diff --git a/dev_tools/autogenerate-bloqs-notebooks-v2.py b/dev_tools/autogenerate-bloqs-notebooks-v2.py
index 0a0dbe453..8ae0b129e 100644
--- a/dev_tools/autogenerate-bloqs-notebooks-v2.py
+++ b/dev_tools/autogenerate-bloqs-notebooks-v2.py
@@ -106,6 +106,7 @@
 import qualtran.bloqs.factoring.ecc
 import qualtran.bloqs.factoring.mod_exp
 import qualtran.bloqs.gf_arithmetic.gf2_addition
+import qualtran.bloqs.gf_arithmetic.gf2_inverse
 import qualtran.bloqs.gf_arithmetic.gf2_multiplication
 import qualtran.bloqs.gf_arithmetic.gf2_square
 import qualtran.bloqs.hamiltonian_simulation.hamiltonian_simulation_by_gqsp
@@ -573,6 +574,11 @@
         module=qualtran.bloqs.gf_arithmetic.gf2_square,
         bloq_specs=[qualtran.bloqs.gf_arithmetic.gf2_square._GF2_SQUARE_DOC],
     ),
+    NotebookSpecV2(
+        title='GF($2^m$) Inverse',
+        module=qualtran.bloqs.gf_arithmetic.gf2_inverse,
+        bloq_specs=[qualtran.bloqs.gf_arithmetic.gf2_inverse._GF2_INVERSE_DOC],
+    ),
 ]
 
 
diff --git a/docs/bloqs/index.rst b/docs/bloqs/index.rst
index 6a543d88b..208f6968b 100644
--- a/docs/bloqs/index.rst
+++ b/docs/bloqs/index.rst
@@ -94,6 +94,7 @@ Bloqs Library
     gf_arithmetic/gf2_multiplication.ipynb
     gf_arithmetic/gf2_addition.ipynb
     gf_arithmetic/gf2_square.ipynb
+    gf_arithmetic/gf2_inverse.ipynb
 
 .. toctree::
     :maxdepth: 2
diff --git a/qualtran/bloqs/gf_arithmetic/__init__.py b/qualtran/bloqs/gf_arithmetic/__init__.py
index 56084da16..d53076128 100644
--- a/qualtran/bloqs/gf_arithmetic/__init__.py
+++ b/qualtran/bloqs/gf_arithmetic/__init__.py
@@ -13,5 +13,6 @@
 #  limitations under the License.
 
 from qualtran.bloqs.gf_arithmetic.gf2_addition import GF2Addition
+from qualtran.bloqs.gf_arithmetic.gf2_inverse import GF2Inverse
 from qualtran.bloqs.gf_arithmetic.gf2_multiplication import GF2Multiplication
 from qualtran.bloqs.gf_arithmetic.gf2_square import GF2Square
diff --git a/qualtran/bloqs/gf_arithmetic/gf2_inverse.ipynb b/qualtran/bloqs/gf_arithmetic/gf2_inverse.ipynb
new file mode 100644
index 000000000..43c4ed58a
--- /dev/null
+++ b/qualtran/bloqs/gf_arithmetic/gf2_inverse.ipynb
@@ -0,0 +1,182 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "25463ba5",
+   "metadata": {
+    "cq.autogen": "title_cell"
+   },
+   "source": [
+    "# GF($2^m$) Inverse"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "38263a0a",
+   "metadata": {
+    "cq.autogen": "top_imports"
+   },
+   "outputs": [],
+   "source": [
+    "from qualtran import Bloq, CompositeBloq, BloqBuilder, Signature, Register\n",
+    "from qualtran import QBit, QInt, QUInt, QAny\n",
+    "from qualtran.drawing import show_bloq, show_call_graph, show_counts_sigma\n",
+    "from typing import *\n",
+    "import numpy as np\n",
+    "import sympy\n",
+    "import cirq"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9054cbbb",
+   "metadata": {
+    "cq.autogen": "GF2Inverse.bloq_doc.md"
+   },
+   "source": [
+    "## `GF2Inverse`\n",
+    "Out of place inversion for elements in GF($2^m$)\n",
+    "\n",
+    "Given a quantum register storing elements from GF($2^m$), this bloq computes the inverse\n",
+    "of the given element in a new output register, out-of-place. Specifically,\n",
+    "it implements the transformation\n",
+    "\n",
+    "$$\n",
+    "    |a\\rangle |0\\rangle \\rightarrow |a\\rangle |a^{-1}\\rangle\n",
+    "$$\n",
+    "\n",
+    "Inverse is computed by using Fermat's little theorem for Finite Fields, which states that\n",
+    "for a finite field $\\mathbb{F}$ with $m$ elements, $\\forall a \\in \\mathbb{F}$\n",
+    "$$\n",
+    "    a^{m} = a\n",
+    "$$\n",
+    "\n",
