diff --git a/poetry.lock b/poetry.lock
index 585f81b53c..1f9f6a070f 100644
--- a/poetry.lock
+++ b/poetry.lock
@@ -211,13 +211,13 @@ files = [
 
 [[package]]
 name = "certifi"
-version = "2024.2.2"
+version = "2024.6.2"
 description = "Python package for providing Mozilla's CA Bundle."
 optional = false
 python-versions = ">=3.6"
 files = [
-    {file = "certifi-2024.2.2-py3-none-any.whl", hash = "sha256:dc383c07b76109f368f6106eee2b593b04a011ea4d55f652c6ca24a754d1cdd1"},
-    {file = "certifi-2024.2.2.tar.gz", hash = "sha256:0569859f95fc761b18b45ef421b1290a0f65f147e92a1e5eb3e635f9a5e4e66f"},
+    {file = "certifi-2024.6.2-py3-none-any.whl", hash = "sha256:ddc6c8ce995e6987e7faf5e3f1b02b302836a0e5d98ece18392cb1a36c72ad56"},
+    {file = "certifi-2024.6.2.tar.gz", hash = "sha256:3cd43f1c6fa7dedc5899d69d3ad0398fd018ad1a17fba83ddaf78aa46c747516"},
 ]
 
 [[package]]
@@ -1414,13 +1414,13 @@ woff = ["brotli (>=1.0.1)", "brotlicffi (>=0.8.0)", "zopfli (>=0.1.4)"]
 
 [[package]]
 name = "fsspec"
-version = "2024.5.0"
+version = "2024.6.0"
 description = "File-system specification"
 optional = false
 python-versions = ">=3.8"
 files = [
-    {file = "fsspec-2024.5.0-py3-none-any.whl", hash = "sha256:e0fdbc446d67e182f49a70b82cf7889028a63588fde6b222521f10937b2b670c"},
-    {file = "fsspec-2024.5.0.tar.gz", hash = "sha256:1d021b0b0f933e3b3029ed808eb400c08ba101ca2de4b3483fbc9ca23fcee94a"},
+    {file = "fsspec-2024.6.0-py3-none-any.whl", hash = "sha256:58d7122eb8a1a46f7f13453187bfea4972d66bf01618d37366521b1998034cee"},
+    {file = "fsspec-2024.6.0.tar.gz", hash = "sha256:f579960a56e6d8038a9efc8f9c77279ec12e6299aa86b0769a7e9c46b94527c2"},
 ]
 
 [package.extras]
@@ -1429,6 +1429,7 @@ adl = ["adlfs"]
 arrow = ["pyarrow (>=1)"]
 dask = ["dask", "distributed"]
 dev = ["pre-commit", "ruff"]
+doc = ["numpydoc", "sphinx", "sphinx-design", "sphinx-rtd-theme", "yarl"]
 dropbox = ["dropbox", "dropboxdrivefs", "requests"]
 full = ["adlfs", "aiohttp (!=4.0.0a0,!=4.0.0a1)", "dask", "distributed", "dropbox", "dropboxdrivefs", "fusepy", "gcsfs", "libarchive-c", "ocifs", "panel", "paramiko", "pyarrow (>=1)", "pygit2", "requests", "s3fs", "smbprotocol", "tqdm"]
 fuse = ["fusepy"]
@@ -1522,13 +1523,13 @@ grpcio-gcp = ["grpcio-gcp (>=0.2.2,<1.0.dev0)"]
 
 [[package]]
 name = "google-auth"
-version = "2.29.0"
+version = "2.30.0"
 description = "Google Authentication Library"
 optional = false
 python-versions = ">=3.7"
 files = [
-    {file = "google-auth-2.29.0.tar.gz", hash = "sha256:672dff332d073227550ffc7457868ac4218d6c500b155fe6cc17d2b13602c360"},
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+    {file = "google-auth-2.30.0.tar.gz", hash = "sha256:ab630a1320f6720909ad76a7dbdb6841cdf5c66b328d690027e4867bdfb16688"},
+    {file = "google_auth-2.30.0-py2.py3-none-any.whl", hash = "sha256:8df7da660f62757388b8a7f249df13549b3373f24388cb5d2f1dd91cc18180b5"},
 ]
 
 [package.dependencies]
@@ -1560,78 +1561,78 @@ six = "*"
 
 [[package]]
 name = "googleapis-common-protos"
-version = "1.63.0"
+version = "1.63.1"
 description = "Common protobufs used in Google APIs"
 optional = false
 python-versions = ">=3.7"
 files = [
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+    {file = "googleapis-common-protos-1.63.1.tar.gz", hash = "sha256:c6442f7a0a6b2a80369457d79e6672bb7dcbaab88e0848302497e3ec80780a6a"},
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 ]
 
