diff --git a/poetry.lock b/poetry.lock
index 3b25b7e..1bb918b 100644
--- a/poetry.lock
+++ b/poetry.lock
@@ -1,4 +1,4 @@
-# This file is automatically @generated by Poetry 2.1.3 and should not be changed by hand.
+# This file is automatically @generated by Poetry 2.1.2 and should not be changed by hand.
 
 [[package]]
 name = "alabaster"
@@ -14,14 +14,14 @@ files = [
 
 [[package]]
 name = "alembic"
-version = "1.16.4"
+version = "1.16.5"
 description = "A database migration tool for SQLAlchemy."
 optional = false
 python-versions = ">=3.9"
 groups = ["main"]
 files = [
-    {file = "alembic-1.16.4-py3-none-any.whl", hash = "sha256:b05e51e8e82efc1abd14ba2af6392897e145930c3e0a2faf2b0da2f7f7fd660d"},
-    {file = "alembic-1.16.4.tar.gz", hash = "sha256:efab6ada0dd0fae2c92060800e0bf5c1dc26af15a10e02fb4babff164b4725e2"},
+    {file = "alembic-1.16.5-py3-none-any.whl", hash = "sha256:e845dfe090c5ffa7b92593ae6687c5cb1a101e91fa53868497dbd79847f9dbe3"},
+    {file = "alembic-1.16.5.tar.gz", hash = "sha256:a88bb7f6e513bd4301ecf4c7f2206fe93f9913f9b48dac3b78babde2d6fe765e"},
 ]
 
 [package.dependencies]
@@ -71,14 +71,14 @@ files = [
 
 [[package]]
 name = "autoray"
-version = "0.7.2"
+version = "0.8.0"
 description = "Abstract your array operations."
 optional = false
 python-versions = ">=3.9"
 groups = ["main"]
 files = [
-    {file = "autoray-0.7.2-py3-none-any.whl", hash = "sha256:f692f0f21a6f187ce877f3cc810f035d601ccd713beac02974811650bb60f580"},
-    {file = "autoray-0.7.2.tar.gz", hash = "sha256:7f8f375af12deea65736b65305b86eff2e11ebaf8c58d3e599752681772ff0d7"},
+    {file = "autoray-0.8.0-py3-none-any.whl", hash = "sha256:fda5e20b072d41818ccdf26c251dfde4c49595e7c323b141dc2640e52f5889c9"},
+    {file = "autoray-0.8.0.tar.gz", hash = "sha256:5d0d71da03cb02d5bc590a1af64e0ba58589352d628843a0ecbcfe90040dc520"},
 ]
 
 [package.extras]
@@ -114,14 +114,14 @@ files = [
 
 [[package]]
 name = "beautifulsoup4"
-version = "4.13.4"
+version = "4.13.5"
 description = "Screen-scraping library"
 optional = false
 python-versions = ">=3.7.0"
 groups = ["docs"]
 files = [
-    {file = "beautifulsoup4-4.13.4-py3-none-any.whl", hash = "sha256:9bbbb14bfde9d79f38b8cd5f8c7c85f4b8f2523190ebed90e950a8dea4cb1c4b"},
-    {file = "beautifulsoup4-4.13.4.tar.gz", hash = "sha256:dbb3c4e1ceae6aefebdaf2423247260cd062430a410e38c66f2baa50a8437195"},
+    {file = "beautifulsoup4-4.13.5-py3-none-any.whl", hash = "sha256:642085eaa22233aceadff9c69651bc51e8bf3f874fb6d7104ece2beb24b47c4a"},
+    {file = "beautifulsoup4-4.13.5.tar.gz", hash = "sha256:5e70131382930e7c3de33450a2f54a63d5e4b19386eab43a5b34d594268f3695"},
 ]
 
 [package.dependencies]
@@ -137,116 +137,103 @@ lxml = ["lxml"]
 
 [[package]]
 name = "certifi"
-version = "2025.7.14"
+version = "2025.8.3"
 description = "Python package for providing Mozilla's CA Bundle."
 optional = false
 python-versions = ">=3.7"
 groups = ["docs"]
 files = [
-    {file = "certifi-2025.7.14-py3-none-any.whl", hash = "sha256:6b31f564a415d79ee77df69d757bb49a5bb53bd9f756cbbe24394ffd6fc1f4b2"},
-    {file = "certifi-2025.7.14.tar.gz", hash = "sha256:8ea99dbdfaaf2ba2f9bac77b9249ef62ec5218e7c2b2e903378ed5fccf765995"},
+    {file = "certifi-2025.8.3-py3-none-any.whl", hash = "sha256:f6c12493cfb1b06ba2ff328595af9350c65d6644968e5d3a2ffd78699af217a5"},
+    {file = "certifi-2025.8.3.tar.gz", hash = "sha256:e564105f78ded564e3ae7c923924435e1daa7463faeab5bb932bc53ffae63407"},
 ]
 