+    "When the finite field is GF($2^m$), Fermat's little theorem can be used to obtain the relation\n",
+    "\n",
+    "$$\n",
+    "    a^{-1} = a^{2^m - 2}\n",
+    "$$\n",
+    "\n",
+    "Thus, the inverse can be obtained via $m - 1$ squaring and multiplication operations.\n",
+    "\n",
+    "#### Parameters\n",
+    " - `bitsize`: The degree $m$ of the galois field $GF(2^m)$. Also corresponds to the number of qubits in the input register whose inverse should be calculated. \n",
+    "\n",
+    "#### Registers\n",
+    " - `x`: Input THRU register of size $m$ that stores elements from $GF(2^m)$.\n",
+    " - `result`: Output RIGHT register of size $m$ that stores $x^{-1}$ from $GF(2^m)$.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b3a43c81",
+   "metadata": {
+    "cq.autogen": "GF2Inverse.bloq_doc.py"
+   },
+   "outputs": [],
+   "source": [
+    "from qualtran.bloqs.gf_arithmetic import GF2Inverse"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b0bf14cc",
+   "metadata": {
+    "cq.autogen": "GF2Inverse.example_instances.md"
+   },
+   "source": [
+    "### Example Instances"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f6ce3222",
+   "metadata": {
+    "cq.autogen": "GF2Inverse.gf16_inverse"
+   },
+   "outputs": [],
+   "source": [
+    "gf16_inverse = GF2Inverse(4)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6c26f44f",
+   "metadata": {
+    "cq.autogen": "GF2Inverse.gf2_inverse_symbolic"
+   },
+   "outputs": [],
+   "source": [
+    "import sympy\n",
+    "\n",
+    "m = sympy.Symbol('m')\n",
+    "gf2_inverse_symbolic = GF2Inverse(m)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c813ddef",
+   "metadata": {
+    "cq.autogen": "GF2Inverse.graphical_signature.md"
+   },
+   "source": [
+    "#### Graphical Signature"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "591221ff",
+   "metadata": {
+    "cq.autogen": "GF2Inverse.graphical_signature.py"
+   },
+   "outputs": [],
+   "source": [
+    "from qualtran.drawing import show_bloqs\n",
+    "show_bloqs([gf16_inverse, gf2_inverse_symbolic],\n",
+    "           ['`gf16_inverse`', '`gf2_inverse_symbolic`'])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c96fb97d",
+   "metadata": {
+    "cq.autogen": "GF2Inverse.call_graph.md"
+   },
+   "source": [
+    "### Call Graph"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "67aeeb6c",
+   "metadata": {
+    "cq.autogen": "GF2Inverse.call_graph.py"
+   },
+   "outputs": [],
+   "source": [
+    "from qualtran.resource_counting.generalizers import ignore_split_join\n",
+    "gf16_inverse_g, gf16_inverse_sigma = gf16_inverse.call_graph(max_depth=1, generalizer=ignore_split_join)\n",
+    "show_call_graph(gf16_inverse_g)\n",
+    "show_counts_sigma(gf16_inverse_sigma)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "name": "python"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/qualtran/bloqs/gf_arithmetic/gf2_inverse.py b/qualtran/bloqs/gf_arithmetic/gf2_inverse.py
new file mode 100644
index 000000000..4ad07410e
--- /dev/null
+++ b/qualtran/bloqs/gf_arithmetic/gf2_inverse.py
@@ -0,0 +1,157 @@
+#  Copyright 2024 Google LLC
+#
+#  Licensed under the Apache License, Version 2.0 (the "License");
+#  you may not use this file except in compliance with the License.
+#  You may obtain a copy of the License at
+#
+#      https://www.apache.org/licenses/LICENSE-2.0
+#
+#  Unless required by applicable law or agreed to in writing, software
+#  distributed under the License is distributed on an "AS IS" BASIS,
+#  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+#  See the License for the specific language governing permissions and
+#  limitations under the License.
