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    "# Executing Quantum Circuits \n",
    "\n",
    "One can excute CUDA Quantum kernels via the `sample` and `observe` function calls. \n",
    "\n",
    "Quantum states collapse upon measurement and hence need to be sampled from many times to gather statistics. The CUDA Quantum `sample` call enables this: \n",
    "\n",
    "## Sample"
   ]
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     "text": [
      "{ 00:490 11:510 }\n"
     ]
    }
   ],
   "source": [
    "import cudaq\n",
    "\n",
    "qubit_count = 2\n",
    "\n",
    "# Define the simulation target.\n",
    "cudaq.set_target(\"qpp-cpu\")\n",
    "\n",
    "# Define a quantum kernel function.\n",
    "kernel = cudaq.make_kernel()\n",
    "\n",
    "# Allocate our `qubit_count` to the kernel.\n",
    "qubits = kernel.qalloc(qubit_count)\n",
    "\n",
    "# 2-qubit GHZ state.\n",
    "kernel.h(qubits[0])\n",
    "for i in range(1, qubit_count):\n",
    "    kernel.cx(qubits[0], qubits[i])\n",
    "\n",
    "# If we dont specify measurements, all qubits are measured in\n",
    "# the Z-basis by default.\n",
    "kernel.mz(qubits)\n",
    "\n",
    "result = cudaq.sample(kernel, shots_count=1000)\n",
    "\n",
    "result.dump()"
   ]
  },
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   "source": [
    "In simulation mode, the quantum state is built once and then sampled from $s$ times where $s$ equals the `shots_count` . In hardware execution mode, the quantum state collapses upon measurement and hence needs to be rebuilt over and over again. \n",
    "\n",
    "## Observe\n",
    "\n",
    "`observe` allows us to gather qubit statistics and calculate expectation values. We must supply a spin operator in the form of a hamiltonian from which we would like to calculate $\\bra{\\psi}H\\ket{\\psi}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
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     "name": "stdout",
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     "text": [
      "<H> = 0.0\n"
     ]
    }
   ],
   "source": [
    "import cudaq\n",
    "from cudaq import spin\n",
    "\n",
    "qubit_count = 2\n",
    "\n",
    "# Define the simulation target.\n",
    "cudaq.set_target(\"qpp-cpu\")\n",
    "\n",
    "# Define a quantum kernel function.\n",
    "kernel = cudaq.make_kernel()\n",
    "\n",
    "# Allocate our `qubit_count` to the kernel.\n",
    "qubits = kernel.qalloc(qubit_count)\n",
    "\n",
    "# 2-qubit GHZ state.\n",
    "kernel.h(qubits[0])\n",
    "\n",
    "for i in range(1, qubit_count):\n",
    "    kernel.cx(qubits[0], qubits[i])\n",
    "\n",
    "# Define a Hamiltonian in terms of Pauli Spin operators.\n",
    "hamiltonian = spin.z(0) + spin.y(1) + spin.x(0) * spin.z(0)\n",
    "\n",
    "# Compute the expectation value given the state prepared by the kernel.\n",
    "result = cudaq.observe(kernel, hamiltonian).expectation()\n",
    "\n",
    "print('<H> =', result)"
   ]
  }
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