Add Harrow-Hassidim-Lloyd (HHL) algorithm

[axif0]

Issue #, if available:
#1194
Description of changes:
Added HHL algorithm implementation and a demonstration notebook. This includes the circuit construction logic, result processing, and examples solving 2x2 linear systems with fidelity checks against classical solutions.
Testing done:
Validated on LocalSimulator with high fidelity results in the provided notebook.

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[sesmart]

Hi @axif0, thanks for the contribution here as well. Mentioned in the other PR - updating from main should fix the readthedocs issue, but you likely need to run tox -e linters as well.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 100.00%. Comparing base (641055c) to head (4a56503).
⚠️ Report is 1 commits behind head on main.

Additional details and impacted files

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  Files           32        34     +2     
  Lines         1358      1535   +177     
  Branches       174       200    +26     
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+ Hits          1358      1535   +177     

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[sesmart]

Maybe would tone down the comment about exponential speedup - I think for HHL it definitely depends on the state preparation, as well as readout.

[yitchen-tim]

Thanks for the update, Can you add the same caveat to the intro of the notebook as well? The place where you mentioned " It can provide an exponential speedup over classical methods for certain classes of problems." without the same caveat.

[yitchen-tim]

@sesmart I have resolved the comments that are solved by @axif0 's update. There are couple still open that I will defer to you.

@axif0 I added one more comment about mirroring the caveat to exponential advantage claim to notebook.

[axif0]

@sesmart, @yitchen-tim hello, Is there anything else for improvement?

[axif0]

I replaced the opaque circ.unitary() calls by leveraging qiskit-braket-provider. I created a new helper function that constructs the controlled-U gate using Qiskit's UnitaryGate(unitary).control(1) and its preset pass manager to compile it down to native 1q/2q basis gates (['cx', 'u', 'id', 'rz', 'sx', 'x']) before translating it back to a Braket Circuit.

Also ~

• Added qiskit-braket-provider as a dependency.
• Added qiskit-braket-provider as a dependency.
• Cleaned up the obsolete unitary matrix construction function and its related tests

@sesmart

[axif0]

Updated and cleaned up a small local lint warning where circ = Circuit() was unused in the pre-existing test_controlled_rotation_zeros .

Is there anything else to be done?

@sesmart @yitchen-tim

[axif0]

@sesmart thank you so much for the marge. And for the conversation.