diff --git a/examples/quimb_intro/quimb_introduction.ipynb b/examples/quimb_intro/quimb_introduction.ipynb
index 73a831b..70fdf45 100644
--- a/examples/quimb_intro/quimb_introduction.ipynb
+++ b/examples/quimb_intro/quimb_introduction.ipynb
@@ -53,7 +53,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 23,
+   "execution_count": 3,
    "id": "64162116-1555-4a68-811c-01593739d622",
    "metadata": {},
    "outputs": [],
@@ -67,10 +67,7 @@
     "# set numpy random seed\n",
     "np.random.seed(42)\n",
     "\n",
-    "quimb_backend.setup_backend_specifics(\n",
-    "    qimb_backend=\"jax\", \n",
-    "    contractions_optimizer=ctg_opt\n",
-    "    )"
+    "quimb_backend.setup_backend_specifics(qimb_backend=\"jax\")"
    ]
   },
   {
@@ -83,7 +80,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 4,
    "id": "4a22a172-f50d-411d-afa3-fa61937c7b3a",
    "metadata": {},
    "outputs": [],
@@ -102,7 +99,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 5,
    "id": "76f23c57-6d08-496b-9a27-52fb63bbfcb1",
    "metadata": {},
    "outputs": [
@@ -124,7 +121,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 6,
    "id": "07b2c097-cea2-42ec-8f1d-b4bbb5b71d98",
    "metadata": {},
    "outputs": [],
@@ -147,7 +144,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 7,
    "id": "2ee03e94-d794-4a51-9e76-01e8d8a259ba",
    "metadata": {},
    "outputs": [],
@@ -175,7 +172,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 8,
    "id": "35a244c3-adba-4b8b-b28c-0ab592b0f7cf",
    "metadata": {},
    "outputs": [
@@ -196,34 +193,32 @@
       "text/plain": [
        "{'nqubits': 4,\n",
        " 'backend': qibotn (quimb),\n",
-       " 'measures': Counter({'1000': 14,\n",
-       "          '1011': 9,\n",
-       "          '0101': 5,\n",
-       "          '0010': 12,\n",
+       " 'measures': Counter({'1010': 9,\n",
+       "          '0100': 8,\n",
+       "          '1101': 15,\n",
+       "          '1011': 4,\n",
+       "          '1111': 12,\n",
+       "          '1000': 13,\n",
        "          '0000': 8,\n",
-       "          '1010': 4,\n",
-       "          '1101': 19,\n",
+       "          '0010': 6,\n",
        "          '0011': 6,\n",
-       "          '0100': 3,\n",
-       "          '1111': 9,\n",
-       "          '0111': 4,\n",
-       "          '0110': 4,\n",
-       "          '1110': 2,\n",
-       "          '1100': 1}),\n",
+       "          '0101': 8,\n",
+       "          '1110': 5,\n",
+       "          '0110': 5,\n",
+       "          '0111': 1}),\n",
        " 'measured_probabilities': {'1101': np.float64(0.12331159869893256),\n",
        "  '1000': np.float64(0.11330883548333587),\n",
-       "  '0010': np.float64(0.09466860481989385),\n",
-       "  '1011': np.float64(0.053499396925872744),\n",
        "  '1111': np.float64(0.10184806171791962),\n",
-       "  '0000': np.float64(0.08390937969317269),\n",
-       "  '0011': np.float64(0.07571277233522114),\n",
-       "  '0101': np.float64(0.05622305772698622),\n",
        "  '1010': np.float64(0.03872758515126756),\n",
-       "  '0111': np.float64(0.04029185074729259),\n",
-       "  '0110': np.float64(0.05146064807369214),\n",
        "  '0100': np.float64(0.07142939529687138),\n",
