Add more 1q and 2q Pauli rotation equivalences (#7407)

* added rx to equevalence library

* fixed sign in ry_to_rx

* added rzz to rxx

* gate correction

Co-authored-by: Julien Gacon <gaconju@gmail.com>

* added ryyrzz, rxxrzz, rzzryy translations, fixed ryrx translation to a more logical variant

* fixed errors

* Add Ryy to Rxx equivalence

* Add release note

---------

Co-authored-by: artemiyburov <artemiy.v.burov@gmail.com>
Co-authored-by: Julien Gacon <gaconju@gmail.com>
Co-authored-by: Jake Lishman <jake.lishman@ibm.com>

qiskit/circuit/library/standard_gates/equivalence_library.py

@@ -354,6 +354,13 @@
 def_ry.append(RGate(theta, pi / 2), [q[0]], [])
 _sel.add_equivalence(RYGate(theta), def_ry)
 
+q = QuantumRegister(1, "q")
+ry_to_rx = QuantumCircuit(q)
+ry_to_rx.sdg(0)
+ry_to_rx.rx(theta, 0)
+ry_to_rx.s(0)
+_sel.add_equivalence(RYGate(theta), ry_to_rx)
+
 # CRYGate
 #
 # q_0: ────■────      q_0: ─────────────■────────────────■──
@@ -451,6 +458,20 @@
     ryy_to_rzz.append(inst, qargs, cargs)
 _sel.add_equivalence(RYYGate(theta), ryy_to_rzz)
 
+# RYY to RXX
+q = QuantumRegister(2, "q")
+theta = Parameter("theta")
+ryy_to_rxx = QuantumCircuit(q)
+for inst, qargs, cargs in [
+    (SdgGate(), [q[0]], []),
+    (SdgGate(), [q[1]], []),
+    (RXXGate(theta), [q[0], q[1]], []),
+    (SGate(), [q[0]], []),
+    (SGate(), [q[1]], []),
+]:
+    ryy_to_rxx.append(inst, qargs, cargs)
+_sel.add_equivalence(RYYGate(theta), ryy_to_rxx)
+
 # RZGate
 #                  global phase: -ϴ/2
 #    ┌───────┐        ┌───────┐
@@ -474,6 +495,13 @@
 rz_to_sxry.sxdg(0)
 _sel.add_equivalence(RZGate(theta), rz_to_sxry)
 
+q = QuantumRegister(1, "q")
+rz_to_rx = QuantumCircuit(q)
+rz_to_rx.h(0)
+rz_to_rx.rx(theta, 0)
+rz_to_rx.h(0)
+_sel.add_equivalence(RZGate(theta), rz_to_rx)
+
 # CRZGate
 #
 # q_0: ────■────     q_0: ─────────────■────────────────■──
@@ -588,7 +616,6 @@
     rzz_to_ryy.append(inst, qargs, cargs)
 _sel.add_equivalence(RZZGate(theta), rzz_to_ryy)
 
-
 # RZXGate
 #
 #      ┌─────────┐

releasenotes/notes/pauli-rotation-equivalences-6b2449c93c042dc9.yaml

@@ -0,0 +1,8 @@
+---
+features:
+  - |
+    The transpiler's built-in :class:`.EquivalenceLibrary` has been taught more Pauli-rotation
+    equivalences between the one-qubit :math:`R_X`, :math:`R_Y` and :math:`R_Z` gates, and between
+    the two-qubit :math:`R_{XX}`, :math:`R_{YY}` and :math:`R_{ZZ}` gates.  This should make
+    simple basis translations more reliable, especially circuits that use :math:`Y` rotations.
+    See `#7332 <https://github.com/Qiskit/qiskit-terra/issues/7332>`__.