Add an equiv method to the StabilizerState class (#9543)

* add equiv method to StabilizerState class

* add a test for equiv method

* add release notes

* add explanantion to the tests

* update comment in tests

* updates following review

* update exp_val calculation

* updates following review

---------

Co-authored-by: mergify[bot] <37929162+mergify[bot]@users.noreply.github.com>

qiskit/quantum_info/states/stabilizerstate.py

@@ -18,7 +18,7 @@
 
 from qiskit.exceptions import QiskitError
 from qiskit.quantum_info.operators.op_shape import OpShape
-from qiskit.quantum_info.operators.symplectic import Clifford, Pauli
+from qiskit.quantum_info.operators.symplectic import Clifford, Pauli, PauliList
 from qiskit.quantum_info.operators.symplectic.clifford_circuits import _append_x
 from qiskit.quantum_info.states.quantum_state import QuantumState
 
@@ -266,6 +266,47 @@ def expectation_value(self, oper, qargs=None):
 
         return pauli_phase
 
+    def equiv(self, other):
+        """Return True if the two generating sets generate the same stabilizer group.
+
+        Args:
+            other (StabilizerState): another StabilizerState.
+
+        Returns:
+            bool: True if other has a generating set that generates the same StabilizerState.
+        """
+        if not isinstance(other, StabilizerState):
+            try:
+                other = StabilizerState(other)
+            except QiskitError:
+                return False
+
+        num_qubits = self.num_qubits
+        if other.num_qubits != num_qubits:
+            return False
+
+        pauli_orig = PauliList.from_symplectic(
+            self._data.stab_z, self._data.stab_x, 2 * self._data.stab_phase
+        )
+        pauli_other = PauliList.from_symplectic(
+            other._data.stab_z, other._data.stab_x, 2 * other._data.stab_phase
+        )
+
+        #  Check that each stabilizer from the original set commutes with each stabilizer
+        #  from the other set
+        if not np.all([pauli.commutes(pauli_other) for pauli in pauli_orig]):
+            return False
+
+        # Compute the expected value of each stabilizer from the original set on the stabilizer state
+        # determined by the other set. The two stabilizer states coincide if and only if the
+        # expected value is +1 for each stabilizer
+        for i in range(num_qubits):
+            exp_val = self.expectation_value(pauli_other[i])
+            if exp_val != 1:
+                return False
+
+        return True
+
     def probabilities(self, qargs=None, decimals=None):
         """Return the subsystem measurement probability vector.
 

releasenotes/notes/add-equiv-stabilizerstate-6ef8790c765690c1.yaml

@@ -0,0 +1,8 @@
+---
+features:
+  - |
+    Added the method :class:`.StabilizerState.equiv`,
+    that checks if the generating sets of two stabilizer states generate the same stabilizer group.
+    For example, the stabilizer group of the two-qubit Bell state contains the four elements
+    :math:`\{II, XX, -YY, ZZ\}` and hence can be generated by either :math:`[XX, ZZ]`,
+    :math:`[XX, -YY]` or :math:`[-YY, ZZ]`.

test/python/quantum_info/states/test_stabilizerstate.py

@@ -936,6 +936,50 @@ def test_expval_random_subsystem(self, num_qubits):
             target = Statevector(qc).expectation_value(op, qargs)
             self.assertAlmostEqual(exp_val, target)
 
+    def test_stabilizer_bell_equiv(self):
+        """Test that two circuits produce the same stabilizer group."""
+
+        qc1 = QuantumCircuit(2)
+        qc1.h(0)
+        qc1.x(1)
+        qc1.cx(0, 1)
+
+        qc2 = QuantumCircuit(2)
+        qc2.h(0)
+        qc2.cx(0, 1)
+        qc2.sdg(0)
+        qc2.sdg(1)
+        qc2.h(0)
+        qc2.h(1)
+
+        qc3 = QuantumCircuit(2)
+        qc3.h(0)
+        qc3.cx(0, 1)
+
+        qc4 = QuantumCircuit(2)
+        qc4.h(0)
+        qc4.cx(0, 1)
+        qc4.s(0)
+        qc4.sdg(1)
+        qc4.h(0)
+        qc4.h(1)
+
+        cliff1 = StabilizerState(qc1)  # ['+XX', '-ZZ']
+        cliff2 = StabilizerState(qc2)  # ['+YY', '+XX']
+        cliff3 = StabilizerState(qc3)  # ['+XX', '+ZZ']
+        cliff4 = StabilizerState(qc4)  # ['-YY', '+XX']
+
+        # [XX, -ZZ] and [XX, YY] both generate the stabilizer group {II, XX, YY, -ZZ}
+        self.assertTrue(cliff1.equiv(cliff2))
+        self.assertEqual(cliff1.probabilities_dict(), cliff2.probabilities_dict())
+
+        # [XX, ZZ] and [XX, -YY] both generate the stabilizer group {II, XX, -YY, ZZ}
+        self.assertTrue(cliff3.equiv(cliff4))
+        self.assertEqual(cliff3.probabilities_dict(), cliff4.probabilities_dict())
+
+        self.assertFalse(cliff1.equiv(cliff3))
+        self.assertFalse(cliff2.equiv(cliff4))
+
 
 if __name__ == "__main__":
     unittest.main()