 [package.dependencies]
-protobuf = ">=3.19.5,<3.20.0 || >3.20.0,<3.20.1 || >3.20.1,<4.21.1 || >4.21.1,<4.21.2 || >4.21.2,<4.21.3 || >4.21.3,<4.21.4 || >4.21.4,<4.21.5 || >4.21.5,<5.0.0.dev0"
+protobuf = ">=3.19.5,<3.20.0 || >3.20.0,<3.20.1 || >3.20.1,<4.21.1 || >4.21.1,<4.21.2 || >4.21.2,<4.21.3 || >4.21.3,<4.21.4 || >4.21.4,<4.21.5 || >4.21.5,<6.0.0.dev0"
 
 [package.extras]
 grpc = ["grpcio (>=1.44.0,<2.0.0.dev0)"]
 
 [[package]]
 name = "grpcio"
-version = "1.64.0"
+version = "1.64.1"
 description = "HTTP/2-based RPC framework"
 optional = false
 python-versions = ">=3.8"
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 [package.extras]
-protobuf = ["grpcio-tools (>=1.64.0)"]
+protobuf = ["grpcio-tools (>=1.64.1)"]
 
 [[package]]
 name = "grpcio-status"
@@ -3111,13 +3112,13 @@ torch = ["torch"]
 
 [[package]]
 name = "packaging"
-version = "24.0"
+version = "24.1"
 description = "Core utilities for Python packages"
 optional = false
-python-versions = ">=3.7"
+python-versions = ">=3.8"
 files = [
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 ]
 
 [[package]]
@@ -3383,13 +3384,13 @@ files = [
 
 [[package]]
 name = "prompt-toolkit"
-version = "3.0.45"
+version = "3.0.47"
 description = "Library for building powerful interactive command lines in Python"
 optional = false
 python-versions = ">=3.7.0"
 files = [
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 [package.dependencies]
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 description = "Tensors and Dynamic neural networks in Python with strong GPU acceleration"
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 python-versions = ">=3.8.0"
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 ]
 
 [package.dependencies]
@@ -5165,7 +5166,7 @@ nvidia-cusparse-cu12 = {version = "12.1.0.106", markers = "platform_system == \"
 nvidia-nccl-cu12 = {version = "2.20.5", markers = "platform_system == \"Linux\" and platform_machine == \"x86_64\""}
 nvidia-nvtx-cu12 = {version = "12.1.105", markers = "platform_system == \"Linux\" and platform_machine == \"x86_64\""}
 sympy = "*"
-triton = {version = "2.3.0", markers = "platform_system == \"Linux\" and platform_machine == \"x86_64\" and python_version < \"3.12\""}
+triton = {version = "2.3.1", markers = "platform_system == \"Linux\" and platform_machine == \"x86_64\" and python_version < \"3.12\""}
 typing-extensions = ">=4.8.0"
 
 [package.extras]
@@ -5229,17 +5230,17 @@ test = ["argcomplete (>=3.0.3)", "mypy (>=1.7.0)", "pre-commit", "pytest (>=7.0,
 
 [[package]]
 name = "triton"
-version = "2.3.0"
+version = "2.3.1"
 description = "A language and compiler for custom Deep Learning operations"
 optional = false
 python-versions = "*"
 files = [
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 ]
 
 [package.dependencies]
@@ -5285,13 +5286,13 @@ files = [
 
 [[package]]
 name = "typing-extensions"
-version = "4.12.0"
+version = "4.12.2"
 description = "Backported and Experimental Type Hints for Python 3.8+"
 optional = false
 python-versions = ">=3.8"
 files = [
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 ]
 
 [[package]]
@@ -5484,18 +5485,18 @@ files = [
 
 [[package]]
 name = "zipp"
-version = "3.19.1"
+version = "3.19.2"
 description = "Backport of pathlib-compatible object wrapper for zip files"
 optional = false
 python-versions = ">=3.8"
 files = [
-    {file = "zipp-3.19.1-py3-none-any.whl", hash = "sha256:2828e64edb5386ea6a52e7ba7cdb17bb30a73a858f5eb6eb93d8d36f5ea26091"},
-    {file = "zipp-3.19.1.tar.gz", hash = "sha256:35427f6d5594f4acf82d25541438348c26736fa9b3afa2754bcd63cdb99d8e8f"},
+    {file = "zipp-3.19.2-py3-none-any.whl", hash = "sha256:f091755f667055f2d02b32c53771a7a6c8b47e1fdbc4b72a8b9072b3eef8015c"},
+    {file = "zipp-3.19.2.tar.gz", hash = "sha256:bf1dcf6450f873a13e952a29504887c89e6de7506209e5b1bcc3460135d4de19"},
 ]
 