+       "  '0000': np.float64(0.08390937969317269),\n",
+       "  '0101': np.float64(0.05622305772698622),\n",
+       "  '0010': np.float64(0.09466860481989385),\n",
+       "  '0011': np.float64(0.07571277233522114),\n",
        "  '1110': np.float64(0.07174919872959985),\n",
-       "  '1100': np.float64(0.013605984872668404)},\n",
+       "  '0110': np.float64(0.05146064807369214),\n",
+       "  '1011': np.float64(0.053499396925872744),\n",
+       "  '0111': np.float64(0.04029185074729259)},\n",
        " 'prob_type': 'default',\n",
        " 'statevector': Array([[ 0.08809624-0.27594998j],\n",
        "        [-0.05174781+0.04471217j],\n",
@@ -243,7 +238,7 @@
        "        [-0.01729754-0.31866732j]], dtype=complex64)}"
       ]
      },
-     "execution_count": 10,
+     "execution_count": 8,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -268,7 +263,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 9,
    "id": "c0443efc-21ef-4ed5-9cf4-785d204a1881",
    "metadata": {},
    "outputs": [
@@ -277,7 +272,7 @@
      "output_type": "stream",
      "text": [
       "Probabilities:\n",
-      " {'1101': np.float64(0.12331159869893256), '1000': np.float64(0.11330883548333587), '0010': np.float64(0.09466860481989385), '1011': np.float64(0.053499396925872744), '1111': np.float64(0.10184806171791962), '0000': np.float64(0.08390937969317269), '0011': np.float64(0.07571277233522114), '0101': np.float64(0.05622305772698622), '1010': np.float64(0.03872758515126756), '0111': np.float64(0.04029185074729259), '0110': np.float64(0.05146064807369214), '0100': np.float64(0.07142939529687138), '1110': np.float64(0.07174919872959985), '1100': np.float64(0.013605984872668404)}\n",
+      " {'1101': np.float64(0.12331159869893256), '1000': np.float64(0.11330883548333587), '1111': np.float64(0.10184806171791962), '1010': np.float64(0.03872758515126756), '0100': np.float64(0.07142939529687138), '0000': np.float64(0.08390937969317269), '0101': np.float64(0.05622305772698622), '0010': np.float64(0.09466860481989385), '0011': np.float64(0.07571277233522114), '1110': np.float64(0.07174919872959985), '0110': np.float64(0.05146064807369214), '1011': np.float64(0.053499396925872744), '0111': np.float64(0.04029185074729259)}\n",
       "\n",
       "State:\n",
       " [[ 0.08809624-0.27594998j]\n",
@@ -307,7 +302,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dd84f1f3-7aa5-4ad1-ae09-81e0aff75b5b",
+   "id": "9531f9d6",
    "metadata": {},
    "source": [
     "### Compute expectation values\n",
@@ -321,57 +316,99 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
-   "id": "37385485-e8a3-4ab0-ad44-bcc4e9da24ca",
+   "execution_count": 10,
+   "id": "647f2073",
    "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "0: ─RY─RZ─o─────X─RY─RZ─o─────X─RY─RZ─o─────X─M─\n",
-      "1: ─RY─RZ─X─o───|─RY─RZ─X─o───|─RY─RZ─X─o───|─M─\n",
-      "2: ─RY─RZ───X─o─|─RY─RZ───X─o─|─RY─RZ───X─o─|─M─\n",
-      "3: ─RY─RZ─────X─o─RY─RZ─────X─o─RY─RZ─────X─o─M─\n"
-     ]
-    }
-   ],
+   "outputs": [],
    "source": [
-    "# We are going to compute the expval of an Hamiltonian\n",
-    "# On the state prepared by the following circuit\n",
-    "circuit.draw()\n",
+    "import numpy as np\n",
