diff --git a/qiskit/circuit/library/standard_gates/i.py b/qiskit/circuit/library/standard_gates/i.py
index a30cfe7643fb..40e0976f69d7 100644
--- a/qiskit/circuit/library/standard_gates/i.py
+++ b/qiskit/circuit/library/standard_gates/i.py
@@ -55,3 +55,7 @@ def inverse(self):
     def __array__(self, dtype=None):
         """Return a numpy.array for the identity gate."""
         return numpy.array([[1, 0], [0, 1]], dtype=dtype)
+
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        return IGate()
diff --git a/qiskit/circuit/library/standard_gates/iswap.py b/qiskit/circuit/library/standard_gates/iswap.py
index 1fe50d74e87a..dcc08c146047 100644
--- a/qiskit/circuit/library/standard_gates/iswap.py
+++ b/qiskit/circuit/library/standard_gates/iswap.py
@@ -13,10 +13,14 @@
 """iSWAP gate."""
 
 from typing import Optional
+
 import numpy as np
+
 from qiskit.circuit.gate import Gate
 from qiskit.circuit.quantumregister import QuantumRegister
 
+from .xx_plus_yy import XXPlusYYGate
+
 
 class iSwapGate(Gate):
     r"""iSWAP gate.
@@ -96,6 +100,7 @@ def _define(self):
         """
         # pylint: disable=cyclic-import
         from qiskit.circuit.quantumcircuit import QuantumCircuit
+
         from .h import HGate
         from .s import SGate
         from .x import CXGate
@@ -118,3 +123,7 @@ def _define(self):
     def __array__(self, dtype=None):
         """Return a numpy.array for the iSWAP gate."""
         return np.array([[1, 0, 0, 0], [0, 0, 1j, 0], [0, 1j, 0, 0], [0, 0, 0, 1]], dtype=dtype)
+
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        return XXPlusYYGate(-np.pi * exponent)
diff --git a/qiskit/circuit/library/standard_gates/p.py b/qiskit/circuit/library/standard_gates/p.py
index 40ee94741a68..12a1528b8bd2 100644
--- a/qiskit/circuit/library/standard_gates/p.py
+++ b/qiskit/circuit/library/standard_gates/p.py
@@ -125,6 +125,11 @@ def __array__(self, dtype=None):
         lam = float(self.params[0])
         return numpy.array([[1, 0], [0, exp(1j * lam)]], dtype=dtype)
 
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        (theta,) = self.params
+        return PhaseGate(exponent * theta)
+
 
 class CPhaseGate(ControlledGate):
     r"""Controlled-Phase gate.
@@ -244,6 +249,11 @@ def __array__(self, dtype=None):
             )
         return numpy.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, eith, 0], [0, 0, 0, 1]], dtype=dtype)
 
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        (theta,) = self.params
+        return CPhaseGate(exponent * theta)
+
 
 class MCPhaseGate(ControlledGate):
     r"""Multi-controlled-Phase gate.
diff --git a/qiskit/circuit/library/standard_gates/r.py b/qiskit/circuit/library/standard_gates/r.py
index 1452ae0b9bcb..292df90264d8 100644
--- a/qiskit/circuit/library/standard_gates/r.py
+++ b/qiskit/circuit/library/standard_gates/r.py
@@ -88,3 +88,8 @@ def __array__(self, dtype=None):
         exp_m = exp(-1j * phi)
         exp_p = exp(1j * phi)
         return numpy.array([[cos, -1j * exp_m * sin], [-1j * exp_p * sin, cos]], dtype=dtype)
+
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        theta, phi = self.params
+        return RGate(exponent * theta, phi)
diff --git a/qiskit/circuit/library/standard_gates/rx.py b/qiskit/circuit/library/standard_gates/rx.py
index 09139674886c..d26a9df60408 100644
--- a/qiskit/circuit/library/standard_gates/rx.py
+++ b/qiskit/circuit/library/standard_gates/rx.py
@@ -106,6 +106,11 @@ def __array__(self, dtype=None):
         sin = math.sin(self.params[0] / 2)
         return numpy.array([[cos, -1j * sin], [-1j * sin, cos]], dtype=dtype)
 