 [package.extras]
 doc = ["furo", "jaraco.packaging (>=9.3)", "jaraco.tidelift (>=1.4)", "rst.linker (>=1.9)", "sphinx (>=3.5)", "sphinx-lint"]
-test = ["big-O", "jaraco.functools", "jaraco.itertools", "jaraco.test", "more-itertools", "pytest (>=6,!=8.1.*)", "pytest-checkdocs (>=2.4)", "pytest-cov", "pytest-enabler (>=2.2)", "pytest-ignore-flaky", "pytest-mypy", "pytest-ruff (>=0.2.1)"]
+test = ["big-O", "importlib-resources", "jaraco.functools", "jaraco.itertools", "jaraco.test", "more-itertools", "pytest (>=6,!=8.1.*)", "pytest-checkdocs (>=2.4)", "pytest-cov", "pytest-enabler (>=2.2)", "pytest-ignore-flaky", "pytest-mypy", "pytest-ruff (>=0.2.1)"]
 
 [extras]
 qinfo = []
diff --git a/pyproject.toml b/pyproject.toml
index 881b2f8b4c..d9d231acb6 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -84,6 +84,7 @@ docs-clean = "make -C doc clean"
 test-docs = "make -C doc doctest"
 
 
+
 [tool.poetry.group.cuda11]
 optional = true
 
@@ -93,6 +94,7 @@ cuquantum-python-cu11 = "^23.3.0"
 qibojit = { git = "https://github.com/qiboteam/qibojit.git" }
 qibotn = { git = "https://github.com/qiboteam/qibotn.git" }
 
+
 [tool.poetry.group.cuda12]
 optional = true
 
diff --git a/src/qibo/models/dbi/double_bracket.py b/src/qibo/models/dbi/double_bracket.py
index 4e1150df6f..e2f8d9b7f2 100644
--- a/src/qibo/models/dbi/double_bracket.py
+++ b/src/qibo/models/dbi/double_bracket.py
@@ -105,29 +105,60 @@ def __init__(
     def __call__(
         self, step: float, mode: DoubleBracketGeneratorType = None, d: np.array = None
     ):
-        """Performs one double bracket rotation."""
+        r"""We use convention that $H' = U^\dagger H U$ where $U=e^{-sW}$ with $W=[D,H]$
+        (or depending on `mode` an approximation, see `eval_dbr_unitary`).
+        If $s>0$ then for $D = \Delta(H)$ the GWW DBR will give a $\sigma$-decrease,
+        see https://arxiv.org/abs/2206.11772."""
+
+        operator = self.eval_dbr_unitary(step, mode, d)
+        operator_dagger = self.backend.cast(
+            np.array(np.matrix(self.backend.to_numpy(operator)).getH())
+        )
+        self.h.matrix = operator_dagger @ self.h.matrix @ operator
+        return operator
+
+    def eval_dbr_unitary(
+        self,
+        step: float,
+        mode: DoubleBracketGeneratorType = None,
+        d: np.array = None,
+    ):
+        """In call we are working in the convention that $H' = U^\\dagger H
+        U$ where $U=e^{-sW}$ with $W=[D,H]$ or an approximation of that by a group commutator.
+        That is handy because if we switch from the DBI in the Heisenberg picture for the
+        Hamiltonian, we get that the transformation of the state is $|\\psi'\rangle = U |\\psi\rangle$
+        so that $\\langle H\rangle_{\\psi'} = \\langle H' \rangle_\\psi$ (i.e. when writing the unitary
+        acting on the state dagger notation is avoided).
+        The group commutator must approximate $U=e^{-s[D,H]}$. This is achieved by setting $r = \\sqrt{s}$ so that
+        $$V = e^{-irH}e^{irD}e^{irH}e^{-irD}$$
+        because
+        $$e^{-irH}De^{irH} = D+ir[D,H]+O(r^2)$$
+        so
+        $$V\approx e^{irD +i^2 r^2[D,H] + O(r^2) -irD} \approx U\\ .$$
+        See the app in https://arxiv.org/abs/2206.11772 for a derivation.
+        """
         if mode is None:
             mode = self.mode
 
