diff --git a/CHANGELOG.md b/CHANGELOG.md
index 19b0fba87..876595e2d 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,5 +1,16 @@
 # Changelog
 
+## v1.62.1 (2023-11-17)
+
+### Bug Fixes and Other Changes
+
+ * Fix broken link to example notebook
+ * update: default no longer returning RETIRED devices from get_devices
+
+### Documentation Changes
+
+ * Add matrix expressions to docstrings
+
 ## v1.62.0 (2023-11-09)
 
 ### Features
diff --git a/doc/examples-hybrid-jobs.rst b/doc/examples-hybrid-jobs.rst
index af873407c..56a1d9ecf 100644
--- a/doc/examples-hybrid-jobs.rst
+++ b/doc/examples-hybrid-jobs.rst
@@ -8,7 +8,7 @@ Learn more about hybrid jobs on Amazon Braket.
     :maxdepth: 2
 
 ************************************************************************************************************************************************************************************************
-`Creating your first Hybrid Job <https://github.com/amazon-braket/amazon-braket-examples/blob/main/examples/hybrid_jobs/0_Creating_your_first_Hybrid_Job/Creating_your_first_Hybrid_Job.ipynb>`_
+`Creating your first Hybrid Job <https://github.com/amazon-braket/amazon-braket-examples/blob/main/examples/hybrid_jobs/0_Creating_your_first_Hybrid_Job/0_Creating_your_first_Hybrid_Job.ipynb>`_
 ************************************************************************************************************************************************************************************************
 
 This tutorial shows how to run your first Amazon Braket Hybrid Job.
diff --git a/src/braket/_sdk/_version.py b/src/braket/_sdk/_version.py
index a7a90664e..3e6932935 100644
--- a/src/braket/_sdk/_version.py
+++ b/src/braket/_sdk/_version.py
@@ -15,4 +15,4 @@
 Version number (major.minor.patch[-label])
 """
 
-__version__ = "1.62.1.dev0"
+__version__ = "1.62.2.dev0"
diff --git a/src/braket/circuits/gates.py b/src/braket/circuits/gates.py
index c3ea7bf77..b226104a1 100644
--- a/src/braket/circuits/gates.py
+++ b/src/braket/circuits/gates.py
@@ -59,7 +59,14 @@
 
 
 class H(Gate):
-    """Hadamard gate."""
+    r"""Hadamard gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{H} = \frac{1}{\sqrt{2}} \begin{bmatrix}
+                1 & 1 \\
+                1 & -1 \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["H"])
@@ -90,7 +97,13 @@ def h(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""Hadamard gate.
+
+        Unitary matrix:
+
+            .. math:: \mathtt{H} = \frac{1}{\sqrt{2}} \begin{bmatrix}
+                    1 & 1 \\
+                    1 & -1 \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s)
@@ -123,8 +136,15 @@ def h(
 Gate.register_gate(H)
 
 
-class I(Gate):  # noqa: E742
-    """Identity gate."""
+class I(Gate):  # noqa: E742, E261
+    r"""Identity gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{I} = \begin{bmatrix}
+                1 & 0 \\
+                0 & 1 \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["I"])
@@ -155,7 +175,13 @@ def i(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""Identity gate.
+
+        Unitary matrix:
+
+            .. math:: \mathtt{I} = \begin{bmatrix}
+                    1 & 0 \\
+                    0 & 1 \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s)
@@ -189,7 +215,15 @@ def i(
 
 
 class X(Gate):
-    """Pauli-X gate."""
+    r"""Pauli-X gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{X} = \begin{bmatrix}
+                0 & 1 \\
+                1 & 0
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["X"])
@@ -220,7 +254,14 @@ def x(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""Pauli-X gate.
+
+        Unitary matrix:
+
+            .. math:: \mathtt{X} = \begin{bmatrix}
+                    0 & 1 \\
+                    1 & 0
+                    \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s)
@@ -254,7 +295,15 @@ def x(
 
 
 class Y(Gate):
-    """Pauli-Y gate."""
+    r"""Pauli-Y gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{Y} = \begin{bmatrix}
+                0 & -i \\
+                i & 0
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["Y"])
@@ -285,7 +334,14 @@ def y(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""Pauli-Y gate.
+
+        Unitary matrix:
+
+            .. math:: \mathtt{Y} = \begin{bmatrix}
+                    0 & -i \\
+                    i & 0
+                    \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s)
@@ -319,7 +375,15 @@ def y(
 
