Add symbolic Gaussian Square Echo to Qiskit Pulse symbolic_pulses library

[miamico]

Summary

Adding the implementation of a symbolic Gaussian Square Echo pulse

Details and comments

This is a first implementation of a symbolic version of Gaussian square echo pulse. Tests and documentation to be added. I have left the definition of the gap between echo pulses as it was but we may want to look at it more closely. Usually, the two echo pulses are separated by the duration of a single qubit gate which allows to echo the control qubit.

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[nkanazawa1989]

I'm glad to see we have other pulse shapes in Qiskit. If this is going to be backend-supported, you may also want to update

qobj converter,
https://github.com/Qiskit/qiskit-terra/blob/f2befd10b1aba6d9369daf4465cffdf75c7da576/qiskit/qobj/converters/pulse_instruction.py#L36-L46

and qpy loader
https://github.com/Qiskit/qiskit-terra/blob/f2befd10b1aba6d9369daf4465cffdf75c7da576/qiskit/qpy/binary_io/schedules.py#L94-L106

[nkanazawa1989]

We need description of the pulse shape (mathematical, or in some sentence) and it would be great if you have some reference paper. Perhaps user may want to know how this differs from GaussianSquare and when they can use this pulse.

Since this description about rise fall parameter is written in GaussianSquare, you can use .. seealso directive to document link to that class and avoid duplication.

[miamico]

I'm glad to see we have other pulse shapes in Qiskit. If this is going to be backend-supported, you may also want to update

qobj converter,

https://github.com/Qiskit/qiskit-terra/blob/f2befd10b1aba6d9369daf4465cffdf75c7da576/qiskit/qobj/converters/pulse_instruction.py#L36-L46

and qpy loader

https://github.com/Qiskit/qiskit-terra/blob/f2befd10b1aba6d9369daf4465cffdf75c7da576/qiskit/qpy/binary_io/schedules.py#L94-L106

I think I have made the changes to those modules to support the GaussianSquareEcho pulse. I have a question though. I see that all the pulses have a corresponding equivalent in the continuous and discrete modules. Should I implement a gaussian square echo pulse there as well?

[nkanazawa1989]

Implementation of waveform counterpart, i.e. continuous and discrete, is not mandatory, because we can get the same array with SymbolicPulse.get_waveform method (but nothing stops you to implement).

[miamico]

Picking this up again since this PR was merged recently. As it is now, GaussianSquareEcho only contains the Gaussian square pulses without the single qubit gates for the echo. There is a delay between the Gaussian Square pulses with positive and negative amplitudes where the second single qubit gate for the echo would fit. I wonder if we should include these single qubit gates or not. If not, how would we add the single qubit gate needed for the echo in the delay between the Gaussian squares?