         if mode is DoubleBracketGeneratorType.canonical:
             operator = self.backend.calculate_matrix_exp(
-                1.0j * step,
+                -1.0j * step,
                 self.commutator(self.diagonal_h_matrix, self.h.matrix),
             )
         elif mode is DoubleBracketGeneratorType.single_commutator:
             if d is None:
                 d = self.diagonal_h_matrix
             operator = self.backend.calculate_matrix_exp(
-                1.0j * step,
+                -1.0j * step,
                 self.commutator(self.backend.cast(d), self.h.matrix),
             )
         elif mode is DoubleBracketGeneratorType.group_commutator:
             if d is None:
                 d = self.diagonal_h_matrix
             operator = (
-                self.h.exp(-step)
+                self.h.exp(step)
                 @ self.backend.calculate_matrix_exp(-step, d)
-                @ self.h.exp(step)
+                @ self.h.exp(-step)
                 @ self.backend.calculate_matrix_exp(step, d)
             )
         elif mode is DoubleBracketGeneratorType.group_commutator_third_order:
@@ -141,11 +172,20 @@ def __call__(
                 @ self.h.exp(-step * (3 - np.sqrt(5)) / 2)
                 @ self.backend.calculate_matrix_exp(-step, d)
             )
-        operator_dagger = self.backend.cast(
-            np.array(np.matrix(self.backend.to_numpy(operator)).getH())
-        )
+            operator = (
+                self.backend.calculate_matrix_exp(step, d)
+                @ self.h.exp(step * (3 - np.sqrt(5)) / 2)
+                @ self.backend.calculate_matrix_exp(-step * (np.sqrt(5) + 1) / 2, d)
+                @ self.h.exp(-step)
+                @ self.backend.calculate_matrix_exp(step * (np.sqrt(5) - 1) / 2, d)
+                @ self.h.exp(step * (np.sqrt(5) - 1) / 2)
+            )
+        else:
+            raise NotImplementedError(
+                f"The mode {mode} is not supported"
+            )  # pragma: no cover
 
-        self.h.matrix = operator @ self.h.matrix @ operator_dagger
+        return operator
 
     @staticmethod
     def commutator(a, b):
diff --git a/src/qibo/models/variational.py b/src/qibo/models/variational.py
index 8ec93b2cbb..aeeaa6352c 100644
--- a/src/qibo/models/variational.py
+++ b/src/qibo/models/variational.py
@@ -96,7 +96,6 @@ def minimize(
             loss = lambda p, c, h: dtype(loss_func(p, c, h))
         elif method != "sgd":
             loss = lambda p, c, h: self.hamiltonian.backend.to_numpy(loss_func(p, c, h))
-
         result, parameters, extra = self.optimizers.optimize(
             loss,
             initial_state,
diff --git a/src/qibo/quantum_info/entropies.py b/src/qibo/quantum_info/entropies.py
index 3d4e7ac027..d6499e15dc 100644
--- a/src/qibo/quantum_info/entropies.py
+++ b/src/qibo/quantum_info/entropies.py
@@ -461,7 +461,7 @@ def von_neumann_entropy(
 
     if purity(state) == 1.0:
         if return_spectrum:
-            return 0.0, backend.cast([1.0], dtype=float)
+            return 0.0, backend.cast([0.0], dtype=float)
 
