Added a warning to ApproxTimeEvolution documentation

[jackyruth]

Context:
Two mathematically equal Hamiltonians H1 and H2 pass all the usual equality checks.

c1 = [0.1, 0.2] 
c2 = [0.3, 0.4]
p1 = [qml.PauliZ(0), qml.PauliZ(1)]
p2 = [qml.PauliX(0), qml.PauliX(1)]
H1 = qml.Hamiltonian(c1 + c2, p1 + p2)
H2 = qml.Hamiltonian(c2 + c1, p2 + p1)

But lead to different time evolutions under qml.ApproxTimeEvolution, due to the order in which the observables are stored in qml.Hamiltonian.

dev = qml.device("default.qubit", wires=2)

@qml.qnode(dev)
def circuit(H, t):
    qml.ApproxTimeEvolution(H, t, 1)
    return qml.expval(qml.PauliX(0) @ qml.PauliX(1))

ts = np.linspace(0, 1, 10)
plt.plot(ts, [circuit(H1, t) for t in ts], "x", label="H1")
plt.plot(ts, [circuit(H2, t) for t in ts], "x", label="H2")

This is a tricky scenario that may confuse many users, so it's prudent to add a warning to the qml.ApproxTimeEvolution documentation.

Description of the Change:
Before

After

Benefits:
Save a few hours of debugging for users who are mislead by the "equality" of the hamiltonians.

Possible Drawbacks:
(Slightly) longer qml.ApproxTimeEvolution documentation.

Related GitHub Issues:
#5092

Related Shortcut Stories:
[sc-55193]

Fixes #5092.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Comparison is base (d5e3cca) 99.69% compared to head (3823bf3) 99.68%.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #5137      +/-   ##
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- Coverage   99.69%   99.68%   -0.01%     
==========================================
  Files         394      394              
  Lines       36118    35846     -272     
==========================================
- Hits        36007    35734     -273     
- Misses        111      112       +1     

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[trbromley]

Thanks @jackyruth!

[codecov-commenter]

Codecov Report

All modified and coverable lines are covered by tests ✅

Comparison is base (249a4bf) 99.68% compared to head (f44d26a) 99.68%.
Report is 1 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #5137      +/-   ##
==========================================
- Coverage   99.68%   99.68%   -0.01%     
==========================================
  Files         396      396              
  Lines       36640    36368     -272     
==========================================
- Hits        36526    36253     -273     
- Misses        114      115       +1     

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📢 Have feedback on the report? Share it here.

[trbromley]

Thanks @jackyruth! Approved, once my comments are resolved.

[Jaybsoni]

Hi @jackyruth,

Thanks for the addition! Indeed you are correct, but actually the source of this is a result of the fundamental theory of the Trotter decomposition, which is what is underlying the implementation of ApproxTimeEvolution.

My suggestion would be to rephrase the warning so that it's clear that this divergent behaviour comes from theory and not as a result of the code specifically. Try something like this:

Warning: The Trotter decomposition depends on the order of terms summed together 
in the Hamiltonian. Mathematically identical, but re-ordered, hamiltonians will 
produced different time evolutions in certain parameter regimes. 

[Jaybsoni]

Thanks 💯 !

[trbromley]

Thanks @jackyruth!

We often tag contributors in our release marketing on Twitter/X. If you would optionally like to be tagged, please could you share your username?

[jackyruth]

Hi Thomas! Thanks so much for reviewing my first pull request. I would like to be tagged under the username @jackyruth <https://github.com/jackyruth> Cheers, Jacky Jiang

…

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