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        (theta,) = self.params
+        return RXGate(exponent * theta)
+
 
 class CRXGate(ControlledGate):
     r"""Controlled-RX gate.
diff --git a/qiskit/circuit/library/standard_gates/rxx.py b/qiskit/circuit/library/standard_gates/rxx.py
index e71e09ca9351..355f6bbcf215 100644
--- a/qiskit/circuit/library/standard_gates/rxx.py
+++ b/qiskit/circuit/library/standard_gates/rxx.py
@@ -119,3 +119,8 @@ def __array__(self, dtype=None):
             [[cos, 0, 0, -isin], [0, cos, -isin, 0], [0, -isin, cos, 0], [-isin, 0, 0, cos]],
             dtype=dtype,
         )
+
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        (theta,) = self.params
+        return RXXGate(exponent * theta)
diff --git a/qiskit/circuit/library/standard_gates/ry.py b/qiskit/circuit/library/standard_gates/ry.py
index f96d75958eb7..a552cff5d087 100644
--- a/qiskit/circuit/library/standard_gates/ry.py
+++ b/qiskit/circuit/library/standard_gates/ry.py
@@ -105,6 +105,11 @@ def __array__(self, dtype=None):
         sin = math.sin(self.params[0] / 2)
         return numpy.array([[cos, -sin], [sin, cos]], dtype=dtype)
 
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        (theta,) = self.params
+        return RYGate(exponent * theta)
+
 
 class CRYGate(ControlledGate):
     r"""Controlled-RY gate.
diff --git a/qiskit/circuit/library/standard_gates/ryy.py b/qiskit/circuit/library/standard_gates/ryy.py
index 343e32947635..b15630aadceb 100644
--- a/qiskit/circuit/library/standard_gates/ryy.py
+++ b/qiskit/circuit/library/standard_gates/ryy.py
@@ -119,3 +119,8 @@ def __array__(self, dtype=None):
             [[cos, 0, 0, isin], [0, cos, -isin, 0], [0, -isin, cos, 0], [isin, 0, 0, cos]],
             dtype=dtype,
         )
+
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        (theta,) = self.params
+        return RYYGate(exponent * theta)
diff --git a/qiskit/circuit/library/standard_gates/rz.py b/qiskit/circuit/library/standard_gates/rz.py
index 32fdc5be157c..f3da059f82ff 100644
--- a/qiskit/circuit/library/standard_gates/rz.py
+++ b/qiskit/circuit/library/standard_gates/rz.py
@@ -117,6 +117,11 @@ def __array__(self, dtype=None):
         ilam2 = 0.5j * float(self.params[0])
         return np.array([[exp(-ilam2), 0], [0, exp(ilam2)]], dtype=dtype)
 
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        (theta,) = self.params
+        return RZGate(exponent * theta)
+
 
 class CRZGate(ControlledGate):
     r"""Controlled-RZ gate.
diff --git a/qiskit/circuit/library/standard_gates/rzx.py b/qiskit/circuit/library/standard_gates/rzx.py
index f1e25dc115b3..94de9513b5d2 100644
--- a/qiskit/circuit/library/standard_gates/rzx.py
+++ b/qiskit/circuit/library/standard_gates/rzx.py
@@ -165,3 +165,8 @@ def __array__(self, dtype=None):
             [[cos, 0, -isin, 0], [0, cos, 0, isin], [-isin, 0, cos, 0], [0, isin, 0, cos]],
             dtype=dtype,
         )
+
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        (theta,) = self.params
+        return RZXGate(exponent * theta)
diff --git a/qiskit/circuit/library/standard_gates/rzz.py b/qiskit/circuit/library/standard_gates/rzz.py
index cd7bf27092e1..952e87c67765 100644
--- a/qiskit/circuit/library/standard_gates/rzz.py
+++ b/qiskit/circuit/library/standard_gates/rzz.py
@@ -132,3 +132,8 @@ def __array__(self, dtype=None):
             ],
             dtype=dtype,
         )
+
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        (theta,) = self.params
+        return RZZGate(exponent * theta)
diff --git a/qiskit/circuit/library/standard_gates/s.py b/qiskit/circuit/library/standard_gates/s.py
index 5bda08fa2a69..eaef481d5961 100644
--- a/qiskit/circuit/library/standard_gates/s.py
+++ b/qiskit/circuit/library/standard_gates/s.py
@@ -13,11 +13,14 @@
 """The S, Sdg, CS and CSdg gates."""
 