 
 class Z(Gate):
-    """Pauli-Z gate."""
+    r"""Pauli-Z gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{Z} = \begin{bmatrix}
+                1 & 0 \\
+                0 & -1
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["Z"])
@@ -350,7 +414,12 @@ def z(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""Pauli-Z gate.
+
+        .. math:: \mathtt{Z} = \begin{bmatrix}
+                1 & 0 \\
+                0 & -1
+                \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s)
@@ -384,7 +453,15 @@ def z(
 
 
 class S(Gate):
-    """S gate."""
+    r"""S gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{S} = \begin{bmatrix}
+                1 & 0 \\
+                0 & i
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["S"])
@@ -415,7 +492,12 @@ def s(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""S gate.
+
+        .. math:: \mathtt{S} = \begin{bmatrix}
+                1 & 0 \\
+                0 & i
+                \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s)
@@ -449,7 +531,15 @@ def s(
 
 
 class Si(Gate):
-    """Conjugate transpose of S gate."""
+    r"""Conjugate transpose of S gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{S}^\dagger = \begin{bmatrix}
+                1 & 0 \\
+                0 & -i
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["Si"])
@@ -480,7 +570,12 @@ def si(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""Conjugate transpose of S gate.
+
+        .. math:: \mathtt{S}^\dagger = \begin{bmatrix}
+                1 & 0 \\
+                0 & -i
+                \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s)
@@ -514,7 +609,15 @@ def si(
 
 
 class T(Gate):
-    """T gate."""
+    r"""T gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{T} = \begin{bmatrix}
+                1 & 0 \\
+                0 & e^{i \pi/4}
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["T"])
@@ -545,7 +648,12 @@ def t(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""T gate.
+
+        .. math:: \mathtt{T} = \begin{bmatrix}
+                1 & 0 \\
+                0 & e^{i \pi/4}
+                \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s)
@@ -579,7 +687,15 @@ def t(
 
 
 class Ti(Gate):
-    """Conjugate transpose of T gate."""
+    r"""Conjugate transpose of T gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{T}^\dagger = \begin{bmatrix}
+                1 & 0 \\
+                0 & e^{-i \pi/4}
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["Ti"])
@@ -610,7 +726,12 @@ def ti(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""Conjugate transpose of T gate.
+
+        .. math:: \mathtt{T}^\dagger = \begin{bmatrix}
+                1 & 0 \\
+                0 & e^{-i \pi/4}
+                \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s)
@@ -644,7 +765,15 @@ def ti(
 
 
 class V(Gate):
-    """Square root of not gate."""
+    r"""Square root of X gate (V gate).
+
+    Unitary matrix:
+
+        .. math:: \mathtt{V} = \frac{1}{2}\begin{bmatrix}
+                1+i & 1-i \\
+                1-i & 1+i
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["V"])
@@ -675,7 +804,12 @@ def v(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""Square root of X gate (V gate).
+
+        .. math:: \mathtt{V} = \frac{1}{2}\begin{bmatrix}
+                1+i & 1-i \\
+                1-i & 1+i
+                \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s)
@@ -709,7 +843,15 @@ def v(
 
 
 class Vi(Gate):
-    """Conjugate transpose of square root of not gate."""
+    r"""Conjugate transpose of square root of X gate (conjugate transpose of V).
+
+    Unitary matrix:
+
+        .. math:: \mathtt{V}^\dagger = \frac{1}{2}\begin{bmatrix}
+                1-i & 1+i \\
+                1+i & 1-i
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["Vi"])
@@ -740,7 +882,12 @@ def vi(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""Conjugate transpose of square root of X gate (conjugate transpose of V).
+
+        .. math:: \mathtt{V}^\dagger = \frac{1}{2}\begin{bmatrix}
+                1-i & 1+i \\
+                1+i & 1-i
+                \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s)
@@ -777,7 +924,14 @@ def vi(
 
 
 class Rx(AngledGate):
-    """X-axis rotation gate.
+    r"""X-axis rotation gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{R_x}(\phi) = \begin{bmatrix}
+                \cos{(\phi/2)} & -i \sin{(\phi/2)} \\
+                -i \sin{(\phi/2)} & \cos{(\phi/2)}
+                \end{bmatrix}.
 
