Optimize parameterized UGate into 1 RXGate + 2 RZGate

[t-imamichi]

What should we add?

Because RXGate is available as a basis gate thanks to the fractional gates support, it would be nice if Qiskit could optimize UGate into 1 RXGate + 2 RZGate as follows.

import numpy as np
from qiskit import QuantumCircuit, generate_preset_pass_manager
from qiskit.circuit import ParameterVector
from qiskit.quantum_info import Operator

x = ParameterVector("x", 3)

qc = QuantumCircuit(1)
qc.u(x[0], x[1], x[2], 0)
pm = generate_preset_pass_manager(optimization_level=3, basis_gates=["rz", "rx"])
tqc = pm.run(qc)
print("optimization level", 3)
print(tqc)
print()

qc2 = QuantumCircuit(1)
qc2.rz(x[2] - np.pi/2, 0)
qc2.rx(x[0], 0)
qc2.rz(x[1] + np.pi/2, 0)
print("hand optimization")
print(qc2)

rng = np.random.default_rng(123)
for _ in range(100):
    params = 2 * rng.random(3) - 1
    op = Operator(qc.assign_parameters(params))
    op2 = Operator(qc2.assign_parameters(params))
    assert op.equiv(op2)
print("pass")

qiskit 1.2.4, 1.3.0rc1, main branch

optimization level 3
global phase: x[1]/2 + x[2]/2
   ┌──────────┐┌─────────┐┌──────────────┐┌─────────┐┌───────────────┐
q: ┤ Rz(x[2]) ├┤ Rx(π/2) ├┤ Rz(x[0] + π) ├┤ Rx(π/2) ├┤ Rz(x[1] + 3π) ├
   └──────────┘└─────────┘└──────────────┘└─────────┘└───────────────┘

hand optimization
   ┌────────────────┐┌──────────┐┌────────────────┐
q: ┤ Rz(x[2] - π/2) ├┤ Rx(x[0]) ├┤ Rz(x[1] + π/2) ├
   └────────────────┘└──────────┘└────────────────┘
pass

[Cryoris]

We currently don't optimize expressions containing parameter expressions, due to the overhead in handling the symbolic expressions. If the angles are bound, this should give the optimal decomposition, though, right?

[jakelishman]

Julien: this is something that we ought to be able to do during basis translation for the given circuit, even for parametric gates. Adding a direct rule to the BasisTranslator's equivalence library should convince it to do it, but that might have the knock-on effect of making other translations worse (probably not, but it's possible) - that's related to the trouble with the BasisTranslator you and I (and Imamichi-san and I) have talked about a bit offline.

[t-imamichi]

Yes, if I bind parameter, the circuit becomes optimal. But users might want to evaluate ansatz with unbound UGates (e.g,. TwoLocal(n, 'u', 'cx')). So, I think it would be nice to optimize or translate UGate in the optimal way when RX is available as a basis gate. But, it's not urgent because the current behavior is not a bug.

   ┌───────────────────┐
q: ┤ U(0.1,0.01,0.001) ├
   └───────────────────┘
optimization level 3
global phase: 0.0055
   ┌─────────────┐┌─────────┐┌────────────┐
q: ┤ Rz(-1.5698) ├┤ Rx(0.1) ├┤ Rz(1.5808) ├
   └─────────────┘└─────────┘└────────────┘

We currently don't optimize expressions containing parameter expressions, due to the overhead in handling the symbolic expressions. If the angles are bound, this should give the optimal decomposition, though, right?