 from typing import Optional, Union
+
 import numpy
-from qiskit.qasm import pi
+
 from qiskit.circuit.controlledgate import ControlledGate
 from qiskit.circuit.gate import Gate
+from qiskit.circuit.library.standard_gates.p import CPhaseGate, PhaseGate
 from qiskit.circuit.quantumregister import QuantumRegister
+from qiskit.qasm import pi
 
 
 class SGate(Gate):
@@ -60,6 +63,7 @@ def _define(self):
         """
         # pylint: disable=cyclic-import
         from qiskit.circuit.quantumcircuit import QuantumCircuit
+
         from .u1 import U1Gate
 
         q = QuantumRegister(1, "q")
@@ -78,6 +82,10 @@ def __array__(self, dtype=None):
         """Return a numpy.array for the S gate."""
         return numpy.array([[1, 0], [0, 1j]], dtype=dtype)
 
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        return PhaseGate(0.5 * numpy.pi * exponent)
+
 
 class SdgGate(Gate):
     r"""Single qubit S-adjoint gate (~Z**0.5).
@@ -119,6 +127,7 @@ def _define(self):
         """
         # pylint: disable=cyclic-import
         from qiskit.circuit.quantumcircuit import QuantumCircuit
+
         from .u1 import U1Gate
 
         q = QuantumRegister(1, "q")
@@ -137,6 +146,10 @@ def __array__(self, dtype=None):
         """Return a numpy.array for the Sdg gate."""
         return numpy.array([[1, 0], [0, -1j]], dtype=dtype)
 
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        return PhaseGate(-0.5 * numpy.pi * exponent)
+
 
 class CSGate(ControlledGate):
     r"""Controlled-S gate.
@@ -194,9 +207,6 @@ def _define(self):
         """
         gate cs a,b { h b; cp(pi/2) a,b; h b; }
         """
-        # pylint: disable=cyclic-import
-        from .p import CPhaseGate
-
         self.definition = CPhaseGate(theta=pi / 2).definition
 
     def inverse(self):
@@ -210,6 +220,10 @@ def __array__(self, dtype=None):
             return numpy.asarray(mat, dtype=dtype)
         return mat
 
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        return CPhaseGate(0.5 * numpy.pi * exponent)
+
 
 class CSdgGate(ControlledGate):
     r"""Controlled-S^\dagger gate.
@@ -273,9 +287,6 @@ def _define(self):
         """
         gate csdg a,b { h b; cp(-pi/2) a,b; h b; }
         """
-        # pylint: disable=cyclic-import
-        from .p import CPhaseGate
-
         self.definition = CPhaseGate(theta=-pi / 2).definition
 
     def inverse(self):
@@ -288,3 +299,7 @@ def __array__(self, dtype=None):
         if dtype is not None:
             return numpy.asarray(mat, dtype=dtype)
         return mat
+
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        return CPhaseGate(-0.5 * numpy.pi * exponent)
diff --git a/qiskit/circuit/library/standard_gates/t.py b/qiskit/circuit/library/standard_gates/t.py
index 9fddfa008e4c..5ca423e54220 100644
--- a/qiskit/circuit/library/standard_gates/t.py
+++ b/qiskit/circuit/library/standard_gates/t.py
@@ -13,10 +13,13 @@
 """T and Tdg gate."""
 import math
 from typing import Optional
+
 import numpy
-from qiskit.qasm import pi
+
 from qiskit.circuit.gate import Gate
+from qiskit.circuit.library.standard_gates.p import PhaseGate
 from qiskit.circuit.quantumregister import QuantumRegister
+from qiskit.qasm import pi
 
 
 class TGate(Gate):
@@ -60,6 +63,7 @@ def _define(self):
         """
         # pylint: disable=cyclic-import
         from qiskit.circuit.quantumcircuit import QuantumCircuit
+
         from .u1 import U1Gate
 
         q = QuantumRegister(1, "q")
@@ -78,6 +82,10 @@ def __array__(self, dtype=None):
         """Return a numpy.array for the T gate."""
         return numpy.array([[1, 0], [0, (1 + 1j) / numpy.sqrt(2)]], dtype=dtype)
 
