gate1 == gate2 should imply f(gate1) == f(gate2)

Example where it doesn’t:

In [2]: z1 = cirq.Z                                                                                                                                  

In [3]: z3 = cirq.Z**3                                                                                                                               

In [4]: z1 == z3                                                                                                                                     
Out[4]: True

In [5]: z1**(1/3) == z3**(1/3)                                                                                                                       
Out[5]: False

This is a consequence of our attempt to preserve the identity

(1)                   (G**a)**b == G**(a*b)

The identity is true for real numbers. It fails when we move on to complex numbers. From then on it continues to fail for unitary operators.

I understand the desire for intuitive behavior, but I fear that an attempt to preserve a false identity will just lead to a surprising behavior elsewhere: since the identity is false, something has to give. In this case, we are giving up a very fundamental and intuitive property of equality: if two gates are equal then transforming each in a given way yields another pair of equal gates.

If we must surprise some users, perhaps it is best to do so in a context that is at least consistent with the rest of the world, which recognizes the failure of identity (1).

Filed from review of #1155.

[Strilanc]

Off the top of my head, I remember that sphere eversion is possible because of conservation of a 3d analogue to the 2d winding number https://www.youtube.com/watch?v=sKqt6e7EcCs

[daxfohl]

A better example of a common place this doesn’t hold. Dictionaries in Python are ordered (now), so:

>>> x = {'a':1, 'b':2}
>>> y = {'b':2, 'a':1}
>>> x == y
True
>>> list(x)
['a', 'b']
>>> list(y)
['b', 'a']
>>> list(x) == list(y)
False