From 5f7a2da1be4036715528566fa03034b834f36a8a Mon Sep 17 00:00:00 2001
From: Jason Saroni <77505813+jsaroni@users.noreply.github.com>
Date: Thu, 29 May 2025 00:50:44 -0500
Subject: [PATCH 1/3] Add files via upload

An example of interfacing scqubits with Qiskit metal.
---
 ...ansmon_qubit_qiskit_metal_interfaced.ipynb | 542 ++++++++++++++++++
 1 file changed, 542 insertions(+)
 create mode 100644 examples/demo_transmon_qubit_qiskit_metal_interfaced.ipynb

diff --git a/examples/demo_transmon_qubit_qiskit_metal_interfaced.ipynb b/examples/demo_transmon_qubit_qiskit_metal_interfaced.ipynb
new file mode 100644
index 0000000..4304ccd
--- /dev/null
+++ b/examples/demo_transmon_qubit_qiskit_metal_interfaced.ipynb
@@ -0,0 +1,542 @@
+{
+  "nbformat": 4,
+  "nbformat_minor": 0,
+  "metadata": {
+    "colab": {
+      "provenance": []
+    },
+    "kernelspec": {
+      "name": "python3",
+      "display_name": "Python 3"
+    },
+    "language_info": {
+      "name": "python"
+    }
+  },
+  "cells": [
+    {
+      "cell_type": "markdown",
+      "source": [
+        "The following tutorial is based on Koch et al. (2007) PRA. It models the transmon qubit in the cooper-pair charge basis, assuming wrapped junction phase variable."
+      ],
+      "metadata": {
+        "id": "EuGcWuEZZCIK"
+      }
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "#Import key modules for this demo"
+      ],
+      "metadata": {
+        "id": "iE0Cjg0nWBSw"
+      }
+    },
+    {
+      "cell_type": "code",
+      "execution_count": 1,
+      "metadata": {
+        "id": "8RKXGlZvHuGp"
+      },
+      "outputs": [],
+      "source": [
+        "%load_ext autoreload\n",
+        "%autoreload 2\n",
+        "%config IPCompleter.greedy = True\n",
+        "%matplotlib inline\n",
+        "%config Completer.use_jedi = False\n",
+        "%config InlineBackend.figure_format = 'svg'"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "!pip install PySide2\n",
+        "!pip install qiskit-metal"
+      ],
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/"
+        },
+        "id": "qoAOt7LRKlNZ",
+        "outputId": "38bf78f4-a14d-4b00-b5d2-8eef0921762b"
+      },
+      "execution_count": 7,
+      "outputs": [
+        {
+          "output_type": "stream",
+          "name": "stdout",
+          "text": [
+            "Requirement already satisfied: PySide2 in /usr/local/lib/python3.11/dist-packages (5.13.2)\n",
+            "Requirement already satisfied: shiboken2==5.13.2 in /usr/local/lib/python3.11/dist-packages (from PySide2) (5.13.2)\n"
+          ]
+        }
+      ]
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "import os\n",
+        "\n",
+        "# Set this environment variable to prevent qiskit-metal from trying to set up a Qt backend\n",
+        "os.environ[\"QISKIT_METAL_HEADLESS\"] = \"1\""
+      ],
+      "metadata": {
+        "id": "0YOEc4VjO7QT"
+      },
+      "execution_count": 11,
+      "outputs": []
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "# Now import qiskit_metal\n",
+        "import qiskit_metal"
+      ],
+      "metadata": {
+        "id": "e-eHuv1PO7S9"
+      },
+      "execution_count": 12,
+      "outputs": []
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "from qiskit_metal.analyses.hamiltonian.transmon_charge_basis import Hcpb\n",
+        "from qiskit_metal.analyses.hamiltonian.transmon_CPB_analytic import Hcpb_analytic\n",
+        "import pandas as pd\n",
+        "import matplotlib.pyplot as plt\n",
+        "import numpy as np\n"
+      ],
+      "metadata": {
+        "id": "YmvJTBQ3O7Ve"
+      },
+      "execution_count": 15,
+      "outputs": []
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "%matplotlib inline\n",
+        "%config InlineBackend.figure_format = 'svg'\n",
+        "\n",
+        "import numpy as np\n",
+        "import scqubits as scq"
+      ],
+      "metadata": {
+        "id": "QpRuTrajapEO"
+      },
+      "execution_count": 74,
+      "outputs": []
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "#The Hamiltonian"
+      ],
+      "metadata": {