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        return PhaseGate(0.25 * numpy.pi * exponent)
+
 
 class TdgGate(Gate):
     r"""Single qubit T-adjoint gate (~Z**0.25).
@@ -119,6 +127,7 @@ def _define(self):
         """
         # pylint: disable=cyclic-import
         from qiskit.circuit.quantumcircuit import QuantumCircuit
+
         from .u1 import U1Gate
 
         q = QuantumRegister(1, "q")
@@ -136,3 +145,7 @@ def inverse(self):
     def __array__(self, dtype=None):
         """Return a numpy.array for the inverse T gate."""
         return numpy.array([[1, 0], [0, (1 - 1j) / math.sqrt(2)]], dtype=dtype)
+
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        return PhaseGate(-0.25 * numpy.pi * exponent)
diff --git a/qiskit/circuit/library/standard_gates/xx_minus_yy.py b/qiskit/circuit/library/standard_gates/xx_minus_yy.py
index 1298c79d29fe..df7fb1f8504f 100644
--- a/qiskit/circuit/library/standard_gates/xx_minus_yy.py
+++ b/qiskit/circuit/library/standard_gates/xx_minus_yy.py
@@ -169,3 +169,8 @@ def __array__(self, dtype=complex):
             ],
             dtype=dtype,
         )
+
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        theta, beta = self.params
+        return XXMinusYYGate(exponent * theta, beta)
diff --git a/qiskit/circuit/library/standard_gates/xx_plus_yy.py b/qiskit/circuit/library/standard_gates/xx_plus_yy.py
index 2d00003652ea..85d63ee625f9 100644
--- a/qiskit/circuit/library/standard_gates/xx_plus_yy.py
+++ b/qiskit/circuit/library/standard_gates/xx_plus_yy.py
@@ -170,3 +170,8 @@ def __array__(self, dtype=complex):
             ],
             dtype=dtype,
         )
+
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        theta, beta = self.params
+        return XXPlusYYGate(exponent * theta, beta)
diff --git a/qiskit/circuit/library/standard_gates/z.py b/qiskit/circuit/library/standard_gates/z.py
index 9fad215878f9..fcf7334df5eb 100644
--- a/qiskit/circuit/library/standard_gates/z.py
+++ b/qiskit/circuit/library/standard_gates/z.py
@@ -13,12 +13,16 @@
 """Z, CZ and CCZ gates."""
 
 from typing import Optional, Union
+
 import numpy
-from qiskit.qasm import pi
+
+from qiskit.circuit._utils import _compute_control_matrix
 from qiskit.circuit.controlledgate import ControlledGate
 from qiskit.circuit.gate import Gate
 from qiskit.circuit.quantumregister import QuantumRegister
-from qiskit.circuit._utils import _compute_control_matrix
+from qiskit.qasm import pi
+
+from .p import PhaseGate
 
 
 class ZGate(Gate):
@@ -74,6 +78,7 @@ def __init__(self, label: Optional[str] = None):
     def _define(self):
         # pylint: disable=cyclic-import
         from qiskit.circuit.quantumcircuit import QuantumCircuit
+
         from .u1 import U1Gate
 
         q = QuantumRegister(1, "q")
@@ -117,6 +122,10 @@ def __array__(self, dtype=None):
         """Return a numpy.array for the Z gate."""
         return numpy.array([[1, 0], [0, -1]], dtype=dtype)
 
+    def power(self, exponent: float):
+        """Raise gate to a power."""
+        return PhaseGate(numpy.pi * exponent)
+
 
 class CZGate(ControlledGate):
     r"""Controlled-Z gate.
@@ -163,6 +172,7 @@ def _define(self):
         """
         # pylint: disable=cyclic-import
         from qiskit.circuit.quantumcircuit import QuantumCircuit
+
         from .h import HGate
         from .x import CXGate
 
