Prevent automatic simplification of qiskit symbolic expressions.

[BryceFuller]

What should we add?

Following a discussion with @doichanj, there is a need for qiskit's symbolic engine to expose an option which prevents the automatic simpliciation of parameter expressions.

For example, there may be situations where A and 1/2*(A+A) lead to different algorithmic outcomes, and it would be preferable if both of these could be expressed using qiskit's symbolic engine.

Sympy addresses this by exposing a flag evaluate=False which can be passed into functions such as Add, Mul, etc. Simplification can then be manually performed later by invoking .simplify() or applying a particular simplification pass.

[jakelishman]

You might want to consider expanding / building on the qiskit.circuit.classical.expr hierarchy rather than ParameterExpression for this. While it's possible to add flags to do delayed simplification to ParameterExpression, the whole system there is still built around the standard complex-number field $\mathbb C$, not non-Abelian operator algebras, and it'd need a whole other evaluation engine writing to support the semantics.

The Expr stuff is explicitly just an IR without any baked-in simplification or mathematical logic, so might be a better short-term fit?

[jakelishman]

There's still complications, in that Expr was intended to be run/near-time evaluated by hardware, whereas ParameterExpression does represent compile-time (or certainly pre-runtime) evaluation, so there's a bit of a "pick your poison" to this. My (very quick and not hugely informed) gut feeling is that Expr will be less painful, since I know we expect to slot this into a different IR framework in the future anyway.

[BryceFuller]

Thanks Jake, even though ultimately I hope to have support for non-Abelian algebras, even then there will still be a need for the same delayed simplification with abelian expressions as well.

That being said, this request is not extremely urgent, but I figured it would be helpful to start making issues for the changes I expect we will eventually need.

[jakelishman]

It is, for sure - thanks for doing it!