         return 0.0
 
diff --git a/src/qibo/transpiler/decompositions.py b/src/qibo/transpiler/decompositions.py
index 33ab35b56b..bae39e38bd 100644
--- a/src/qibo/transpiler/decompositions.py
+++ b/src/qibo/transpiler/decompositions.py
@@ -466,3 +466,23 @@ def _u3_to_gpi2(t, p, l):
     gates.ECR, [gates.S(0), gates.SX(1), gates.CNOT(0, 1), gates.X(0)]
 )
 standard_decompositions.add(gates.CCZ, [gates.H(2), gates.TOFFOLI(0, 1, 2), gates.H(2)])
+standard_decompositions.add(
+    gates.TOFFOLI,
+    [
+        gates.H(2),
+        gates.CNOT(1, 2),
+        gates.TDG(2),
+        gates.CNOT(0, 2),
+        gates.T(2),
+        gates.CNOT(1, 2),
+        gates.T(1),
+        gates.TDG(2),
+        gates.CNOT(0, 2),
+        gates.CNOT(0, 1),
+        gates.T(2),
+        gates.T(0),
+        gates.TDG(1),
+        gates.H(2),
+        gates.CNOT(0, 1),
+    ],
+)
diff --git a/src/qibo/transpiler/pipeline.py b/src/qibo/transpiler/pipeline.py
index 7d9b68e130..04c1072d0a 100644
--- a/src/qibo/transpiler/pipeline.py
+++ b/src/qibo/transpiler/pipeline.py
@@ -225,7 +225,7 @@ def __call__(self, circuit):
         physical (keys) to logical (values) qubit. If `int_qubit_name` is `True`
         each key `i` correspond to the `i-th` qubit in the graph.
         """
-        final_layout = self.initial_layout
+        self.initial_layout = None
         for transpiler_pass in self.passes:
             if isinstance(transpiler_pass, Optimizer):
                 transpiler_pass.connectivity = self.connectivity
@@ -234,6 +234,9 @@ def __call__(self, circuit):
                 transpiler_pass.connectivity = self.connectivity
                 if self.initial_layout == None:
                     self.initial_layout = transpiler_pass(circuit)
+                    final_layout = (
+                        self.initial_layout
+                    )  # This way the final layout will be the same as the initial layout if no router is used
                 else:
                     raise_error(
                         TranspilerPipelineError,
diff --git a/tests/test_callbacks.py b/tests/test_callbacks.py
index 5ff73f66a5..bf2ac7d2d0 100644
--- a/tests/test_callbacks.py
+++ b/tests/test_callbacks.py
@@ -97,7 +97,7 @@ def test_entropy_in_circuit(backend, density_matrix, base):
     values = [backend.to_numpy(x) for x in entropy]
     backend.assert_allclose(values, target, atol=PRECISION_TOL)
 
-    target_spectrum = [1.0] + list([0, 0, np.log(2), np.log(2)] / np.log(base))
+    target_spectrum = [0.0] + list([0, 0, np.log(2), np.log(2)] / np.log(base))
     entropy_spectrum = np.ravel(np.concatenate(entropy.spectrum)).tolist()
     backend.assert_allclose(entropy_spectrum, target_spectrum, atol=PRECISION_TOL)
 
diff --git a/tests/test_cirq.py b/tests/test_cirq.py
index 636581656d..8ff8498d7a 100644
--- a/tests/test_cirq.py
+++ b/tests/test_cirq.py
@@ -4,8 +4,9 @@
 import numpy as np
 import pytest
 
-from qibo import gates, models
+from qibo import Circuit, gates, matrices
 from qibo.backends import NumpyBackend
+from qibo.models import QFT
 from qibo.quantum_info import random_statevector, random_unitary
 
 numpy_backend = NumpyBackend()
@@ -54,7 +55,7 @@ def assert_gates_equivalent(
     if ndevices is not None:
         accelerators = {"/GPU:0": ndevices}
 
-    c = models.Circuit(nqubits, accelerators)
+    c = Circuit(nqubits, accelerators)
     c.add(qibo_gate)
     assert c.depth == target_depth
     if accelerators and not backend.supports_multigpu:
@@ -307,7 +308,7 @@ def test_unitary_matrix_gate_controlled_by(backend, nqubits, ntargets, ndevices)
 
 @pytest.mark.parametrize("nqubits", [5, 6, 7, 11, 12])
 def test_qft(backend, accelerators, nqubits):
-    c = models.QFT(nqubits, accelerators=accelerators)
+    c = QFT(nqubits, accelerators=accelerators)
     initial_state = random_statevector(2**nqubits, backend=numpy_backend)
     final_state = backend.execute_circuit(c, np.copy(initial_state)).state()
     final_state = backend.cast(final_state, dtype=final_state.dtype)
diff --git a/tests/test_gates_gates.py b/tests/test_gates_gates.py
index 3246413c31..2b2ae20973 100644
--- a/tests/test_gates_gates.py
+++ b/tests/test_gates_gates.py
@@ -6,6 +6,7 @@
 from qibo import Circuit, gates, matrices
 from qibo.parameter import Parameter
 from qibo.quantum_info import random_hermitian, random_statevector, random_unitary
+from qibo.transpiler.decompositions import standard_decompositions
 
 
 def apply_gates(backend, gatelist, nqubits=None, initial_state=None):
@@ -1205,6 +1206,13 @@ def test_toffoli(backend, applyx):
     assert not gates.TOFFOLI(0, 1, 2).clifford
     assert gates.TOFFOLI(0, 1, 2).unitary
 
+    # test decomposition
+    decomposition = Circuit(3)
+    decomposition.add(standard_decompositions(gates.TOFFOLI(0, 1, 2)))
+    decomposition = decomposition.unitary(backend)
+
+    backend.assert_allclose(decomposition, backend.cast(matrices.TOFFOLI), atol=1e-10)
+
 
 def test_ccz(backend):
     nqubits = 3