+    "import jax\n",
+    "from qibo.backends import construct_backend\n",
+    "from qibo import Circuit, gates"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "74c63a41",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# construct qibotn backend\n",
+    "quimb_backend = construct_backend(backend=\"qibotn\", platform=\"quimb\")\n",
     "\n",
-    "circuit.set_parameters(\n",
-    "    np.random.randn(len(circuit.get_parameters()))\n",
+    "quimb_backend.setup_backend_specifics(\n",
+    "    qimb_backend    =\"jax\", \n",
+    "    contractions_optimizer='auto-hq'\n",
+    "    )\n",
+    "\n",
+    "quimb_backend.configure_tn_simulation(\n",
+    "    max_bond_dimension=10\n",
     ")"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 20,
-   "id": "9712783d",
+   "execution_count": 19,
+   "id": "b2a0decb",
    "metadata": {},
    "outputs": [],
    "source": [
     "# define Hamiltonian\n",
-    "operators = [\"zz\", \"xz\", \"z\"]\n",
-    "qubits = [\"01\", \"02\", \"3\"]\n",
-    "coefficients = [\"0.5\", \"-1.5\", \"1\"]\n",
+    "operators = [\"xzy\", \"yxzy\", \"zy\"]\n",
+    "qubits = [\"011\", \"0112\", \"01\"]\n",
+    "coefficients = [\"1\", \"2\", \"j\"]\n",
     "hamiltonian = (operators, qubits, coefficients)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 25,
-   "id": "163b70a3-814a-4a62-a98a-2ffca933a544",
+   "execution_count": 18,
+   "id": "bd734be8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# define circuit\n",
+    "def build_circuit(nqubits, nlayers):\n",
+    "    circ = Circuit(nqubits)\n",
+    "    for layer in range(nlayers):\n",
+    "        for q in range(nqubits):\n",
+    "            circ.add(gates.RY(q=q, theta=0.))\n",
+    "            circ.add(gates.RZ(q=q, theta=0.))\n",
+    "            circ.add(gates.RX(q=q, theta=0.))\n",
+    "        for q in range(nqubits - 1):\n",
+    "            circ.add(gates.CNOT(q, q + 1))\n",
+    "            circ.add(gates.SWAP(q, q + 1))\n",
+    "    circ.add(gates.M(*range(nqubits)))\n",
+    "    return circ\n",
+    "\n",
+    "def build_circuit_problematic(nqubits, nlayers):\n",
+    "    circ = Circuit(nqubits)\n",
+    "    for _ in range(nlayers):\n",
+    "        for q in range(nqubits):\n",
+    "            circ.add(gates.RY(q=q, theta=0.))\n",
+    "            circ.add(gates.RZ(q=q, theta=0.))\n",
+    "        [circ.add(gates.CNOT(q%nqubits, (q+1)%nqubits) for q in range(nqubits))]\n",
+    "    circ.add(gates.M(*range(nqubits)))\n",
+    "    return circ\n",
+    "\n",
+    "\n",
+    "nqubits = 4\n",
+    "circuit = build_circuit(nqubits=nqubits, nlayers=3)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "id": "fe63ff24",
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Expectation value: 0.7143570184707642\n",
-      "Elapsed time: 0.1498 seconds\n"
+      "Expectation value: 0.0\n",
+      "Elapsed time: 0.1071 seconds\n"
      ]
     }
    ],
@@ -391,7 +428,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "90663e28",
+   "id": "d976a849",
    "metadata": {},
    "source": [
     "Try with Qibo (which is by default using the Qibojit backend)\n"
@@ -399,52 +436,31 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
-   "id": "c8760074",