 [[package]]
 name = "charset-normalizer"
-version = "3.4.2"
+version = "3.4.3"
 description = "The Real First Universal Charset Detector. Open, modern and actively maintained alternative to Chardet."
 optional = false
 python-versions = ">=3.7"
 groups = ["docs"]
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 [[package]]
@@ -413,100 +400,116 @@ test = ["altair", "baytune", "chocolate", "cmaes", "dask", "distributed", "kahyp
 
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 ]
 
 [package.extras]
 docs = ["Sphinx", "furo"]
-test = ["objgraph", "psutil"]
+test = ["objgraph", "psutil", "setuptools"]
 
 [[package]]
 name = "h5py"
@@ -1209,104 +1228,125 @@ i18n = ["Babel (>=2.7)"]
 
 [[package]]
 name = "joblib"
-version = "1.5.1"
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 description = "Lightweight pipelining with Python functions"
 optional = false
 python-versions = ">=3.9"
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 ]
 
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 name = "kiwisolver"
-version = "1.4.8"
+version = "1.4.9"
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 optional = false
 python-versions = ">=3.10"
 groups = ["main"]
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 [[package]]
@@ -1323,33 +1363,33 @@ files = [
 
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+    {file = "numpy-2.3.3.tar.gz", hash = "sha256:ddc7c39727ba62b80dfdbedf400d1c10ddfa8eefbd7ec8dcb118be8b56d31029"},
 ]
 
 [[package]]
@@ -1689,14 +1769,14 @@ tests = ["pytest (>=6.0)", "pyyaml"]
 
 [[package]]
 name = "optuna"
-version = "4.4.0"
+version = "4.5.0"
 description = "A hyperparameter optimization framework"
 optional = false
 python-versions = ">=3.8"
 groups = ["main"]
 files = [
-    {file = "optuna-4.4.0-py3-none-any.whl", hash = "sha256:fad8d9c5d5af993ae1280d6ce140aecc031c514a44c3b639d8c8658a8b7920ea"},
-    {file = "optuna-4.4.0.tar.gz", hash = "sha256:a9029f6a92a1d6c8494a94e45abd8057823b535c2570819072dbcdc06f1c1da4"},
+    {file = "optuna-4.5.0-py3-none-any.whl", hash = "sha256:5b8a783e84e448b0742501bc27195344a28d2c77bd2feef5b558544d954851b0"},
+    {file = "optuna-4.5.0.tar.gz", hash = "sha256:264844da16dad744dea295057d8bc218646129c47567d52c35a201d9f99942ba"},
 ]
 
 [package.dependencies]
@@ -1710,10 +1790,10 @@ tqdm = "*"
 
 [package.extras]
 benchmark = ["asv (>=0.5.0)", "cma", "virtualenv"]
-checking = ["black", "blackdoc", "flake8", "isort", "mypy", "mypy_boto3_s3", "types-PyYAML", "types-redis", "types-setuptools", "types-tqdm", "typing_extensions (>=3.10.0.0)"]
-document = ["ase", "cmaes (>=0.10.0)", "fvcore", "kaleido (<0.4)", "lightgbm", "matplotlib (!=3.6.0)", "pandas", "pillow", "plotly (>=4.9.0)", "scikit-learn", "sphinx", "sphinx-copybutton", "sphinx-gallery", "sphinx-notfound-page", "sphinx_rtd_theme (>=1.2.0)", "torch", "torchvision"]
-optional = ["boto3", "cmaes (>=0.10.0)", "google-cloud-storage", "grpcio", "matplotlib (!=3.6.0)", "pandas", "plotly (>=4.9.0)", "protobuf (>=5.28.1)", "redis", "scikit-learn (>=0.24.2)", "scipy", "torch ; python_version <= \"3.12\""]
-test = ["coverage", "fakeredis[lua]", "grpcio", "kaleido (<0.4)", "moto", "protobuf (>=5.28.1)", "pytest", "scipy (>=1.9.2)", "torch ; python_version <= \"3.12\""]
+checking = ["black", "blackdoc", "flake8", "isort", "mypy", "mypy_boto3_s3", "scipy-stubs ; python_version >= \"3.10\"", "types-PyYAML", "types-redis", "types-setuptools", "types-tqdm", "typing_extensions (>=3.10.0.0)"]
+document = ["ase", "cmaes (>=0.12.0)", "fvcore", "kaleido (<0.4)", "lightgbm", "matplotlib (!=3.6.0)", "pandas", "pillow", "plotly (>=4.9.0)", "scikit-learn", "sphinx", "sphinx-copybutton", "sphinx-gallery", "sphinx-notfound-page", "sphinx_rtd_theme (>=1.2.0)", "torch", "torchvision"]
+optional = ["boto3", "cmaes (>=0.12.0)", "google-cloud-storage", "grpcio", "matplotlib (!=3.6.0)", "pandas", "plotly (>=4.9.0)", "protobuf (>=5.28.1)", "redis", "scikit-learn (>=0.24.2)", "scipy", "torch"]
+test = ["coverage", "fakeredis[lua]", "grpcio", "kaleido (<0.4)", "moto", "protobuf (>=5.28.1)", "pytest", "pytest-xdist", "scipy (>=1.9.2)", "torch"]
 