     Args:
         angle (Union[FreeParameterExpression, float]): angle in radians.
@@ -824,7 +978,12 @@ def rx(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""X-axis rotation gate.
+
+        .. math:: \mathtt{R_x}(\phi) = \begin{bmatrix}
+                \cos{(\phi/2)} & -i \sin{(\phi/2)} \\
+                -i \sin{(\phi/2)} & \cos{(\phi/2)}
+                \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s).
@@ -858,7 +1017,14 @@ def rx(
 
 
 class Ry(AngledGate):
-    """Y-axis rotation gate.
+    r"""Y-axis rotation gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{R_y}(\phi) = \begin{bmatrix}
+                \cos{(\phi/2)} & -\sin{(\phi/2)} \\
+                \sin{(\phi/2)} & \cos{(\phi/2)}
+                \end{bmatrix}.
 
     Args:
         angle (Union[FreeParameterExpression, float]): angle in radians.
@@ -905,7 +1071,12 @@ def ry(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""Y-axis rotation gate.
+
+        .. math:: \mathtt{R_y}(\phi) = \begin{bmatrix}
+                \cos{(\phi/2)} & -\sin{(\phi/2)} \\
+                \sin{(\phi/2)} & \cos{(\phi/2)}
+                \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s).
@@ -924,6 +1095,7 @@ def ry(
         Returns:
             Iterable[Instruction]: Rx instruction.
 
+
         Examples:
             >>> circ = Circuit().ry(0, 0.15)
         """
@@ -939,7 +1111,14 @@ def ry(
 
 
 class Rz(AngledGate):
-    """Z-axis rotation gate.
+    r"""Z-axis rotation gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{R_z}(\phi) = \begin{bmatrix}
+                e^{-i \phi/2} & 0 \\
+                0 & e^{i \phi/2}
+                \end{bmatrix}.
 
     Args:
         angle (Union[FreeParameterExpression, float]): angle in radians.
@@ -981,7 +1160,12 @@ def rz(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""Z-axis rotation gate.
+
+        .. math:: \mathtt{R_z}(\phi) = \begin{bmatrix}
+                e^{-i \phi/2} & 0 \\
+                0 & e^{i \phi/2}
+                \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s).
@@ -1015,7 +1199,14 @@ def rz(
 
 
 class PhaseShift(AngledGate):
-    """Phase shift gate.
+    r"""Phase shift gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{PhaseShift}(\phi) = \begin{bmatrix}
+                1 & 0 \\
+                0 & e^{i \phi}
+                \end{bmatrix}
 
     Args:
         angle (Union[FreeParameterExpression, float]): angle in radians.
@@ -1055,7 +1246,12 @@ def phaseshift(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""Phase shift gate.
+
+        .. math:: \mathtt{PhaseShift}(\phi) = \begin{bmatrix}
+                1 & 0 \\
+                0 & e^{i \phi}
+                \end{bmatrix}
 
         Args:
             target (QubitSetInput): Target qubit(s).
@@ -1096,7 +1292,17 @@ def phaseshift(
 
 
 class CNot(Gate):
-    """Controlled NOT gate."""
+    r"""Controlled NOT gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{CNOT} = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 0 & 1 \\
+                0 & 0 & 1 & 0 \\
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["C", "X"])
@@ -1129,7 +1335,14 @@ def fixed_qubit_count() -> int:
     @staticmethod
     @circuit.subroutine(register=True)
     def cnot(control: QubitSetInput, target: QubitInput, power: float = 1) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Controlled NOT gate.
+
+        .. math:: \mathtt{CNOT} = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 0 & 1 \\
+                0 & 0 & 1 & 0 \\
+                \end{bmatrix}.
 
         Args:
             control (QubitSetInput): Control qubit(s). The last control qubit
@@ -1156,7 +1369,17 @@ def cnot(control: QubitSetInput, target: QubitInput, power: float = 1) -> Instru
 
 
 class Swap(Gate):
-    """Swap gate."""
+    r"""Swap gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{SWAP} = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 0 & 1 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 0 & 1 \\
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["SWAP", "SWAP"])
@@ -1196,7 +1419,14 @@ def swap(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Swap gate.
+
+        .. math:: \mathtt{SWAP} = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 0 & 1 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 0 & 1 \\
+                \end{bmatrix}.
 