@@ -241,6 +251,7 @@ def _define(self):
         """
         # pylint: disable=cyclic-import
         from qiskit.circuit.quantumcircuit import QuantumCircuit
+
         from .h import HGate
         from .x import CCXGate
 
diff --git a/releasenotes/notes/efficient-gate-power-effa21e3ee4581ee.yaml b/releasenotes/notes/efficient-gate-power-effa21e3ee4581ee.yaml
new file mode 100644
index 000000000000..46b00ebd68f0
--- /dev/null
+++ b/releasenotes/notes/efficient-gate-power-effa21e3ee4581ee.yaml
@@ -0,0 +1,25 @@
+---
+features:
+  - |
+    Overrides some gate power methods to make them more efficient and less lossy:
+    - Powering a :class:`~.CPhaseGate` gives a :class:`~.CPhaseGate`.
+    - Powering a :class:`~.CSGate` gives a :class:`~.CPhaseGate`.
+    - Powering a :class:`~.CSdgGate` gives a :class:`~.CPhaseGate`.
+    - Powering an :class:`~.IGate` gives an :class:`~.IGate`.
+    - Powering a :class:`~.PhaseGate` gives a :class:`~.PhaseGate`.
+    - Powering an :class:`~.RGate` gives an :class:`~.RGate`.
+    - Powering an :class:`~.RXGate` gives an :class:`~.RXGate`.
+    - Powering an :class:`~.RXXGate` gives an :class:`~.RXXGate`.
+    - Powering an :class:`~.RYGate` gives an :class:`~.RYGate`.
+    - Powering an :class:`~.RYYGate` gives an :class:`~.RYYGate`.
+    - Powering an :class:`~.RZGate` gives an :class:`~.RZGate`.
+    - Powering an :class:`~.RZXGate` gives an :class:`~.RZXGate`.
+    - Powering an :class:`~.RZZGate` gives an :class:`~.RZZGate`.
+    - Powering a :class:`~.SdgGate` gives a :class:`~.PhaseGate`.
+    - Powering an :class:`~.SGate` gives a :class:`~.PhaseGate`.
+    - Powering a :class:`~.TdgGate` gives a :class:`~.PhaseGate`.
+    - Powering a :class:`~.TGate` gives a :class:`~.PhaseGate`.
+    - Powering an :class:`~.XXMinusYYGate` gives an :class:`~.XXMinusYYGate`.
+    - Powering an :class:`~.XXPlusYYGate` gives an :class:`~.XXPlusYYGate`.
+    - Powering a :class:`~.ZGate` gives a :class:`~.PhaseGate`.
+    - Powering an :class:`~.iSwapGate` gives an :class:`~.XXPlusYYGate`.
\ No newline at end of file
diff --git a/test/python/circuit/test_gate_power.py b/test/python/circuit/test_gate_power.py
index 3ea821dd5546..43de932ed246 100644
--- a/test/python/circuit/test_gate_power.py
+++ b/test/python/circuit/test_gate_power.py
@@ -14,13 +14,40 @@
 """Test Qiskit's power instruction operation."""
 
 import unittest
-from ddt import ddt, data
-from numpy import array, eye
+from typing import Type
+
+import numpy as np
+import scipy.linalg
+from ddt import data, ddt, unpack
 
-from qiskit.test import QiskitTestCase
-from qiskit.extensions import SGate, UnitaryGate, CXGate
 from qiskit.circuit import Gate, QuantumCircuit
+from qiskit.extensions import (
+    CPhaseGate,
+    CSdgGate,
+    CSGate,
+    CXGate,
+    IGate,
+    PhaseGate,
+    RGate,
+    RXGate,
+    RXXGate,
+    RYGate,
+    RYYGate,
+    RZGate,
+    RZXGate,
+    RZZGate,
+    SdgGate,
+    SGate,
+    TdgGate,
+    TGate,
+    UnitaryGate,
+    XXMinusYYGate,
+    XXPlusYYGate,
+    ZGate,
+    iSwapGate,
+)
 from qiskit.quantum_info.operators import Operator
+from qiskit.test import QiskitTestCase
 