+        "id": "ZMh9LZ6JbZ-I"
+      }
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "The Hamiltonian of the transmon qubit in the can be written in the charge basis as:"
+      ],
+      "metadata": {
+        "id": "W1u0TH73WuXf"
+      }
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "\\begin{align}\n",
+        "        H = 4E_C(\\hat{n} - n_g)^2 - \\frac{1}{2}E_J\\sum_n\\left(\n",
+        "          \\left| n \\right \\rangle \\left \\langle n+1 \\right | + \\text{h.c.}  \\right)\n",
+        "\\end{align}"
+      ],
+      "metadata": {
+        "id": "qReRvMBJX2ah"
+      }
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "tmon = scq.Transmon.create()\n",
+        "print(tmon)"
+      ],
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/"
+        },
+        "id": "lxBDWrwTcwy6",
+        "outputId": "b2b64eff-53af-47ff-bcfd-d3ed0d38562a"
+      },
+      "execution_count": 76,
+      "outputs": [
+        {
+          "output_type": "stream",
+          "name": "stderr",
+          "text": [
+            "WARNING:py.warnings:Warning: Starting with v3.2, scqubits uses the optional package 'ipyuetify' for graphical user interfaces. To use this functionality, add the package via `conda install -c conda-forge ipyvuetify` or `pip install ipyvuetify`.\n",
+            "For use with jupyter lab, additionally execute `jupyter labextension install jupyter-vuetify`.\n",
+            "\n",
+            " /usr/local/lib/python3.11/dist-packages/scqubits/utils/misc.py: 143\n"
+          ]
+        },
+        {
+          "output_type": "stream",
+          "name": "stdout",
+          "text": [
+            "Transmon------------| [Transmon_9]\n",
+            "                    | EJ: 15.0\n",
+            "                    | EC: 0.3\n",
+            "                    | ng: 0.0\n",
+            "                    | ncut: 30\n",
+            "                    | truncated_dim: 10\n",
+            "                    |\n",
+            "                    | dim: 61\n",
+            "\n"
+          ]
+        }
+      ]
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "x = np.linspace(-2.0,2.0,101) # this represents the charging energy (ng)\n",
+        "H_norm = Hcpb(nlevels=2, Ej=1000.0, Ec=1000.0, ng=0.5) # Hamiltonian definition\n",
+        "norm = H_norm.fij(0,1) # normalization constant"
+      ],
+      "metadata": {
+        "id": "AX8Vj7EZcw2R"
+      },
+      "execution_count": 79,
+      "outputs": []
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "E0, E1, E2 = [], [], []\n",
+        "\n",
+        "# For a given value of offset charge (ng, represented by x) we will calculate the CPB Hamiltonian using the previously assigned values of E_J and E_C. Then we calculate the eigenvalue for a given value of m.\n",
+        "for i in x:\n",
+        "    H = Hcpb(nlevels=30, Ej=15.0, Ec=0.3, ng=i)\n",
+        "    E0.append(H.evalue_k(0)/norm)\n",
+        "    E1.append(H.evalue_k(1)/norm)\n",
+        "    E2.append(H.evalue_k(2)/norm)\n"
+      ],
+      "metadata": {
+        "id": "5qaCI49Zapnw"
+      },
+      "execution_count": 80,
+      "outputs": []
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "#Verifying Orthonormality of the Wavefunctions\n",
+        "\n"
+      ],
+      "metadata": {
+        "id": "STp-uA8laevC"
+      }
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "Psi0, theta0 = H.psi_k(0)\n",
+        "Psi1, theta1 = H.psi_k(1)\n",
+        "print(np.dot(Psi0,Psi1.conj()))\n",
+        "print(np.dot(Psi0, Psi0.conj()))\n",
+        "print(np.dot(Psi1, Psi1.conj()))"
+      ],
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/"
+        },
+        "id": "YFkasdgOad-7",
+        "outputId": "c52e5b87-5ad4-4315-9b98-5c1ca8a152c7"
+      },
+      "execution_count": 81,
+      "outputs": [
+        {
+          "output_type": "stream",
+          "name": "stdout",
+          "text": [
+            "(8.326672684688674e-17-7.392060725770952e-18j)\n",
+            "(0.9999999999999996+0j)\n",
+            "(0.9999999999999998+0j)\n"
+          ]
+        }