         Args:
             target1 (QubitInput): Target qubit 1 index.
@@ -1231,7 +1461,17 @@ def swap(
 
 
 class ISwap(Gate):
-    """ISwap gate."""
+    r"""ISwap gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{iSWAP} = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 0 & i & 0 \\
+                0 & i & 0 & 0 \\
+                0 & 0 & 0 & 1 \\
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["ISWAP", "ISWAP"])
@@ -1271,7 +1511,14 @@ def iswap(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""ISwap gate.
+
+        .. math:: \mathtt{iSWAP} = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 0 & i & 0 \\
+                0 & i & 0 & 0 \\
+                0 & 0 & 0 & 1 \\
+                \end{bmatrix}.
 
         Args:
             target1 (QubitInput): Target qubit 1 index.
@@ -1306,7 +1553,16 @@ def iswap(
 
 
 class PSwap(AngledGate):
-    """PSwap gate.
+    r"""PSwap gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{PSWAP}(\phi) = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 0 & e^{i \phi} & 0 \\
+                0 & e^{i \phi} & 0 & 0 \\
+                0 & 0 & 0 & 1 \\
+                \end{bmatrix}.
 
     Args:
         angle (Union[FreeParameterExpression, float]): angle in radians.
@@ -1358,7 +1614,14 @@ def pswap(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""PSwap gate.
+
+        .. math:: \mathtt{PSWAP}(\phi) = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 0 & e^{i \phi} & 0 \\
+                0 & e^{i \phi} & 0 & 0 \\
+                0 & 0 & 0 & 1 \\
+                \end{bmatrix}.
 
         Args:
             target1 (QubitInput): Target qubit 1 index.
@@ -1394,10 +1657,20 @@ def pswap(
 
 
 class XY(AngledGate):
-    """XY gate.
+    r"""XY gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{XY}(\phi) = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & \cos{(\phi/2)} & i\sin{(\phi/2)} & 0 \\
+                0 & i\sin{(\phi/2)} & \cos{(\phi/2)} & 0 \\
+                0 & 0 & 0 & 1 \\
+            \end{bmatrix}.
 
     Reference: https://arxiv.org/abs/1912.04424v1
 
+
     Args:
         angle (Union[FreeParameterExpression, float]): angle in radians.
     """
@@ -1455,7 +1728,14 @@ def xy(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""XY gate.
+
+        .. math:: \mathtt{XY}(\phi) = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & \cos{(\phi/2)} & i\sin{(\phi/2)} & 0 \\
+                0 & i\sin{(\phi/2)} & \cos{(\phi/2)} & 0 \\
+                0 & 0 & 0 & 1 \\
+            \end{bmatrix}.
 
         Args:
             target1 (QubitInput): Target qubit 1 index.
@@ -1491,7 +1771,16 @@ def xy(
 
 
 class CPhaseShift(AngledGate):
-    """Controlled phase shift gate.
+    r"""Controlled phase shift gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{CPhaseShift}(\phi) = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 1 & 0 \\
+                0 & 0 & 0 & e^{i \phi}
+            \end{bmatrix}.
 
     Args:
         angle (Union[FreeParameterExpression, float]): angle in radians.
@@ -1529,7 +1818,14 @@ def cphaseshift(
         angle: Union[FreeParameterExpression, float],
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Controlled phase shift gate.
+
+        .. math:: \mathtt{CPhaseShift}(\phi) = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 1 & 0 \\
+                0 & 0 & 0 & e^{i \phi}
+            \end{bmatrix}.
 
         Args:
             control (QubitSetInput): Control qubit(s). The last control qubit
@@ -1560,7 +1856,16 @@ def cphaseshift(
 
 
 class CPhaseShift00(AngledGate):
-    r"""Controlled phase shift gate for phasing the \\|00> state.
+    r"""Controlled phase shift gate for phasing the \|00> state.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{CPhaseShift00}(\phi) = \begin{bmatrix}
+                e^{i \phi} & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 1 & 0 \\
+                0 & 0 & 0 & 1
+                \end{bmatrix}.
 