 [[package]]
 name = "packaging"
@@ -1729,36 +1809,20 @@ files = [
 
 [[package]]
 name = "parso"
-version = "0.8.4"
+version = "0.8.5"
 description = "A Python Parser"
 optional = false
 python-versions = ">=3.6"
 groups = ["dev"]
 files = [
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+    {file = "parso-0.8.5-py2.py3-none-any.whl", hash = "sha256:646204b5ee239c396d040b90f9e272e9a8017c630092bf59980beb62fd033887"},
+    {file = "parso-0.8.5.tar.gz", hash = "sha256:034d7354a9a018bdce352f48b2a8a450f05e9d6ee85db84764e9b6bd96dafe5a"},
 ]
 
 [package.extras]
 qa = ["flake8 (==5.0.4)", "mypy (==0.971)", "types-setuptools (==67.2.0.1)"]
 testing = ["docopt", "pytest"]
 
-[[package]]
-name = "pbr"
-version = "6.1.1"
-description = "Python Build Reasonableness"
-optional = true
-python-versions = ">=2.6"
-groups = ["main"]
-markers = "extra == \"qmatchatea\""
-files = [
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-]
-
-[package.dependencies]
-setuptools = "*"
-
 [[package]]
 name = "pexpect"
 version = "4.9.0"
@@ -1914,14 +1978,14 @@ xmp = ["defusedxml"]
 
 [[package]]
 name = "platformdirs"
-version = "4.3.8"
+version = "4.4.0"
 description = "A small Python package for determining appropriate platform-specific dirs, e.g. a `user data dir`."
 optional = false
 python-versions = ">=3.9"
 groups = ["analysis"]
 files = [
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+    {file = "platformdirs-4.4.0-py3-none-any.whl", hash = "sha256:abd01743f24e5287cd7a5db3752faf1a2d65353f38ec26d98e25a6db65958c85"},
+    {file = "platformdirs-4.4.0.tar.gz", hash = "sha256:ca753cf4d81dc309bc67b0ea38fd15dc97bc30ce419a7f58d13eb3bf14c4febf"},
 ]
 
 [package.extras]
@@ -1947,14 +2011,14 @@ testing = ["coverage", "pytest", "pytest-benchmark"]
 
 [[package]]
 name = "prompt-toolkit"
-version = "3.0.51"
+version = "3.0.52"
 description = "Library for building powerful interactive command lines in Python"
 optional = false
 python-versions = ">=3.8"
 groups = ["dev"]
 files = [
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+    {file = "prompt_toolkit-3.0.52-py3-none-any.whl", hash = "sha256:9aac639a3bbd33284347de5ad8d68ecc044b91a762dc39b7c21095fcd6a19955"},
+    {file = "prompt_toolkit-3.0.52.tar.gz", hash = "sha256:28cde192929c8e7321de85de1ddbe736f1375148b02f2e17edd840042b1be855"},
 ]
 
 [package.dependencies]
@@ -1962,27 +2026,26 @@ wcwidth = "*"
 
 [[package]]
 name = "psutil"
-version = "7.0.0"
-description = "Cross-platform lib for process and system monitoring in Python.  NOTE: the syntax of this script MUST be kept compatible with Python 2.7."
+version = "7.1.0"
+description = "Cross-platform lib for process and system monitoring."
 optional = false
 python-versions = ">=3.6"
 groups = ["main"]
 files = [
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+    {file = "psutil-7.1.0.tar.gz", hash = "sha256:655708b3c069387c8b77b072fc429a57d0e214221d01c0a772df7dfedcb3bcd2"},
 ]
 
