Add equivalence-library rules between rzz and cp

These gates are locally equivalence (as are all the Ising-interaction
gates), and this simple additional rule lets things like QFT, which are
defined by Qiskit's default constructor in terms of `cp`, get converted
into `rzz` or `rzx`.

One `ControlledGate` test needed a case removing, because the underlying
control mechanism now works correctly.

qiskit/circuit/library/standard_gates/equivalence_library.py

@@ -189,6 +189,21 @@ def _cnot_rxx_decompose(plus_ry: bool = True, plus_rxx: bool = True):
 cphase_to_cu1.append(CU1Gate(theta), [0, 1])
 _sel.add_equivalence(CPhaseGate(theta), cphase_to_cu1)
 
+# CPhaseGate
+#
+#                  global phase: ϴ/4
+#                                  ┌─────────┐
+#  q_0: ─■────     q_0: ─■─────────┤ Rz(ϴ/2) ├
+#        │P(ϴ)  ≡        │ZZ(-ϴ/2) ├─────────┤
+#  q_1: ─■────     q_1: ─■─────────┤ Rz(ϴ/2) ├
+#                                  └─────────┘
+theta = Parameter("theta")
+cphase_to_rzz = QuantumCircuit(2, global_phase=theta / 4)
+cphase_to_rzz.rzz(-theta / 2, 0, 1)
+cphase_to_rzz.rz(theta / 2, 0)
+cphase_to_rzz.rz(theta / 2, 1)
+_sel.add_equivalence(CPhaseGate(theta), cphase_to_rzz)
+
 # RGate
 #
 #    ┌────────┐        ┌───────────────────────┐
@@ -394,6 +409,19 @@ def _cnot_rxx_decompose(plus_ry: bool = True, plus_rxx: bool = True):
     def_rzx.append(inst, qargs, cargs)
 _sel.add_equivalence(RZXGate(theta), def_rzx)
 
+# RZXGate to RZZGate
+#      ┌─────────┐
+# q_0: ┤0        ├     q_0: ──────■───────────
+#      │  Rzx(ϴ) │  ≡       ┌───┐ │ZZ(ϴ) ┌───┐
+# q_1: ┤1        ├     q_1: ┤ H ├─■──────┤ H ├
+#      └─────────┘          └───┘        └───┘
+theta = Parameter("theta")
+rzx_to_rzz = QuantumCircuit(2)
+rzx_to_rzz.h(1)
+rzx_to_rzz.rzz(theta, 0, 1)
+rzx_to_rzz.h(1)
+_sel.add_equivalence(RZXGate(theta), rzx_to_rzz)
+
 
 # RYGate
 #
@@ -654,6 +682,21 @@ def _cnot_rxx_decompose(plus_ry: bool = True, plus_rxx: bool = True):
 rzz_to_rzx.h(1)
 _sel.add_equivalence(RZZGate(theta), rzz_to_rzx)
 
+# RZZ to CPhase
+#
+#                 global phase: ϴ/2
+#                                ┌───────┐
+#  q_0: ─■─────   q_0: ─■────────┤ Rz(ϴ) ├
+#        │ZZ(ϴ) ≡       │P(-2*ϴ) ├───────┤
+#  q_1: ─■─────   q_1: ─■────────┤ Rz(ϴ) ├
+#                                └───────┘
+theta = Parameter("theta")
+rzz_to_cphase = QuantumCircuit(2, global_phase=theta / 2)
+rzz_to_cphase.cp(-theta * 2, 0, 1)
+rzz_to_cphase.rz(theta, 0)
+rzz_to_cphase.rz(theta, 1)
+_sel.add_equivalence(RZZGate(theta), rzz_to_cphase)
+
 # RZZ to RYY
 q = QuantumRegister(2, "q")
 theta = Parameter("theta")

releasenotes/notes/cphase-rzz-equivalence-e8afc37b71a74366.yaml

@@ -0,0 +1,6 @@
+---
+features_circuits:
+  - |
+    New equivalence rules in the standard equivalence library mean that the transpiler can now
+    directly convert between two-qubit continuous Pauli rotations, rather than always decomposing
+    into a discrete two-CX-based solution.

test/python/circuit/test_controlled_gate.py

@@ -1438,7 +1438,6 @@ def test_control_zero_operand_gate(self, num_ctrl_qubits):
     @data(
         RXGate,
         RYGate,
-        RZGate,
         RXXGate,
         RYYGate,
         RZXGate,