     Args:
         angle (Union[FreeParameterExpression, float]): angle in radians.
@@ -1598,7 +1903,14 @@ def cphaseshift00(
         angle: Union[FreeParameterExpression, float],
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Controlled phase shift gate for phasing the \|00> state.
+
+        .. math:: \mathtt{CPhaseShift00}(\phi) = \begin{bmatrix}
+                e^{i \phi} & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 1 & 0 \\
+                0 & 0 & 0 & 1
+                \end{bmatrix}.
 
         Args:
             control (QubitSetInput): Control qubit(s). The last control qubit
@@ -1629,7 +1941,16 @@ def cphaseshift00(
 
 
 class CPhaseShift01(AngledGate):
-    r"""Controlled phase shift gate for phasing the \\|01> state.
+    r"""Controlled phase shift gate for phasing the \|01> state.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{CPhaseShift01}(\phi) = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & e^{i \phi} & 0 & 0 \\
+                0 & 0 & 1 & 0 \\
+                0 & 0 & 0 & 1
+            \end{bmatrix}.
 
     Args:
         angle (Union[FreeParameterExpression, float]): angle in radians.
@@ -1667,7 +1988,14 @@ def cphaseshift01(
         angle: Union[FreeParameterExpression, float],
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Controlled phase shift gate for phasing the \|01> state.
+
+        .. math:: \mathtt{CPhaseShift01}(\phi) = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & e^{i \phi} & 0 & 0 \\
+                0 & 0 & 1 & 0 \\
+                0 & 0 & 0 & 1
+            \end{bmatrix}.
 
         Args:
             control (QubitSetInput): Control qubit(s). The last control qubit
@@ -1700,6 +2028,15 @@ def cphaseshift01(
 class CPhaseShift10(AngledGate):
     r"""Controlled phase shift gate for phasing the \\|10> state.
 
+    Unitary matrix:
+
+        .. math:: \mathtt{CPhaseShift10}(\phi) = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & e^{i \phi} & 0 \\
+                0 & 0 & 0 & 1
+                \end{bmatrix}.
+
     Args:
         angle (Union[FreeParameterExpression, float]): angle in radians.
     """
@@ -1736,7 +2073,14 @@ def cphaseshift10(
         angle: Union[FreeParameterExpression, float],
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Controlled phase shift gate for phasing the \\|10> state.
+
+        .. math:: \mathtt{CPhaseShift10}(\phi) = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & e^{i \phi} & 0 \\
+                0 & 0 & 0 & 1
+                \end{bmatrix}.
 
         Args:
             control (QubitSetInput): Control qubit(s). The last control qubit
@@ -1767,7 +2111,17 @@ def cphaseshift10(
 
 
 class CV(Gate):
-    """Controlled Sqrt of NOT gate."""
+    r"""Controlled Sqrt of X gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{CV} = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 0.5+0.5i & 0.5-0.5i \\
+                0 & 0 & 0.5-0.5i & 0.5+0.5i
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["C", "V"])
@@ -1800,7 +2154,14 @@ def fixed_qubit_count() -> int:
     @staticmethod
     @circuit.subroutine(register=True)
     def cv(control: QubitSetInput, target: QubitInput, power: float = 1) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Controlled Sqrt of X gate.
+
+        .. math:: \mathtt{CV} = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 0.5+0.5i & 0.5-0.5i \\
+                0 & 0 & 0.5-0.5i & 0.5+0.5i
+                \end{bmatrix}.
 
         Args:
             control (QubitSetInput): Control qubit(s). The last control qubit
@@ -1827,7 +2188,17 @@ def cv(control: QubitSetInput, target: QubitInput, power: float = 1) -> Instruct
 
 
 class CY(Gate):
-    """Controlled Pauli-Y gate."""
+    r"""Controlled Pauli-Y gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{CY} = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 0 & -i \\
+                0 & 0 & i & 0
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["C", "Y"])
@@ -1860,7 +2231,14 @@ def fixed_qubit_count() -> int:
     @staticmethod
     @circuit.subroutine(register=True)
     def cy(control: QubitSetInput, target: QubitInput, power: float = 1) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Controlled Pauli-Y gate.
+
+        .. math:: \mathtt{CY} = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 0 & -i \\
+                0 & 0 & i & 0
+                \end{bmatrix}.
 