 [package.extras]
-dev = ["abi3audit", "black (==24.10.0)", "check-manifest", "coverage", "packaging", "pylint", "pyperf", "pypinfo", "pytest", "pytest-cov", "pytest-xdist", "requests", "rstcheck", "ruff", "setuptools", "sphinx", "sphinx_rtd_theme", "toml-sort", "twine", "virtualenv", "vulture", "wheel"]
-test = ["pytest", "pytest-xdist", "setuptools"]
+dev = ["abi3audit", "black", "check-manifest", "coverage", "packaging", "pylint", "pyperf", "pypinfo", "pyreadline ; os_name == \"nt\"", "pytest", "pytest-cov", "pytest-instafail", "pytest-subtests", "pytest-xdist", "pywin32 ; os_name == \"nt\" and platform_python_implementation != \"PyPy\"", "requests", "rstcheck", "ruff", "setuptools", "sphinx", "sphinx_rtd_theme", "toml-sort", "twine", "virtualenv", "vulture", "wheel", "wheel ; os_name == \"nt\" and platform_python_implementation != \"PyPy\"", "wmi ; os_name == \"nt\" and platform_python_implementation != \"PyPy\""]
+test = ["pytest", "pytest-instafail", "pytest-subtests", "pytest-xdist", "pywin32 ; os_name == \"nt\" and platform_python_implementation != \"PyPy\"", "setuptools", "wheel ; os_name == \"nt\" and platform_python_implementation != \"PyPy\"", "wmi ; os_name == \"nt\" and platform_python_implementation != \"PyPy\""]
 
 [[package]]
 name = "ptyprocess"
@@ -2050,14 +2113,14 @@ windows-terminal = ["colorama (>=0.4.6)"]
 
 [[package]]
 name = "pylint"
-version = "3.3.7"
+version = "3.3.8"
 description = "python code static checker"
 optional = false
 python-versions = ">=3.9.0"
 groups = ["analysis"]
 files = [
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-    {file = "pylint-3.3.7.tar.gz", hash = "sha256:2b11de8bde49f9c5059452e0c310c079c746a0a8eeaa789e5aa966ecc23e4559"},
+    {file = "pylint-3.3.8-py3-none-any.whl", hash = "sha256:7ef94aa692a600e82fabdd17102b73fc226758218c97473c7ad67bd4cb905d83"},
+    {file = "pylint-3.3.8.tar.gz", hash = "sha256:26698de19941363037e2937d3db9ed94fb3303fdadf7d98847875345a8bb6b05"},
 ]
 
 [package.dependencies]
@@ -2078,14 +2141,14 @@ testutils = ["gitpython (>3)"]
 
 [[package]]
 name = "pyparsing"
-version = "3.2.3"
-description = "pyparsing module - Classes and methods to define and execute parsing grammars"
+version = "3.2.5"
+description = "pyparsing - Classes and methods to define and execute parsing grammars"
 optional = false
 python-versions = ">=3.9"
 groups = ["main"]
 files = [
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+    {file = "pyparsing-3.2.5.tar.gz", hash = "sha256:2df8d5b7b2802ef88e8d016a2eb9c7aeaa923529cd251ed0fe4608275d4105b6"},
 ]
 
 [package.extras]
@@ -2093,14 +2156,14 @@ diagrams = ["jinja2", "railroad-diagrams"]
 
 [[package]]
 name = "pytest"
-version = "8.4.1"
+version = "8.4.2"
 description = "pytest: simple powerful testing with Python"
 optional = false
 python-versions = ">=3.9"
 groups = ["tests"]
 files = [
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+    {file = "pytest-8.4.2-py3-none-any.whl", hash = "sha256:872f880de3fc3a5bdc88a11b39c9710c3497a547cfa9320bc3c5e62fbf272e79"},
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 ]
 
 [package.dependencies]
@@ -2230,49 +2293,54 @@ files = [
 
 [[package]]
 name = "qibo"
-version = "0.2.20"
+version = "0.2.21"
 description = "A framework for quantum computing with hardware acceleration."
 optional = false
-python-versions = "<3.14,>=3.10"
+python-versions = ">=3.10,<3.14"
 groups = ["main"]
-files = [
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-]
+files = []
+develop = false
 
