MPSPrep adding extra empty qubit

[LuisAlfredoNu]

Context:
Lightning Tensor can use an MPS as input but should be passed through the circuit operation qml.MPSPrep but must have the same shape as the allocated MPS.

Description of the Change:
This PR adds the following features to Lightning Tensor:

• Allow qml.MPSPrep to accept an MPS with len(MPS) = n_wires-1
• Expand the input MPS to fit with the allocated MPS.

The following algorithm was implemented

• Padding with zeros the tensor $B_i$ to fit the tensor shape $A_{i+1}$ up to $i = N/2$ where $N$ is the total number of tensors in $B$
• Add the identity matrix with shape (1,2,2) at the beginning of $B$
• Restore the $B$ MPS into a right-canonical form to spread the new site information across the entire MPS.

Benefits:
More flexibility to work with LT and provide custom MPS.

Possible Drawbacks:

Related GitHub Issues:

[sc-84509]

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.36%. Comparing base (dcebe62) to head (09a31c5).
Report is 1 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1064      +/-   ##
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+ Coverage   96.09%   97.36%   +1.27%     
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  Files         232      111     -121     
  Lines       39079    16976   -22103     
==========================================
- Hits        37552    16529   -21023     
+ Misses       1527      447    -1080     

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[github-actions]

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[maliasadi]

Thanks @LuisAlfredoNu! Nice work ⚡

[multiphaseCFD]

Thank you @LuisAlfredoNu for your excellent work. This enhancement will improve the performance of the LT process with block2 data and provides a way to verify the canonical form of MPS site data.

My only question for discussion is whether LT is the most appropriate location for this feature. I'm concerned about the potential risk due to the current lack of safety checks. If we keep this functionality within LT, there's a possibility of issues arising. For example, a user might intend to analyze an n_wires system and correctly provide n_wires MPS site data to MPSPrep. However, if they accidentally provide n_wires-1 MPS site data, the LT process might not detect the discrepancy and throw an error beforehand. I'd like to open this up for discussion with the team.

CC: @maliasadi @josephleekl @AmintorDusko

[LuisAlfredoNu]

My only question for discussion is whether LT is the most appropriate location for this feature. I'm concerned about the potential risk due to the current lack of safety checks. If we keep this functionality within LT, there's a possibility of issues arising. For example, a user might intend to analyze an n_wires system and correctly provide n_wires MPS site data to MPSPrep. However, if they accidentally provide n_wires-1 MPS site data, the LT process might not detect the discrepancy and throw an error beforehand. I'd like to open this up for discussion with the team.

Thank you @multiphaseCFD for your review. The situation that you mention throws a proper error to the user, if they try to do that then LT will show the error:

Error in PennyLane Lightning: The incoming MPS does not have the correct layout for lightning.tensor.

because the input MPS and the target MPS don't match in dimension. I will add an extra test to clarify this situation.

[maliasadi]

Nice work @LuisAlfredoNu! 🎸 Happy to approve after updating docstrings ✍️

[AmintorDusko]

Nice job! Thank you!

[maliasadi]

Thanks @LuisAlfredoNu! Happy to approve the impl 🙌