         Args:
             control (QubitSetInput): Control qubit(s). The last control qubit
@@ -1887,7 +2265,17 @@ def cy(control: QubitSetInput, target: QubitInput, power: float = 1) -> Instruct
 
 
 class CZ(Gate):
-    """Controlled Pauli-Z gate."""
+    r"""Controlled Pauli-Z gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{CZ} = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 1 & 0 \\
+                0 & 0 & 0 & -1
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["C", "Z"])
@@ -1912,7 +2300,14 @@ def fixed_qubit_count() -> int:
     @staticmethod
     @circuit.subroutine(register=True)
     def cz(control: QubitSetInput, target: QubitInput, power: float = 1) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Controlled Pauli-Z gate.
+
+        .. math:: \mathtt{CZ} = \begin{bmatrix}
+                1 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 \\
+                0 & 0 & 1 & 0 \\
+                0 & 0 & 0 & -1
+                \end{bmatrix}.
 
         Args:
             control (QubitSetInput): Control qubit(s). The last control qubit
@@ -1939,7 +2334,17 @@ def cz(control: QubitSetInput, target: QubitInput, power: float = 1) -> Instruct
 
 
 class ECR(Gate):
-    """An echoed RZX(pi/2) gate."""
+    r"""An echoed RZX(pi/2) gate (ECR gate).
+
+    Unitary matrix:
+
+        .. math:: \mathtt{ECR} = \begin{bmatrix}
+                0 & 0 & 1 & i \\
+                0 & 0 & i & 1 \\
+                1 & -i & 0 & 0 \\
+                -i & 1 & 0 & 0
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["ECR", "ECR"])
@@ -1978,7 +2383,14 @@ def ecr(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""An echoed RZX(pi/2) gate (ECR gate).
+
+        .. math:: \mathtt{ECR} = \begin{bmatrix}
+                0 & 0 & 1 & i \\
+                0 & 0 & i & 1 \\
+                1 & -i & 0 & 0 \\
+                -i & 1 & 0 & 0
+                \end{bmatrix}.
 
         Args:
             target1 (QubitInput): Target qubit 1 index.
@@ -2013,7 +2425,16 @@ def ecr(
 
 
 class XX(AngledGate):
-    """Ising XX coupling gate.
+    r"""Ising XX coupling gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{XX}(\phi) = \begin{bmatrix}
+                \cos{(\phi/2)} & 0 & 0 & -i \sin{(\phi/2)} \\
+                0 & \cos{(\phi/2)} & -i \sin{(\phi/2)} & 0 \\
+                0 & -i \sin{(\phi/2)} & \cos{(\phi/2)} & 0 \\
+                -i \sin{(\phi/2)} & 0 & 0 & \cos{(\phi/2)}
+                \end{bmatrix}.
 
     Reference: https://arxiv.org/abs/1707.06356
 
@@ -2074,7 +2495,14 @@ def xx(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Ising XX coupling gate.
+
+        .. math:: \mathtt{XX}(\phi) = \begin{bmatrix}
+                \cos{(\phi/2)} & 0 & 0 & -i \sin{(\phi/2)} \\
+                0 & \cos{(\phi/2)} & -i \sin{(\phi/2)} & 0 \\
+                0 & -i \sin{(\phi/2)} & \cos{(\phi/2)} & 0 \\
+                -i \sin{(\phi/2)} & 0 & 0 & \cos{(\phi/2)}
+                \end{bmatrix}.
 
         Args:
             target1 (QubitInput): Target qubit 1 index.
@@ -2110,7 +2538,16 @@ def xx(
 
 
 class YY(AngledGate):
-    """Ising YY coupling gate.
+    r"""Ising YY coupling gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{YY}(\phi) = \begin{bmatrix}
+                \cos{(\phi/2)} & 0 & 0 & i \sin{(\phi/2)} \\
+                0 & \cos{(\phi/2)} & -i \sin{(\phi/2)} & 0 \\
+                0 & -i \sin{(\phi/2)} & \cos{(\phi/2)} & 0 \\
+                i \sin{(\phi/2)} & 0 & 0 & \cos{(\phi/2)}
+                \end{bmatrix}.
 