 [package.dependencies]
-cma = ">=3.4.0,<4.0.0"
-joblib = ">=1.4.2,<2.0.0"
-networkx = ">=3.2.1,<4.0.0"
-numpy = ">=2.0.0,<3.0.0"
-openqasm3 = {version = ">=1.0.1,<2.0.0", extras = ["parser"]}
-optuna = ">=4.2.1,<5.0.0"
-scipy = ">=1.15.2,<2.0.0"
-sympy = ">=1.13.1,<2.0.0"
-tabulate = ">=0.9.0,<0.10.0"
+cma = "^3.4.0"
+joblib = "^1.4.2"
+networkx = "^3.2.1"
+numpy = "^2.0.0"
+openqasm3 = {version = "^1.0.1", extras = ["parser"]}
+optuna = "^4.2.1"
+scipy = "^1.15.2"
+sympy = "^1.13.1"
+tabulate = "^0.9.0"
 
 [package.extras]
+qinfo = []
 qulacs = ["qulacs (>=0.6.4,<0.7.0) ; python_version < \"3.13\""]
 
+[package.source]
+type = "git"
+url = "https://github.com/qiboteam/qibo"
+reference = "expectation"
+resolved_reference = "41b52af8faba1974c776a074fc0aba3461aeb6fe"
+
 [[package]]
 name = "qiskit"
-version = "1.4.3"
+version = "1.4.4"
 description = "An open-source SDK for working with quantum computers at the level of extended quantum circuits, operators, and primitives."
 optional = true
 python-versions = ">=3.9"
 groups = ["main"]
 markers = "extra == \"qmatchatea\""
 files = [
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+    {file = "qiskit-1.4.4.tar.gz", hash = "sha256:96251c94b14175455809016c08c013dfb615020289e2920a9878fddf35bdda86"},
 ]
 
 [package.dependencies]
@@ -2361,14 +2429,14 @@ scipy = "*"
 
 [[package]]
 name = "quimb"
-version = "1.11.1"
+version = "1.11.2"
 description = "Quantum information and many-body library."
 optional = false
 python-versions = ">=3.9"
 groups = ["main"]
 files = [
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 ]
 
 [package.dependencies]
@@ -2391,14 +2459,14 @@ tests = ["coverage", "pytest", "pytest-cov"]
 
 [[package]]
 name = "requests"
-version = "2.32.4"
+version = "2.32.5"
 description = "Python HTTP for Humans."
 optional = false
-python-versions = ">=3.8"
+python-versions = ">=3.9"
 groups = ["docs"]
 files = [
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 ]
 
 [package.dependencies]
@@ -2413,24 +2481,25 @@ use-chardet-on-py3 = ["chardet (>=3.0.2,<6)"]
 
 [[package]]
 name = "rustworkx"
-version = "0.16.0"
-description = "A python graph library implemented in Rust"
+version = "0.17.1"
+description = "A High-Performance Graph Library for Python"
 optional = true
 python-versions = ">=3.9"
 groups = ["main"]
 markers = "extra == \"qmatchatea\""
 files = [
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 ]
 
 [package.dependencies]
@@ -2443,67 +2512,73 @@ mpl = ["matplotlib (>=3.0)"]
 
 [[package]]
 name = "scipy"
-version = "1.16.1"
+version = "1.16.2"
 description = "Fundamental algorithms for scientific computing in Python"
 optional = false
 python-versions = ">=3.11"
 groups = ["main"]
 files = [
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 [[package]]
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 description = "Easily download, build, install, upgrade, and uninstall Python packages"
 optional = false
 python-versions = ">=3.9"
-groups = ["main", "dev"]
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+    {file = "sqlalchemy-2.0.43-py3-none-any.whl", hash = "sha256:1681c21dd2ccee222c2fe0bef671d1aef7c504087c9c4e800371cfcc8ac966fc"},
+    {file = "sqlalchemy-2.0.43.tar.gz", hash = "sha256:788bfcef6787a7764169cfe9859fe425bf44559619e1d9f56f5bddf2ebf6f417"},
 ]
 
 [package.dependencies]
@@ -2878,20 +2952,17 @@ sqlcipher = ["sqlcipher3_binary"]
 
 [[package]]
 name = "stevedore"
-version = "5.4.1"
+version = "5.5.0"
 description = "Manage dynamic plugins for Python applications"
 optional = true
 python-versions = ">=3.9"
 groups = ["main"]
 markers = "extra == \"qmatchatea\""
 files = [
-    {file = "stevedore-5.4.1-py3-none-any.whl", hash = "sha256:d10a31c7b86cba16c1f6e8d15416955fc797052351a56af15e608ad20811fcfe"},
-    {file = "stevedore-5.4.1.tar.gz", hash = "sha256:3135b5ae50fe12816ef291baff420acb727fcd356106e3e9cbfa9e5985cd6f4b"},
+    {file = "stevedore-5.5.0-py3-none-any.whl", hash = "sha256:18363d4d268181e8e8452e71a38cd77630f345b2ef6b4a8d5614dac5ee0d18cf"},
+    {file = "stevedore-5.5.0.tar.gz", hash = "sha256:d31496a4f4df9825e1a1e4f1f74d19abb0154aff311c3b376fcc89dae8fccd73"},
 ]
 