 
 @ddt
@@ -31,16 +58,14 @@ class TestPowerSgate(QiskitTestCase):
     def test_sgate_int(self, n):
         """Test Sgate.power(n) method with n as integer."""
         result = SGate().power(n)
-
-        self.assertEqual(result.label, "s^%s" % n)
-        self.assertIsInstance(result, UnitaryGate)
+        self.assertIsInstance(result, PhaseGate)
         self.assertEqual(Operator(result), Operator(SGate()).power(n))
 
     results = {
-        1.5: array([[1, 0], [0, -0.70710678 + 0.70710678j]], dtype=complex),
-        0.1: array([[1, 0], [0, 0.98768834 + 0.15643447j]], dtype=complex),
-        -1.5: array([[1, 0], [0, -0.70710678 - 0.70710678j]], dtype=complex),
-        -0.1: array([[1, 0], [0, 0.98768834 - 0.15643447j]], dtype=complex),
+        1.5: np.array([[1, 0], [0, -0.70710678 + 0.70710678j]], dtype=complex),
+        0.1: np.array([[1, 0], [0, 0.98768834 + 0.15643447j]], dtype=complex),
+        -1.5: np.array([[1, 0], [0, -0.70710678 - 0.70710678j]], dtype=complex),
+        -0.1: np.array([[1, 0], [0, 0.98768834 - 0.15643447j]], dtype=complex),
     }
 
     @data(1.5, 0.1, -1.5, -0.1)
@@ -49,8 +74,7 @@ def test_sgate_float(self, n):
         result = SGate().power(n)
 
         expected = self.results[n]
-        self.assertEqual(result.label, "s^%s" % n)
-        self.assertIsInstance(result, UnitaryGate)
+        self.assertIsInstance(result, PhaseGate)
         self.assertEqual(Operator(result), Operator(expected))
 
 
@@ -59,11 +83,11 @@ class TestPowerIntCX(QiskitTestCase):
     """Test CX.power() with integer"""
 
     results = {
-        2: array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], dtype=complex),
-        1: array([[1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0], [0, 1, 0, 0]], dtype=complex),
-        0: array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], dtype=complex),
-        -1: array([[1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0], [0, 1, 0, 0]], dtype=complex),
-        -2: array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], dtype=complex),
+        2: np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], dtype=complex),
+        1: np.array([[1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0], [0, 1, 0, 0]], dtype=complex),
+        0: np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], dtype=complex),
+        -1: np.array([[1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0], [0, 1, 0, 0]], dtype=complex),
+        -2: np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], dtype=complex),
     }
 
     @data(2, 1, 0, -2)
@@ -81,7 +105,7 @@ class TestGateSqrt(QiskitTestCase):
 
     def test_unitary_sqrt(self):
         """Test UnitaryGate.power(1/2) method."""
-        expected = array([[0.5 + 0.5j, 0.5 + 0.5j], [-0.5 - 0.5j, 0.5 + 0.5j]], dtype=complex)
+        expected = np.array([[0.5 + 0.5j, 0.5 + 0.5j], [-0.5 - 0.5j, 0.5 + 0.5j]], dtype=complex)
 
         result = UnitaryGate([[0, 1j], [-1j, 0]]).power(1 / 2)
 
@@ -92,11 +116,10 @@ def test_unitary_sqrt(self):
 
     def test_standard_sqrt(self):
         """Test standard Gate.power(1/2) method."""
-        expected = array([[1, 0], [0, 0.70710678118 + 0.70710678118j]], dtype=complex)
+        expected = np.array([[1, 0], [0, 0.70710678118 + 0.70710678118j]], dtype=complex)
 
         result = SGate().power(1 / 2)
 
-        self.assertEqual(result.label, "s^0.5")
         self.assertEqual(len(result.definition), 1)
         self.assertIsInstance(result, Gate)
         self.assertEqual(Operator(result), Operator(expected))
@@ -104,7 +127,6 @@ def test_standard_sqrt(self):
     def test_composite_sqrt(self):
         """Test composite Gate.power(1/2) method."""
         circ = QuantumCircuit(1, name="my_gate")
-        import numpy as np
 
         thetaz = 0.1
         thetax = 0.2
@@ -141,7 +163,6 @@ def test_direct_root(self, degree):
         """Test nth root"""
         result = SGate().power(1 / degree)
 
