Repository: Qiskit/qiskit

Issue:
`Pauli.evolve()` should recognize ECR gates as Clifford gates
### Environment

- **Qiskit version**: 1.0.2
- **Python version**: 3.11
- **Operating system**: macos


### What is happening?

```
qc = QuantumCircuit(2)
qc.ecr(0,1)
qi.Pauli('XX').evolve(qc)
```

The above call to evolve() raises an error even though qc contains only Clifford gates:

```
QiskitError: 'Cannot apply Instruction: u'
```

### How can we reproduce the issue?

```
qc = QuantumCircuit(2)
qc.ecr(0,1)
qi.Pauli('XX').evolve(qc)
```

### What should happen?

ECR gates should be recognized as Clifford gates, and the Pauli should evolve successfully.

Related: #12085 

### Any suggestions?

_No response_


Discussion:
Currently, the `ECRGate` is a Clifford gate, and can be transformed into a `Clifford` operator, so this code should work:
```
from qiskit.quantum_info import *
qc = QuantumCircuit(2)
qc.ecr(0,1)
cliff = Clifford(qc)
Pauli('XX').evolve(cliff)
```
but since the `ECRGate` [definition](https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.ECRGate) contains non-Clifford gates, when appending it to a `QuantumCircuit`, it's being decomposed into non-Clifford gates, and that's why your code doesn't work. 

Resolving patch:
diff --git a/qiskit/quantum_info/operators/symplectic/base_pauli.py b/qiskit/quantum_info/operators/symplectic/base_pauli.py
--- a/qiskit/quantum_info/operators/symplectic/base_pauli.py
+++ b/qiskit/quantum_info/operators/symplectic/base_pauli.py
@@ -541,75 +541,50 @@ def _append_circuit(self, circuit, qargs=None):
         if qargs is None:
             qargs = list(range(self.num_qubits))
 
-        if isinstance(circuit, QuantumCircuit):
-            gate = circuit.to_instruction()
-        else:
+        if not isinstance(circuit, QuantumCircuit):
             gate = circuit
 
-        # Basis Clifford Gates
-        basis_1q = {
-            "i": _evolve_i,
-            "id": _evolve_i,
-            "iden": _evolve_i,
-            "x": _evolve_x,
-            "y": _evolve_y,
-            "z": _evolve_z,
-            "h": _evolve_h,
-            "s": _evolve_s,
-            "sdg": _evolve_sdg,
-            "sinv": _evolve_sdg,
-        }
-        basis_2q = {"cx": _evolve_cx, "cz": _evolve_cz, "cy": _evolve_cy, "swap": _evolve_swap}
-
-        # Non-Clifford gates
-        non_clifford = ["t", "tdg", "ccx", "ccz"]
-
-        if isinstance(gate, str):
-            # Check if gate is a valid Clifford basis gate string
-            if gate not in basis_1q and gate not in basis_2q:
-                raise QiskitError(f"Invalid Clifford gate name string {gate}")
-            name = gate
-        else:
-            # Assume gate is an Instruction
-            name = gate.name
-
-        # Apply gate if it is a Clifford basis gate
-        if name in non_clifford:
-            raise QiskitError(f"Cannot update Pauli with non-Clifford gate {name}")
-        if name in basis_1q:
-            if len(qargs) != 1:
-                raise QiskitError("Invalid qubits for 1-qubit gate.")
-            return basis_1q[name](self, qargs[0])
-        if name in basis_2q:
-            if len(qargs) != 2:
-                raise QiskitError("Invalid qubits for 2-qubit gate.")
-            return basis_2q[name](self, qargs[0], qargs[1])
-
-        # If not a Clifford basis gate we try to unroll the gate and
-        # raise an exception if unrolling reaches a non-Clifford gate.
-        if gate.definition is None:
-            raise QiskitError(f"Cannot apply Instruction: {gate.name}")
-        if not isinstance(gate.definition, QuantumCircuit):
-            raise QiskitError(
-                "{} instruction definition is {}; expected QuantumCircuit".format(
-                    gate.name, type(gate.definition)
+            if isinstance(gate, str):
+                # Check if gate is a valid Clifford basis gate string
+                if gate not in _basis_1q and gate not in _basis_2q:
+                    raise QiskitError(f"Invalid Clifford gate name string {gate}")
+                name = gate
+            else:
+                # Assume gate is an Instruction
+                name = gate.name
+
+            # Apply gate if it is a Clifford basis gate
+            if name in _non_clifford:
+                raise QiskitError(f"Cannot update Pauli with non-Clifford gate {name}")
+            if name in _basis_1q:
+                if len(qargs) != 1:
+                    raise QiskitError("Invalid qubits for 1-qubit gate.")
+                return _basis_1q[name](self, qargs[0])
+            if name in _basis_2q:
+                if len(qargs) != 2:
+                    raise QiskitError("Invalid qubits for 2-qubit gate.")
+                return _basis_2q[name](self, qargs[0], qargs[1])
+
+            # If not a Clifford basis gate we try to unroll the gate and
+            # raise an exception if unrolling reaches a non-Clifford gate.
+            if gate.definition is None:
+                raise QiskitError(f"Cannot apply Instruction: {gate.name}")
+            if not isinstance(gate.definition, QuantumCircuit):
+                raise QiskitError(
+                    "{} instruction definition is {}; expected QuantumCircuit".format(
+                        gate.name, type(gate.definition)
+                    )
                 )
-            )
 
