Say I build a circuit using the power and control modifier on gates.

from qiskit.circuit.library import *
from qiskit import *
from qiskit.quantum_info import random_unitary
from qiskit.extensions import UnitaryGate
from numpy import pi

θ = pi/8
U = random_unitary(2)

circuit = QuantumCircuit(3)
circuit.append(PhaseGate(θ).power(3), [0])
circuit.append(PhaseGate(θ).control(2), [0, 1, 2])
circuit.append(UnitaryGate(U).control(), [0, 2])

It is good that the circuit drawers support these modifiers natively:

     ┌─────┐
q_0: ┤ p^3 ├─■────────────■─────
     └─────┘ │            │
q_1: ────────■────────────┼─────
             │P(π/8) ┌────┴────┐
q_2: ────────■───────┤ Unitary ├
                     └─────────┘

The following issues exist:

• I can't directly control the quantum_info.random_unitary U, have to build a UnitaryGate out of it first. i.e. quantum_info objects are not first class citizens in circuits.

• PhaseGate(θ).control(2) yields a MCPhaseGate under the hood (that's why the circuit drawer knows to draw it like that). I'm not sure we need this MCPhaseGate class, since PhaseGate(θ).control(2) already has well-defined semantics. The synthesis routine should know that this is a diagonal gate and thus decompose it appropriately. The drawer should also know that controlled phase gates can be drawn symmetrically.

• The decomposition gets greedily computed and stored in the gate itself when you modify a gate via power or control or inverse (https://github.com/Qiskit/qiskit-terra/blob/bdedeae59a5341b6ea4f399fe4a4f03c051e2e13/qiskit/circuit/library/standard_gates/p.py#L109). Instead we can keep them as powered- or controlled- or inverted- gates and make the rest of Qiskit work with that (e.g. quantum_info should be able to simulate this circuit at this high level without caring about the decomposition).

• Similarly all gate optimization routines should work the same way even if the base gates happen to be controlled. For example optimize_1q_decomposition should work when we have ctrl-U and ctrl-V back-to-back (I did this in an old PR but never merged).