Simplify Product block encoding decomposition

[charlesyuan314]

This change refactors the Product block encoding, following suggestion in discussion, to perform fewer partitions of the ancilla and resource registers up front and to rely on AutoPartition to simplify the circuit diagram of each constituent block.

[mpharrigan]

I'm staring at the product_block_encoding_properties.decompose_bloq() diagram and I think this mostly makes sense. Why are the "ancilla" and "resource" registers still being split with different ones being fed into B[U_a] vs B[U_b]?

[charlesyuan314]

With ancilla, we are doing just one split at the beginning into the flag_bits and the inner ancilla that is then fed to each of the constituents. The constituents use AutoPartition to ignore the unused bits. I have updated a comment to make it clearer that there are not separate individual ancilla registers.

With resource, there is a question of whether these bits are being returned to zero. If so, the constituents can share the resource bits in the same way as ancilla but without need for the flag bits. I have gone ahead with this assumption for now and changed the code to not split the resource register and rely only on AutoPartition. This cleans up the diagram.

It is possible that some block encodings will leave their resource state dirty and thus require flag bits separate resource registers here. I think properly supporting this behavior without using excessive bits could be done via a flag attribute in the BlockEncoding interface, analogous to the check of whether the signal_state is identity.

I can open an issue tracking the general fix above and add a code comment referencing it. Does that sound like a reasonable plan?

[mpharrigan]

Thanks for the careful thought! Issue sounds like a good plan