2 Z2]
+ (1) [Z4 Z4]
+ (-2) [Z0 Z2]
+ (-2) [Z0 Z4]
+ (-2) [Z1 Z2]
+ (-2) [Z1 Z4]
+ (2) [Z0 Z1]
+ (2) [Z2 Z4]

The desired output of the above example would be:

  (4) [I0]
+ (-2) [Z0 Z2]
+ (-2) [Z0 Z4]
+ (-2) [Z1 Z2]
+ (-2) [Z1 Z4]
+ (2) [Z0 Z1]
+ (2) [Z2 Z4]

This may no longer be needed when related issues on Hamiltonian simplification are closed

[github-actions]

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[codecov]

Codecov Report

Merging #1213 (9905194) into master (2c6de3c) will increase coverage by 0.00%.
The diff coverage is 100.00%.

@@           Coverage Diff           @@
##           master    #1213   +/-   ##
=======================================
  Coverage   98.13%   98.13%           
=======================================
  Files         147      147           
  Lines       11260    11277   +17     
=======================================
+ Hits        11050    11067   +17     
  Misses        210      210           

Impacted Files	Coverage Δ
pennylane/qaoa/cycle.py	100.00% <100.00%> (ø)

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[trbromley]

Thanks @anthayes92! Looks good, could we come up with two minimal working examples that motivate why we need these functions (to be sure it's worth adding them in)?

E.g., could we have a simple Hamiltonian H where squaring with H @ H doesn't work as intended? Likewise a Hamiltonian H where calling the simplify() method doesn't work as intended?

This will also help us understand the shortfalls of the current approach in qml.Hamiltonian. Also, it can help us decide whether we could just make a quick change to the qml.Hamiltonian.simplify() method instead of having these improvements just available in the cycle module.

[trbromley]

Awesome!