-[package.dependencies]
-pbr = ">=2.0.0"
-
 [[package]]
 name = "symengine"
 version = "0.13.0"
@@ -3042,14 +3113,14 @@ test = ["argcomplete (>=3.0.3)", "mypy (>=1.7.0)", "pre-commit", "pytest (>=7.0,
 
 [[package]]
 name = "typing-extensions"
-version = "4.14.1"
+version = "4.15.0"
 description = "Backported and Experimental Type Hints for Python 3.9+"
 optional = false
 python-versions = ">=3.9"
 groups = ["main", "docs"]
 files = [
-    {file = "typing_extensions-4.14.1-py3-none-any.whl", hash = "sha256:d1e1e3b58374dc93031d6eda2420a48ea44a36c2b4766a4fdeb3710755731d76"},
-    {file = "typing_extensions-4.14.1.tar.gz", hash = "sha256:38b39f4aeeab64884ce9f74c94263ef78f3c22467c8724005483154c26648d36"},
+    {file = "typing_extensions-4.15.0-py3-none-any.whl", hash = "sha256:f0fa19c6845758ab08074a0cfa8b7aecb71c999ca73d62883bc25cc018c4e548"},
+    {file = "typing_extensions-4.15.0.tar.gz", hash = "sha256:0cea48d173cc12fa28ecabc3b837ea3cf6f38c6d1136f85cbaaf598984861466"},
 ]
 
 [[package]]
@@ -3072,14 +3143,14 @@ zstd = ["zstandard (>=0.18.0)"]
 
 [[package]]
 name = "wcwidth"
-version = "0.2.13"
+version = "0.2.14"
 description = "Measures the displayed width of unicode strings in a terminal"
 optional = false
-python-versions = "*"
+python-versions = ">=3.6"
 groups = ["dev"]
 files = [
-    {file = "wcwidth-0.2.13-py2.py3-none-any.whl", hash = "sha256:3da69048e4540d84af32131829ff948f1e022c1c6bdb8d6102117aac784f6859"},
-    {file = "wcwidth-0.2.13.tar.gz", hash = "sha256:72ea0c06399eb286d978fdedb6923a9eb47e1c486ce63e9b4e64fc18303972b5"},
+    {file = "wcwidth-0.2.14-py2.py3-none-any.whl", hash = "sha256:a7bb560c8aee30f9957e5f9895805edd20602f2d7f720186dfd906e82b4982e1"},
+    {file = "wcwidth-0.2.14.tar.gz", hash = "sha256:4d478375d31bc5395a3c55c40ccdf3354688364cd61c4f6adacaa9215d0b3605"},
 ]
 
 [extras]
@@ -3089,4 +3160,4 @@ qmatchatea = ["qiskit", "qmatchatea", "qtealeaves"]
 [metadata]
 lock-version = "2.1"
 python-versions = ">=3.11,<3.14"
-content-hash = "fa9982c6a663449880e90e3a0d7ec5782283d27a324d9b41c42d4616ded66048"
+content-hash = "1a588cc02000984b68fc1342301a1e2873b804a82e1b9ba3639ed3a01e637d4b"
diff --git a/pyproject.toml b/pyproject.toml
index efb39ca..a311eff 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -21,7 +21,7 @@ packages = [{ include = "qibotn", from = "src" }]
 
 [tool.poetry.dependencies]
 python = ">=3.11,<3.14"
-qibo = "^0.2.17"
+qibo = { git="https://github.com/qiboteam/qibo", branch="expectation"}
 quimb = { version = "^1.10.0", extras = ["tensor"] }
 cupy-cuda11x = { version = "^13.1.0", optional = true }
 cuquantum-python-cu11 = { version = "^24.1.0", optional = true }
diff --git a/src/qibotn/backends/quimb.py b/src/qibotn/backends/quimb.py
index 8a11bb9..6e4629b 100644
--- a/src/qibotn/backends/quimb.py
+++ b/src/qibotn/backends/quimb.py
@@ -1,63 +1,55 @@
 from collections import Counter
 
 import numpy as np
-
-from qibo.models import Circuit
-from qibo.backends import NumpyBackend
-from qibo.config import raise_error
-from qibotn.backends.abstract import QibotnBackend
-from qibotn.result import TensorNetworkResult
-from qibo.gates.abstract import ParametrizedGate
-
 import quimb as qu
 import quimb.tensor as qtn
+from qibo.backends import NumpyBackend
+from qibo.config import raise_error
+from qibo.gates.abstract import ParametrizedGate
+from qibo.models import Circuit
 