     Reference: https://arxiv.org/abs/1707.06356
 
@@ -2171,7 +2608,14 @@ def yy(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Ising YY coupling gate.
+
+        .. math:: \mathtt{YY}(\phi) = \begin{bmatrix}
+                \cos{(\phi/2)} & 0 & 0 & i \sin{(\phi/2)} \\
+                0 & \cos{(\phi/2)} & -i \sin{(\phi/2)} & 0 \\
+                0 & -i \sin{(\phi/2)} & \cos{(\phi/2)} & 0 \\
+                i \sin{(\phi/2)} & 0 & 0 & \cos{(\phi/2)}
+                \end{bmatrix}.
 
         Args:
             target1 (QubitInput): Target qubit 1 index.
@@ -2207,7 +2651,16 @@ def yy(
 
 
 class ZZ(AngledGate):
-    """Ising ZZ coupling gate.
+    r"""Ising ZZ coupling gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{ZZ}(\phi) = \begin{bmatrix}
+                e^{-i\phi/2} & 0 & 0 & 0 \\
+                0 & e^{i\phi/2} & 0 & 0 \\
+                0 & 0 & e^{i\phi/2} & 0 \\
+                0 & 0 & 0 & e^{-i\phi/2}
+            \end{bmatrix}.
 
     Reference: https://arxiv.org/abs/1707.06356
 
@@ -2261,7 +2714,14 @@ def zz(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Ising ZZ coupling gate.
+
+        .. math:: \mathtt{ZZ}(\phi) = \begin{bmatrix}
+                e^{-i\phi/2} & 0 & 0 & 0 \\
+                0 & e^{i\phi/2} & 0 & 0 \\
+                0 & 0 & e^{i\phi/2} & 0 \\
+                0 & 0 & 0 & e^{-i\phi/2}
+            \end{bmatrix}.
 
         Args:
             target1 (QubitInput): Target qubit 1 index.
@@ -2300,7 +2760,21 @@ def zz(
 
 
 class CCNot(Gate):
-    """CCNOT gate or Toffoli gate."""
+    r"""CCNOT gate or Toffoli gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{CCNOT} = \begin{bmatrix}
+                1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\
+                0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
+                0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
+                0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
+                0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
+                0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
+                0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["C", "C", "X"])
@@ -2345,7 +2819,18 @@ def ccnot(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""CCNOT gate or Toffoli gate.
+
+        .. math:: \mathtt{CCNOT} = \begin{bmatrix}
+                1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\
+                0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
+                0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
+                0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
+                0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
+                0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
+                0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
+                \end{bmatrix}.
 
         Args:
             control1 (QubitInput): Control qubit 1 index.
@@ -2383,7 +2868,21 @@ def ccnot(
 
 
 class CSwap(Gate):
-    """Controlled Swap gate."""
+    r"""Controlled Swap gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{CSWAP} = \begin{bmatrix}
+                1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\
+                0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
+                0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
+                0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
+                0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
+                0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
+                0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
+                \end{bmatrix}.
+    """
 
     def __init__(self):
         super().__init__(qubit_count=None, ascii_symbols=["C", "SWAP", "SWAP"])
@@ -2425,7 +2924,18 @@ def cswap(
         target2: QubitInput,
         power: float = 1,
     ) -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Controlled Swap gate.
+
+        .. math:: \mathtt{CSWAP} = \begin{bmatrix}
+                1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+                0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\
+                0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
+                0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
+                0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
+                0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
+                0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
+                0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
+                \end{bmatrix}.
 
         Args:
             control (QubitSetInput): Control qubit(s). The last control qubit
@@ -2456,7 +2966,14 @@ def cswap(
 
 
 class GPi(AngledGate):
-    """IonQ GPi gate.
+    r"""IonQ GPi gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{GPi}(\phi) = \begin{bmatrix}
+                0 & e^{-i \phi} \\
+                e^{i \phi} & 0
+                \end{bmatrix}.
 