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/latex": [
-       "$\\displaystyle - 1.5 X_{0} Z_{2} + 0.5 Z_{0} Z_{1} + Z_{3}$"
-      ],
-      "text/plain": [
-       "-1.5*X0*Z2 + 0.5*Z0*Z1 + Z3"
-      ]
-     },
-     "execution_count": 18,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "from qibo.symbols import Z, X, I\n",
-    "# We can create a symbolic Hamiltonian\n",
-    "form = 0.5 * Z(0) * Z(1) +- 1.5 *  X(0) * Z(2) + Z(3)\n",
-    "hamiltonian = hamiltonians.SymbolicHamiltonian(form)\n",
-    "\n",
-    "#  Let's show it\n",
-    "hamiltonian.form"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 19,
-   "id": "e2d05707",
+   "execution_count": 22,
+   "id": "fb1436c8",
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Expectation value: 0.7143570920618565\n",
-      "Elapsed time: 0.5597 seconds\n"
+      "Expectation value: 1.5\n",
+      "Elapsed time: 0.0501 seconds\n"
      ]
     }
    ],
    "source": [
+    "from qibo.symbols import Z, X, I\n",
+    "# We can create a symbolic Hamiltonian\n",
+    "form = 0.5 * Z(0) * Z(1) +- 1.5 *  X(0) * Z(2) + Z(3)\n",
+    "sym_hamiltonian = hamiltonians.SymbolicHamiltonian(form)\n",
+    "\n",
+    "#  Let's show it\n",
+    "sym_hamiltonian.form\n",
+    "\n",
+    "# Compute expectation value\n",
     "start = time.time()\n",
-    "result = hamiltonian.expectation(circuit().state())\n",
+    "result = sym_hamiltonian.expectation(circuit().state())\n",
     "elapsed = time.time() - start\n",
     "print(f\"Expectation value: {result}\")\n",
     "print(f\"Elapsed time: {elapsed:.4f} seconds\")"
@@ -452,7 +468,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "94df291c-9ddc-4b2e-8442-5fca00784bd8",
+   "id": "77bef077",
    "metadata": {},
    "source": [
     "They match! 🥳"
@@ -460,107 +476,35 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d2d119fc",
+   "id": "50130ae6",
    "metadata": {},
    "source": [
-    "### Derivative of the extimation function"
+    "We can also compute gradient of expectation function"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
-   "id": "8df55c5f",
+   "execution_count": 23,
+   "id": "6a3b26e4",
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "0: ─RY─RZ─o─────X─RY─RZ─o─────X─RY─RZ─o─────X─M─\n",
-      "1: ─RY─RZ─X─o───|─RY─RZ─X─o───|─RY─RZ─X─o───|─M─\n",
-      "2: ─RY─RZ───X─o─|─RY─RZ───X─o─|─RY─RZ───X─o─|─M─\n",
-      "3: ─RY─RZ─────X─o─RY─RZ─────X─o─RY─RZ─────X─o─M─\n"
+      "[ 8.19939339e-10 -3.14190913e-08 -2.99498648e-09 -1.03641796e-07\n",
+      "  8.48652704e-10  1.00297093e-07 -6.75429277e-08 -9.78565140e-09\n",
+      " -5.11915417e-08  1.29225235e-08 -7.44280655e-08 -3.49115048e-08\n",
+      " -4.98508879e-09  6.80729357e-08 -3.29755920e-08  4.20008526e-08\n",
+      " -2.89742630e-08  1.18602941e-07 -2.88252178e-08  5.57985391e-09\n",
+      " -3.17434115e-08 -1.03342952e-08  1.34079716e-08 -7.05437886e-09\n",
+      " -4.34059650e-08 -2.18019203e-08 -5.36932561e-08 -6.38544009e-08\n",
+      "  5.85312279e-08  8.45709067e-08 -1.12777876e-09 -6.41545981e-08\n",
+      "  7.25317406e-08  4.10035668e-08 -1.29046382e-08  6.07501676e-08]\n"
      ]
     }
    ],
    "source": [
-    "def build_circuit_A(nqubits, nlayers):\n",