+from qibotn.backends.abstract import QibotnBackend
+from qibotn.result import TensorNetworkResult
 
 GATE_MAP = {
-        "h": "H",
-        "x": "X",
-        "y": "Y",
-        "z": "Z",
-        "s": "S",
-        "t": "T",
+    "h": "H",
+    "x": "X",
+    "y": "Y",
+    "z": "Z",
+    "s": "S",
+    "t": "T",
+    "rx": "RX",
+    "ry": "RY",
+    "rz": "RZ",
+    "u3": "U3",  # TODO: check
+    "cx": "CX",
+    "cnot": "CNOT",
+    "cy": "CY",
+    "cz": "CZ",
+    "iswap": "ISWAP",
+    "swap": "SWAP",
+    "ccx": "CCX",
+    "ccy": "CCY",
+    "ccz": "CCZ",
+    "toffoli": "TOFFOLI",
+    "cswap": "CSWAP",
+    "fredkin": "FREDKIN",
+    "fsim": "fsim",
+    "measure": "measure",
+}
 
-        "rx": "RX",
-        "ry": "RY",
-        "rz": "RZ",
-
-        "u3": "U3", # TODO: check
-
-        "cx": "CX",
-        "cnot": "CNOT",
-        "cy": "CY",
-        "cz": "CZ",
-
-        "iswap": "ISWAP",
-        "swap": "SWAP",
-
-        "ccx": "CCX",
-        "ccy": "CCY",
-        "ccz": "CCZ",
-
-        "toffoli": "TOFFOLI",
-        "cswap": "CSWAP",
-        "fredkin": "FREDKIN",
-
-        "fsim": "fsim",
-
-        "measure": "measure"
-    }
 
 class QuimbBackend(QibotnBackend, NumpyBackend):
 
-    def __init__(self):
+    def __init__(self, engine="numpy"):
         super().__init__()
 
         self.name = "qibotn"
         self.platform = "quimb"
+        self.engine = engine
 
         self.configure_tn_simulation()
-        self.setup_backend_specifics()
+        self.setup_backend_specifics(quimb_backend=engine)
 
     def configure_tn_simulation(
         self,
@@ -71,7 +63,7 @@ class QuimbBackend(QibotnBackend, NumpyBackend):
         Args:
             ansatz : str, optional
                 The tensor network ansatz to use. Default is `None` and, in this case, a
-                generic Circuit Quimb class is used.            
+                generic Circuit Quimb class is used.
                 max_bond_dimension : int, optional
                 The maximum bond dimension for the MPS ansatz. Default is 10.
 
@@ -83,7 +75,9 @@ class QuimbBackend(QibotnBackend, NumpyBackend):
         self.max_bond_dimension = max_bond_dimension
         self.n_most_frequent_states = n_most_frequent_states
 
-    def setup_backend_specifics(self, qimb_backend="numpy", contractions_optimizer="auto-hq"):
+    def setup_backend_specifics(
+        self, quimb_backend="numpy", contractions_optimizer="auto-hq"
+    ):
         """Setup backend specifics.
         Args:
             qimb_backend: str
@@ -91,19 +85,22 @@ class QuimbBackend(QibotnBackend, NumpyBackend):
             contractions_optimizer: str, optional
                 The contractions_optimizer to use for the quimb tensor network simulation.
         """
-        if qimb_backend == "jax":
+        if quimb_backend == "jax":
             import jax.numpy as jnp
+
             self.np = jnp
-        elif qimb_backend == "numpy":
+        elif quimb_backend == "numpy":
             import numpy as np
+
             self.np = np
-        elif qimb_backend == "torch":
+        elif quimb_backend == "torch":
             import torch
+
             self.np = torch
         else:
-            raise_error(ValueError, f"Unsupported quimb backend: {qimb_backend}")
+            raise_error(ValueError, f"Unsupported quimb backend: {quimb_backend}")
 