-        flat_instr = gate.definition
-        bit_indices = {
-            bit: index
-            for bits in [flat_instr.qubits, flat_instr.clbits]
-            for index, bit in enumerate(bits)
-        }
+            circuit = gate.definition
 
-        for instruction in flat_instr:
+        for instruction in circuit:
             if instruction.clbits:
                 raise QiskitError(
                     f"Cannot apply Instruction with classical bits: {instruction.operation.name}"
                 )
             # Get the integer position of the flat register
-            new_qubits = [qargs[bit_indices[tup]] for tup in instruction.qubits]
+            new_qubits = [qargs[circuit.find_bit(qb)[0]] for qb in instruction.qubits]
             self._append_circuit(instruction.operation, new_qubits)
 
         # Since the individual gate evolution functions don't take mod
@@ -715,6 +690,42 @@ def _evolve_swap(base_pauli, q1, q2):
     return base_pauli
 
 
+def _evolve_ecr(base_pauli, q1, q2):
+    """Update P -> ECR.P.ECR"""
+    base_pauli = _evolve_s(base_pauli, q1)
+    base_pauli = _evolve_h(base_pauli, q2)
+    base_pauli = _evolve_s(base_pauli, q2)
+    base_pauli = _evolve_h(base_pauli, q2)
+    base_pauli = _evolve_cx(base_pauli, q1, q2)
+    base_pauli = _evolve_x(base_pauli, q1)
+    return base_pauli
+
+
 def _count_y(x, z, dtype=None):
     """Count the number of I Paulis"""
     return (x & z).sum(axis=1, dtype=dtype)
+
+
+# Basis Clifford Gates
+_basis_1q = {
+    "i": _evolve_i,
+    "id": _evolve_i,
+    "iden": _evolve_i,
+    "x": _evolve_x,
+    "y": _evolve_y,
+    "z": _evolve_z,
+    "h": _evolve_h,
+    "s": _evolve_s,
+    "sdg": _evolve_sdg,
+    "sinv": _evolve_sdg,
+}
+_basis_2q = {
+    "cx": _evolve_cx,
+    "cz": _evolve_cz,
+    "cy": _evolve_cy,
+    "swap": _evolve_swap,
+    "ecr": _evolve_ecr,
+}
+
+# Non-Clifford gates
+_non_clifford = ["t", "tdg", "ccx", "ccz"]


Test patch:
diff --git a/test/python/quantum_info/operators/symplectic/test_pauli.py b/test/python/quantum_info/operators/symplectic/test_pauli.py
--- a/test/python/quantum_info/operators/symplectic/test_pauli.py
+++ b/test/python/quantum_info/operators/symplectic/test_pauli.py
@@ -36,6 +36,7 @@
     CZGate,
     CYGate,
     SwapGate,
+    ECRGate,
     EfficientSU2,
 )
 from qiskit.circuit.library.generalized_gates import PauliGate
@@ -410,7 +411,11 @@ def test_evolve_clifford1(self, gate, label):
         self.assertEqual(value, value_h)
         self.assertEqual(value_inv, value_s)
 
-    @data(*it.product((CXGate(), CYGate(), CZGate(), SwapGate()), pauli_group_labels(2, False)))
+    @data(
+        *it.product(
+            (CXGate(), CYGate(), CZGate(), SwapGate(), ECRGate()), pauli_group_labels(2, False)
+        )
+    )
     @unpack
     def test_evolve_clifford2(self, gate, label):
         """Test evolve method for 2-qubit Clifford gates."""
@@ -439,6 +444,7 @@ def test_evolve_clifford2(self, gate, label):
                 CYGate(),
                 CZGate(),
                 SwapGate(),
+                ECRGate(),
             ),
             [int, np.int8, np.uint8, np.int16, np.uint16, np.int32, np.uint32, np.int64, np.uint64],
         )
@@ -468,6 +474,13 @@ def test_evolve_clifford_qargs(self):
         self.assertEqual(value, value_h)
         self.assertEqual(value_inv, value_s)
 
+    @data("s", "h")
+    def test_evolve_with_misleading_name(self, frame):
+        """Test evolve by circuit contents, not by name (fixed bug)."""
+        circ = QuantumCircuit(2, name="cx")
+        p = Pauli("IX")
+        self.assertEqual(p, p.evolve(circ, frame=frame))
+
     def test_barrier_delay_sim(self):
         """Test barrier and delay instructions can be simulated"""
         target_circ = QuantumCircuit(2)
diff --git a/test/python/quantum_info/operators/symplectic/test_pauli_list.py b/test/python/quantum_info/operators/symplectic/test_pauli_list.py
--- a/test/python/quantum_info/operators/symplectic/test_pauli_list.py
+++ b/test/python/quantum_info/operators/symplectic/test_pauli_list.py
@@ -33,6 +33,7 @@
     XGate,
     YGate,
     ZGate,
+    ECRGate,
 )
 from qiskit.quantum_info.operators import (
     Clifford,
@@ -1997,10 +1998,12 @@ def test_evolve_clifford1(self, gate):
             CYGate(),
             CZGate(),
             SwapGate(),
+            ECRGate(),
             Clifford(CXGate()),
             Clifford(CYGate()),
             Clifford(CZGate()),
             Clifford(SwapGate()),
+            Clifford(ECRGate()),
         )
     )
     def test_evolve_clifford2(self, gate):
@@ -2033,6 +2036,7 @@ def test_phase_dtype_evolve_clifford(self):
             CYGate(),
             CZGate(),
             SwapGate(),
+            ECRGate(),
         )
         dtypes = [
             int,