-    "    \"\"\"Construct a parametric quantum circuit.\"\"\"\n",
-    "    circ = Circuit(nqubits)\n",
-    "    for _ in range(nlayers):\n",
-    "        for q in range(nqubits):\n",
-    "            circ.add(gates.RY(q=q, theta=0.))\n",
-    "            circ.add(gates.RZ(q=q, theta=0.))\n",
-    "        [circ.add(gates.CNOT(q%nqubits, (q+1)%nqubits) for q in range(nqubits))]\n",
-    "    circ.add(gates.M(*range(nqubits)))\n",
-    "    return circ\n",
-    "\n",
-    "circuit = build_circuit(nqubits=nqubits, nlayers=3)\n",
-    "circuit.draw()"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "id": "b02de56b",
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "0: ─RY─RZ─RX─o─x─────────RY─RZ─RX─o─x─────────RY─RZ─RX─o─x─────────M─\n",
-      "1: ─RY─RZ─RX─X─x─o─x─────RY─RZ─RX─X─x─o─x─────RY─RZ─RX─X─x─o─x─────M─\n",
-      "2: ─RY─RZ─RX─────X─x─o─x─RY─RZ─RX─────X─x─o─x─RY─RZ─RX─────X─x─o─x─M─\n",
-      "3: ─RY─RZ─RX─────────X─x─RY─RZ─RX─────────X─x─RY─RZ─RX─────────X─x─M─\n"
-     ]
-    }
-   ],
-   "source": [
-    "def build_circuit_B(nqubits, nlayers):\n",
-    "    circ = Circuit(nqubits)\n",
-    "    for _ in range(nlayers):\n",
-    "        for q in range(nqubits):\n",
-    "            circ.add(gates.RY(q=q, theta=0.))\n",
-    "            circ.add(gates.RZ(q=q, theta=0.))\n",
-    "            circ.add(gates.RX(q=q, theta=0.))\n",
-    "        for q in range(nqubits - 1):\n",
-    "            circ.add(gates.CNOT(q, q + 1))\n",
-    "            circ.add(gates.SWAP(q, q + 1))\n",
-    "    circ.add(gates.M(*range(nqubits)))\n",
-    "    return circ\n",
-    "\n",
-    "circuit = build_circuit(nqubits=nqubits, nlayers=3)\n",
-    "circuit.draw()"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "id": "0943482e",
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "[ 2.57090405e-02 -2.24703997e-02 -2.82387018e-01 -1.02654770e-01\n",
-      "  1.03672206e-01  1.03572123e-01 -9.93297935e-01  3.26367974e-01\n",
-      " -8.58561993e-01 -3.14284384e-01 -1.22645356e-01 -2.13570029e-01\n",
-      " -4.55642402e-01  1.11669600e-02 -2.92290837e-01 -1.91316485e-01\n",
-      "  2.78813928e-01  8.80600572e-01  5.07975101e-01 -1.97107181e-01\n",
-      " -4.69740361e-01 -8.50831568e-02  4.45045829e-01  3.42172906e-02\n",
-      " -8.43066633e-01  1.86891228e-01  4.52477366e-01  7.36747682e-03\n",
-      " -6.28291368e-01 -9.38566178e-02  5.43581992e-02  3.57441790e-02\n",
-      "  5.15162945e-04  2.55566716e-01 -3.20922613e-01  4.96513635e-01]\n",
-      "Elapsed time: 11.3724 seconds\n"
-     ]
-    }
-   ],
-   "source": [
-    "import jax\n",
-    "import time\n",
-    "\n",
     "def f(circuit, hamiltonian, params):\n",
     "    circuit.set_parameters(params)\n",
     "    return quimb_backend.expectation(\n",
@@ -571,11 +515,7 @@
     "    )\n",
     "\n",
     "parameters = np.random.uniform(-np.pi, np.pi, size=len(circuit.get_parameters()))\n",
-    "start = time.time()\n",
-    "grad = jax.grad(f, argnums=2)(circuit, hamiltonian, parameters)\n",
-    "elapsed = time.time() - start\n",
-    "print(grad)\n",
-    "print(f\"Elapsed time: {elapsed:.4f} seconds\")"
+    "print(jax.grad(f, argnums=2)(circuit, hamiltonian, parameters))\n"
    ]
   }
  ],