-        self.backend = qimb_backend
+        self.backend = quimb_backend
         self.contractions_optimizer = contractions_optimizer
 
     def execute_circuit(
@@ -180,7 +177,9 @@ class QuimbBackend(QibotnBackend, NumpyBackend):
             measured_probabilities = None
 
         statevector = (
-            circ_quimb.to_dense(backend=self.backend, optimize=self.contractions_optimizer)
+            circ_quimb.to_dense(
+                backend=self.backend, optimize=self.contractions_optimizer
+            )
             if return_array
             else None
         )
@@ -193,14 +192,16 @@ class QuimbBackend(QibotnBackend, NumpyBackend):
             statevector=statevector,
         )
 
-    def expectation(self, circuit, operators_list, sites_list, coeffs_list):
+    def expectation_observable_symbolic_from_state(
+        self, circuit, operators_list, sites_list, coeffs_list, nqubits
+    ):
         """
         Compute the expectation value of a symbolic Hamiltonian on a quantum circuit using tensor network contraction.
         This method takes a Qibo circuit, converts it to a Quimb tensor network circuit, and evaluates the expectation value
         of a Hamiltonian specified by three lists of strings: operators, sites, and coefficients.
         The expectation value is computed by summing the contributions from each term in the Hamiltonian, where each term's
         expectation is calculated using Quimb's `local_expectation` function.
-        
+
         Parameters
         ----------
         circuit : qibo.models.Circuit
@@ -216,27 +217,30 @@ class QuimbBackend(QibotnBackend, NumpyBackend):
         float
             The real part of the expectation value of the Hamiltonian on the given circuit state.
         """
-        quimb_circuit = self._qibo_circuit_to_quimb(circuit, quimb_circuit_type=qtn.Circuit)
-            
+        quimb_circuit = self._qibo_circuit_to_quimb(
+            circuit, quimb_circuit_type=qtn.Circuit
+        )
+
         expectation_value = 0.0
-        for opstr, sitesstr, coeffstr in zip(operators_list, sites_list, coeffs_list):
-            
+        for opstr, sites, coeff in zip(operators_list, sites_list, coeffs_list):
+
             ops = self._string_to_quimb_operator(opstr)
-            coeff = self._parse_coefficient(coeffstr)
-            sites = tuple(int(q) for q in sitesstr)
-            
+            coeff = coeff.real
+
             exp_values = quimb_circuit.local_expectation(
                 ops,
                 where=sites,
                 backend=self.backend,
-                optimize=self.contractions_optimizer
+                optimize=self.contractions_optimizer,
             )
 
             expectation_value = expectation_value + coeff * exp_values
-        
-        return np.real(expectation_value)
 
-    def _qibo_circuit_to_quimb(self, qibo_circ, quimb_circuit_type=qtn.Circuit, **circuit_kwargs):
+        return self.np.real(expectation_value)
+
+    def _qibo_circuit_to_quimb(
+        self, qibo_circ, quimb_circuit_type=qtn.Circuit, **circuit_kwargs
+    ):
         """
         Convert a Qibo Circuit to a Quimb Circuit. Measurement gates are ignored. If are given gates not supported by Quimb, an error is raised.
 
@@ -267,18 +271,12 @@ class QuimbBackend(QibotnBackend, NumpyBackend):
 
             params = getattr(gate, "parameters", ())
             qubits = getattr(gate, "qubits", ())
-            
-            is_parametrized = (
-                isinstance(gate, ParametrizedGate)
-                and getattr(gate, "trainable", True)
-                )
+
+            is_parametrized = isinstance(gate, ParametrizedGate) and getattr(
+                gate, "trainable", True
+            )
             if is_parametrized:
-                circ.apply_gate(
-                    qname,
-                    *params,
-                    *qubits,
-                    parametrized=is_parametrized
-                )
+                circ.apply_gate(qname, *params, *qubits, parametrized=is_parametrized)
             else:
                 circ.apply_gate(
                     qname,
@@ -287,28 +285,6 @@ class QuimbBackend(QibotnBackend, NumpyBackend):
                 )
         return circ
 
-    def _parse_coefficient(self, s):
-        """Parse a coefficient from string to float, int, or complex.
-        Args:
-            s: str
-                The string representation of the coefficient.
-        Returns:
-            The coefficient as float, int, or complex.
-        """ 
-        try:
-            return float(s)
-        except ValueError:
-            pass
-        if s == "j":
-            return 1j
-        elif s == "-j":
-            return -1j
-        else:
-            try:
-                return complex(s)
-            except ValueError:
-                raise ValueError(f"Cannot parse coefficient: {s}")
-
     def _string_to_quimb_operator(self, op_str):
         """
         Convert a Pauli string (e.g. 'xzy') to a Quimb operator using '&' chaining.