     Args:
         angle (Union[FreeParameterExpression, float]): angle in radians.
@@ -2501,7 +3018,12 @@ def gpi(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""IonQ GPi gate.
+
+        .. math:: \mathtt{GPi}(\phi) = \begin{bmatrix}
+                0 & e^{-i \phi} \\
+                e^{i \phi} & 0
+                \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s).
@@ -2535,7 +3057,14 @@ def gpi(
 
 
 class GPi2(AngledGate):
-    """IonQ GPi2 gate.
+    r"""IonQ GPi2 gate.
+
+    Unitary matrix:
+
+        .. math:: \mathtt{GPi2}(\phi) = \begin{bmatrix}
+                1 & -i e^{-i \phi} \\
+                -i e^{i \phi} & 1
+            \end{bmatrix}.
 
     Args:
         angle (Union[FreeParameterExpression, float]): angle in radians.
@@ -2580,7 +3109,12 @@ def gpi2(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""IonQ GPi2 gate.
+
+        .. math:: \mathtt{GPi2}(\phi) = \begin{bmatrix}
+                1 & -i e^{-i \phi} \\
+                -i e^{i \phi} & 1
+            \end{bmatrix}.
 
         Args:
             target (QubitSetInput): Target qubit(s).
@@ -2614,12 +3148,26 @@ def gpi2(
 
 
 class MS(TripleAngledGate):
-    """IonQ Mølmer-Sørenson gate.
+    r"""IonQ Mølmer-Sørensen gate.
+
+    Unitary matrix:
+
+            .. math:: &\mathtt{MS}(\phi_0, \phi_1, \theta) =\\ &\begin{bmatrix}
+                    \cos{\frac{\theta}{2}} & 0 &
+                    0 & -ie^{-i (\phi_0 + \phi_1)}\sin{\frac{\theta}{2}} \\
+                    0 & \cos{\frac{\theta}{2}} &
+                    -ie^{-i (\phi_0 - \phi_1)}\sin{\frac{\theta}{2}} & 0 \\
+                    0 & -ie^{i (\phi_0 - \phi_1)}\sin{\frac{\theta}{2}} &
+                    \cos{\frac{\theta}{2}} & 0 \\
+                    -ie^{i (\phi_0 + \phi_1)}\sin{\frac{\theta}{2}} & 0
+                    & 0 & \cos{\frac{\theta}{2}}
+                    \end{bmatrix}.
 
     Args:
         angle_1 (Union[FreeParameterExpression, float]): angle in radians.
         angle_2 (Union[FreeParameterExpression, float]): angle in radians.
         angle_3 (Union[FreeParameterExpression, float]): angle in radians.
+        Default value is angle_3=pi/2.
     """
 
     def __init__(
@@ -2693,7 +3241,18 @@ def ms(
         control_state: Optional[BasisStateInput] = None,
         power: float = 1,
     ) -> Iterable[Instruction]:
-        r"""Registers this function into the circuit class.
+        r"""IonQ Mølmer-Sørensen gate.
+
+        .. math:: &\mathtt{MS}(\phi_0, \phi_1, \theta) =\\ &\begin{bmatrix}
+                    \cos{\frac{\theta}{2}} & 0 &
+                    0 & -ie^{-i (\phi_0 + \phi_1)}\sin{\frac{\theta}{2}} \\
+                    0 & \cos{\frac{\theta}{2}} &
+                    -ie^{-i (\phi_0 - \phi_1)}\sin{\frac{\theta}{2}} & 0 \\
+                    0 & -ie^{i (\phi_0 - \phi_1)}\sin{\frac{\theta}{2}} &
+                    \cos{\frac{\theta}{2}} & 0 \\
+                    -ie^{i (\phi_0 + \phi_1)}\sin{\frac{\theta}{2}} & 0
+                    & 0 & \cos{\frac{\theta}{2}}
+                    \end{bmatrix}.
 
         Args:
             target1 (QubitInput): Target qubit 1 index.
@@ -2733,7 +3292,7 @@ def ms(
 
 
 class Unitary(Gate):
-    """Arbitrary unitary gate
+    """Arbitrary unitary gate.
 
     Args:
         matrix (numpy.ndarray): Unitary matrix which defines the gate.
@@ -2796,7 +3355,7 @@ def _transform_matrix_to_ir(matrix: np.ndarray) -> list:
     @staticmethod
     @circuit.subroutine(register=True)
     def unitary(targets: QubitSet, matrix: np.ndarray, display_name: str = "U") -> Instruction:
-        r"""Registers this function into the circuit class.
+        r"""Arbitrary unitary gate.
 
         Args:
             targets (QubitSet): Target qubits.