-        self.assertEqual(result.label, "s^" + str(1 / degree))
         self.assertEqual(len(result.definition), 1)
         self.assertIsInstance(result, Gate)
         self.assertEqual(Operator(result).power(degree), Operator(SGate()))
@@ -151,7 +172,6 @@ def test_float_gt_one(self, exponent):
         """Test greater-than-one exponents"""
         result = SGate().power(exponent)
 
-        self.assertEqual(result.label, "s^" + str(exponent))
         self.assertEqual(len(result.definition), 1)
         self.assertIsInstance(result, Gate)
         # SGate().to_matrix() is diagonal so `**` is equivalent.
@@ -161,11 +181,10 @@ def test_minus_zero_two(self, exponent=-0.2):
         """Test Sgate^(-0.2)"""
         result = SGate().power(exponent)
 
-        self.assertEqual(result.label, "s^" + str(exponent))
         self.assertEqual(len(result.definition), 1)
         self.assertIsInstance(result, Gate)
         self.assertEqual(
-            Operator(array([[1, 0], [0, 0.95105652 - 0.30901699j]], dtype=complex)),
+            Operator(np.array([[1, 0], [0, 0.95105652 - 0.30901699j]], dtype=complex)),
             Operator(result),
         )
 
@@ -178,7 +197,6 @@ class TestPowerInvariant(QiskitTestCase):
     def test_invariant1_int(self, n):
         """Test (op^(1/n))^(n) == op, integer n"""
         result = SGate().power(1 / n).power(n)
-        self.assertEqual(result.label, "unitary^" + str(n))
         self.assertEqual(len(result.definition), 1)
         self.assertIsInstance(result, Gate)
         self.assertTrue(Operator(SGate()), Operator(result))
@@ -187,11 +205,49 @@ def test_invariant1_int(self, n):
     def test_invariant2(self, n):
         """Test op^(n) * op^(-n) == I"""
         result = Operator(SGate()).power(n) & Operator(SGate()).power(-n)
-        expected = Operator(eye(2))
+        expected = Operator(np.eye(2))
 
         self.assertEqual(len(result.data), len(expected.data))
         self.assertEqual(result, expected)
 
 
+@ddt
+class TestEfficientGatePowering(QiskitTestCase):
+    """Test gate powering is efficient where expected."""
+
+    @data(
+        (CPhaseGate(0.1), CPhaseGate),
+        (CSdgGate(), CPhaseGate),
+        (CSGate(), CPhaseGate),
+        (IGate(), IGate),
+        (PhaseGate(-0.1), PhaseGate),
+        (RGate(0.1, 0.1), RGate),
+        (RXGate(0.1), RXGate),
+        (RXXGate(-0.1), RXXGate),
+        (RYGate(-0.1), RYGate),
+        (RYYGate(0.1), RYYGate),
+        (RZGate(0.1), RZGate),
+        (RZXGate(-0.1), RZXGate),
+        (RZZGate(-0.1), RZZGate),
+        (SdgGate(), PhaseGate),
+        (SGate(), PhaseGate),
+        (TGate(), PhaseGate),
+        (TdgGate(), PhaseGate),
+        (XXMinusYYGate(-0.1, 0.1), XXMinusYYGate),
+        (XXPlusYYGate(2.1, 0.1), XXPlusYYGate),
+        (ZGate(), PhaseGate),
+        (iSwapGate(), XXPlusYYGate),
+    )
+    @unpack
+    def test_efficient_gate_powering(self, gate: Gate, output_gate_type: Type[Gate]):
+        """Test efficient gate powering."""
+        exponents = (-5, -0.5, -0.1, 0, 0.1, 0.5, 5)
+        for exponent in exponents:
+            result = gate.power(exponent)
+            self.assertIsInstance(result, output_gate_type)
+            expected = scipy.linalg.fractional_matrix_power(np.array(gate), exponent)
+            np.testing.assert_allclose(np.array(result), expected, atol=1e-8)
+
+
 if __name__ == "__main__":
     unittest.main()