+from functools import cached_property
+from typing import Dict, Set, TYPE_CHECKING, Union
+
+import attrs
+import numpy as np
+
+from qualtran import (
+    Bloq,
+    bloq_example,
+    BloqDocSpec,
+    DecomposeTypeError,
+    QGF,
+    Register,
+    Side,
+    Signature,
+)
+from qualtran.bloqs.gf_arithmetic.gf2_addition import GF2Addition
+from qualtran.bloqs.gf_arithmetic.gf2_multiplication import GF2Multiplication
+from qualtran.bloqs.gf_arithmetic.gf2_square import GF2Square
+from qualtran.symbolics import is_symbolic, SymbolicInt
+
+if TYPE_CHECKING:
+    from qualtran import BloqBuilder, Soquet
+    from qualtran.resource_counting import BloqCountDictT, BloqCountT, SympySymbolAllocator
+    from qualtran.simulation.classical_sim import ClassicalValT
+
+
+@attrs.frozen
+class GF2Inverse(Bloq):
+    r"""Out of place inversion for elements in GF($2^m$)
+
+    Given a quantum register storing elements from GF($2^m$), this bloq computes the inverse
+    of the given element in a new output register, out-of-place. Specifically,
+    it implements the transformation
+
+    $$
+        |a\rangle |0\rangle \rightarrow |a\rangle |a^{-1}\rangle
+    $$
+
+    Inverse is computed by using Fermat's little theorem for Finite Fields, which states that
+    for a finite field $\mathbb{F}$ with $m$ elements, $\forall a \in \mathbb{F}$
+    $$
+        a^{m} = a
+    $$
+
+    When the finite field is GF($2^m$), Fermat's little theorem can be used to obtain the relation
+
+    $$
+        a^{-1} = a^{2^m - 2}
+    $$
+
+    Thus, the inverse can be obtained via $m - 1$ squaring and multiplication operations.
+
+    Args:
+        bitsize: The degree $m$ of the galois field $GF(2^m)$. Also corresponds to the number of
+            qubits in the input register whose inverse should be calculated.
+
+    Registers:
+        x: Input THRU register of size $m$ that stores elements from $GF(2^m)$.
+        result: Output RIGHT register of size $m$ that stores $x^{-1}$ from $GF(2^m)$.
+    """
+
+    bitsize: SymbolicInt
+
+    @cached_property
+    def signature(self) -> 'Signature':
+        junk_reg = (
+            [Register('junk', dtype=self.qgf, shape=(self.bitsize - 2,), side=Side.RIGHT)]
+            if is_symbolic(self.bitsize) or self.bitsize > 2
+            else []
+        )
+        return Signature(
+            [
+                Register('x', dtype=self.qgf),
+                Register('result', dtype=self.qgf, side=Side.RIGHT),
+                *junk_reg,
+            ]
+        )
+
+    @cached_property
+    def qgf(self) -> QGF:
+        return QGF(characteristic=2, degree=self.bitsize)
+
+    def build_composite_bloq(self, bb: 'BloqBuilder', *, x: 'Soquet') -> Dict[str, 'Soquet']:
+        if is_symbolic(self.bitsize):
+            raise DecomposeTypeError(f"Cannot decompose symbolic {self}")
+        result = bb.allocate(dtype=self.qgf)
+        if self.bitsize == 1:
+            x, result = bb.add(GF2Addition(self.bitsize), x=x, y=result)
+            return {'x': x, 'result': result}
+
+        x = bb.add(GF2Square(self.bitsize), x=x)
+        x, result = bb.add(GF2Addition(self.bitsize), x=x, y=result)
+
+        junk = []
+        for i in range(2, self.bitsize):
+            x = bb.add(GF2Square(self.bitsize), x=x)
+            x, result, new_result = bb.add(GF2Multiplication(self.bitsize), x=x, y=result)
+            junk.append(result)
+            result = new_result
+        x = bb.add(GF2Square(self.bitsize), x=x)
+        return {'x': x, 'result': result} | ({'junk': np.array(junk)} if junk else {})
+
+    def build_call_graph(
+        self, ssa: 'SympySymbolAllocator'
+    ) -> Union['BloqCountDictT', Set['BloqCountT']]:
+        if is_symbolic(self.bitsize) or self.bitsize > 2:
+            return {
+                GF2Addition(self.bitsize): 1,
+                GF2Square(self.bitsize): self.bitsize - 1,
