Allow `qpy.dump` and `qpy.load` to serialize a list of `ParameterExpression`s

[ihincks]

What should we add?

As a part of the samplex serialization format, samplomatic/executor need to send a list of parameter expressions over the wire. In order to use only the public qpy interface, we do this by creating a single-qubit circuit and placing each parameter expression into an rz gate. With this strategy, because it's common to have tens of thousands of parameters in a circuit, our samplex payloads are often dominated in size by these parameter expressions, rather than the actual DAG of the samplex. For example, in one of our 5k benchmarks, 70% of the samplex serialization is the parameter expression table.

It would be nice if there were a way to serialize and deserialize this more directly.
Here is the object I'd like to serialize: https://github.com/Qiskit/samplomatic/blob/main/samplomatic/samplex/parameter_expression_table.py
Here is how we currently do it: https://github.com/Qiskit/samplomatic/blob/main/samplomatic/serialization/parameter_expression_serializer.py

[gadial]

This change will require an udpate to the QPY format to allow for non-circuit elements as part of the qpy. There is, however, an easier alternative I can implement right away - I can add functions like write_parameter_expression_list and read_parameter_expression_list to the qpy module and you can use those directly. This is similar to how one can serialize a circuit directly, without going through dump, although the result is not something that can be read via load but has to be read using read_circuit. Which options you prefer?

[ihincks]

Your suggestion would work just as well as load/dump for this application, so long as the qpy version is configurable.

Is there any difference between what py_write_circuit writes vs what dump writes?

[gadial]

Basically dump can pack multiple circuits and stores the qpy version as well. py_write_circuit stores no metadata and simply serializes the given circuit, with the qpy version given as a parameter to the write/load functions and not stored explicitly in the file.

I'll prepare a quick prototype if you want something to experiment with and tell me if the interface is good enough.

[ihincks]

Okay, that should work, thanks! The serializer I linked to would then itself save the QPY version beside your binary blob to keep it safe.

[gadial]

I opened #15740. Will you be able to test it locally and confirm it works as you request? The TestWriteReadValueList test suite has usage examples.