Merge `cirq.Pauli.X` and `cirq.X` into one object (do same for Y and Z)

Currently, there is an unnecessary distinction between cirq.Pauli.X (an instance of cirq.Pauli) and cirq.X (an instance of cirq.RotXGate). Some code wants the pauli x, some wants the x rotation. We need to unify these two things into one thing.

I’m not exactly sure how this should be done.

One way would be to introduce a PauliXGate that is the unification of both concepts, and falls back into one or the other as methods are applied (e.g. cirq.X would be a PauliGate but cirq.X**0.5 would again be a RotXGate).

Another way is to destroy the Pauli class and replace it with a type such as Union[cirq.X, cirq.Y, cirq.Z], and all the class methods because just functions that can be applied to that type.

I’m labelling this “good part time project” because I think it’s reasonably “meaty” design-wise and amount-of-touched-code-wise. I expect it to require iteration to hit something that works well.

[Strilanc]

Some example use cases that I think should work:

Creating and exponentiating pauli products to create PauliStringPhasor operations that can be put into a circuit:

xy = cirq.X(q) * cirq.Y(p)
sqrt_xy = xy**0.5
c = cirq.Circuit.from_ops(sqrt_xy)

The future possibility of creating linear combinations of pauli products, which defines a QubitOperator (a term from openfermion):

hamiltonian = 0.5 * cirq.X(q) * cirq.Y(p) - cirq.Z(r)

[viathor]

Clarification on the status of cirq.Pauli: it still exists, but as abstract class. It has no public subclasses and only three instances: cirq.X, cirq.Y and cirq.Z, see #1249.

The class is used in type annotations and for its static methods that enable index-based access to the three instances, e.g. in the definition of SingleQubitCliffordGate class.

In [2]: isinstance(cirq.X, cirq.Pauli)
Out[2]: True

In [3]: isinstance(cirq.T, cirq.Pauli)
Out[3]: False

In [4]: cirq.X == cirq.Pauli.by_index(0)
Out[4]: True

In [5]: cirq.X == cirq.Pauli.by_index(1)
Out[5]: False

In [6]: cirq.Y == cirq.Pauli.by_index(1)
Out[6]: True