+                GF2Multiplication(self.bitsize): self.bitsize - 2,
+            }
+        return {GF2Addition(self.bitsize): 1} | (
+            {GF2Square(self.bitsize): 1} if self.bitsize == 2 else {}
+        )
+
+    def on_classical_vals(self, *, x) -> Dict[str, 'ClassicalValT']:
+        assert isinstance(x, self.qgf.gf_type)
+        x_temp = x**2
+        result = x_temp
+        junk = []
+        for i in range(2, int(self.bitsize)):
+            junk.append(result)
+            x_temp = x_temp * x_temp
+            result = result * x_temp
+        return {'x': x, 'result': x ** (-1), 'junk': np.array(junk)}
+
+
+@bloq_example
+def _gf16_inverse() -> GF2Inverse:
+    gf16_inverse = GF2Inverse(4)
+    return gf16_inverse
+
+
+@bloq_example
+def _gf2_inverse_symbolic() -> GF2Inverse:
+    import sympy
+
+    m = sympy.Symbol('m')
+    gf2_inverse_symbolic = GF2Inverse(m)
+    return gf2_inverse_symbolic
+
+
+_GF2_INVERSE_DOC = BloqDocSpec(bloq_cls=GF2Inverse, examples=(_gf16_inverse, _gf2_inverse_symbolic))
diff --git a/qualtran/bloqs/gf_arithmetic/gf2_inverse_test.py b/qualtran/bloqs/gf_arithmetic/gf2_inverse_test.py
new file mode 100644
index 000000000..4fceffa8b
--- /dev/null
+++ b/qualtran/bloqs/gf_arithmetic/gf2_inverse_test.py
@@ -0,0 +1,46 @@
+#  Copyright 2024 Google LLC
+#
+#  Licensed under the Apache License, Version 2.0 (the "License");
+#  you may not use this file except in compliance with the License.
+#  You may obtain a copy of the License at
+#
+#      https://www.apache.org/licenses/LICENSE-2.0
+#
+#  Unless required by applicable law or agreed to in writing, software
+#  distributed under the License is distributed on an "AS IS" BASIS,
+#  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+#  See the License for the specific language governing permissions and
+#  limitations under the License.
+
+import pytest
+from galois import GF
+
+from qualtran.bloqs.gf_arithmetic.gf2_inverse import (
+    _gf2_inverse_symbolic,
+    _gf16_inverse,
+    GF2Inverse,
+)
+from qualtran.testing import assert_consistent_classical_action
+
+
+def test_gf16_multiplication(bloq_autotester):
+    bloq_autotester(_gf16_inverse)
+
+
+def test_gf2_multiplication_symbolic(bloq_autotester):
+    bloq_autotester(_gf2_inverse_symbolic)
+
+
+def test_gf2_multiplication_classical_sim_quick():
+    m = 1
+    bloq = GF2Inverse(m)
+    GFM = GF(2**m)
+    assert_consistent_classical_action(bloq, x=GFM.elements[1:])
+
+
+@pytest.mark.slow
+@pytest.mark.parametrize('m', [2, 3, 4, 5])
+def test_gf2_multiplication_classical_sim(m):
+    bloq = GF2Inverse(m)
+    GFM = GF(2**m)
+    assert_consistent_classical_action(bloq, x=GFM.elements[1:])
diff --git a/qualtran/conftest.py b/qualtran/conftest.py
index 8a4e7a0c1..f3eeec1b3 100644
--- a/qualtran/conftest.py
+++ b/qualtran/conftest.py
@@ -96,6 +96,8 @@ def assert_bloq_example_serializes_for_pytest(bloq_ex: BloqExample):
         'gf2_multiplication_symbolic',  # cannot serialize QGF
         'gf16_square',  # cannot serialize QGF
         'gf2_square_symbolic',  # cannot serialize QGF
+        'gf16_inverse',  # cannot serialize QGF
+        'gf2_inverse_symbolic',  # cannot serialize QGF
         'gqsp_1d_ising',
         'auto_partition',
         'unitary_block_encoding',
diff --git a/qualtran/testing.py b/qualtran/testing.py
index fbb268c69..8cc89bfde 100644
--- a/qualtran/testing.py
+++ b/qualtran/testing.py
@@ -708,4 +708,6 @@ def assert_consistent_classical_action(
         call_with = {p: v for p, v in zip(parameter_names, vals)}
         bloq_res = bloq.call_classically(**call_with)
         decomposed_res = cb.call_classically(**call_with)
-        assert bloq_res == decomposed_res, f'{bloq=} {call_with=} {bloq_res=} {decomposed_res=}'
+        np.testing.assert_equal(
+            bloq_res, decomposed_res, err_msg=f'{bloq=} {call_with=} {bloq_res=} {decomposed_res=}'
+        )