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "#Charge dispersion"
+      ],
+      "metadata": {
+        "id": "UocxZpEnhFIO"
+      }
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "The peak-to-peak value of the charge dispersion for the mth energy level is given by the expression: $\\epsilon_m = E_m(n_g=0.5)-E_m(n_g=0.0)$. We can plot $\\epsilon_m/E_{01}$ as a function of $E_J/E_C$ for the first few energy levels and reproduce the figure published in Phys. Rev. A 76, 042319 (2007) (Figure 4(a)). We can start by defining a value of charging energy and creating empty lists for $\\epsilon_0$ through $\\epsilon_4$"
+      ],
+      "metadata": {
+        "id": "VunqiRmbcAUp"
+      }
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "E_c=100.0 # charging energy\n",
+        "epsilon0, epsilon1, epsilon2, epsilon3 = [], [], [], []    # charge dispersion for m=0 through m=4\n",
+        "x = np.linspace(1,140,101)"
+      ],
+      "metadata": {
+        "id": "dypU9-Yab0s0"
+      },
+      "execution_count": 83,
+      "outputs": []
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "#Anharmonicity"
+      ],
+      "metadata": {
+        "id": "ga7GZViWbPcS"
+      }
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "We know that for the transmon qubit, having the Josephson Energy much larger than the charging energy ($E_J>>E_C$) results in a decrease in anharmonicity. The latter is critical for a functional qubit in which the energy difference between the lowest two states ($E_{01}$) is sufficiently different than the energy difference between the second and third states, ($E_{12}$). The absolute anharmonicity is defined as $\\alpha = E_{12}-E_{01}$, while the relative anharmonicity is defined as\n",
+        ".\n",
+        "\n",
+        "We can easily make a plot of the anharmonicity as a function of $E_J/E_C$ using the Hcpb class. Let’s have the ratio of $E_J/E_C$ (which we’ll call x) vary from 0 to 80, and then we’ll create empty lists for $\\alpha$ and $\\alpha_r$\n",
+        "\n"
+      ],
+      "metadata": {
+        "id": "4T74-QzHj9J-"
+      }
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "x = np.linspace(0,80,101)   #EJ/EC\n",
+        "alpha = []\n",
+        "alpha_r = []"
+      ],
+      "metadata": {
+        "id": "-bKdEjF-WtST"
+      },
+      "execution_count": 82,
+      "outputs": []
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "for i in x:\n",
+        "    H_anharm = Hcpb(nlevels=15, Ej=i*E_c, Ec=E_c, ng=0.5)\n",
+        "    alpha.append(H_anharm.anharm())\n",
+        "    alpha_r.append(H_anharm.anharm()/H_anharm.fij(0,1))"
+      ],
+      "metadata": {
+        "id": "i6fXf09UWtXc"
+      },
+      "execution_count": 84,
+      "outputs": []
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "plt.figure(1)\n",
+        "plt.subplot(131)\n",
+        "plt.plot(x,alpha)\n",
+        "plt.xlabel(\"Ej/Ec\")\n",
+        "plt.ylabel(\"alpha\")\n",
+        "plt.subplot(133)\n",
+        "plt.plot(x,alpha_r)\n",
+        "plt.ylim(-0.2, 1.0)\n",
+        "plt.xlabel(\"Ej/Ec\")\n",
+        "plt.ylabel(\"alpha_r\")"
+      ],
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/",
+          "height": 458
+        },
+        "id": "iA_iKSoqbnfe",
+        "outputId": "9598e78b-85f9-4545-b33f-e4860d86fcc6"
+      },
+      "execution_count": 85,
+      "outputs": [
+        {
+          "output_type": "execute_result",
+          "data": {
+            "text/plain": [
+              "Text(0, 0.5, 'alpha_r')"
+            ]
+          },
+          "metadata": {},
+          "execution_count": 85
+        },
+        {
+          "output_type": "display_data",
+          "data": {
+            "text/plain": [
+              "<Figure size 640x480 with 2 Axes>"
+            ],
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+          },
+          "metadata": {}
+        }
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "#Dephasing Time (T2)"
+      ],
+      "metadata": {
+        "id": "oRYWjpV7cUxt"
+      }
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "We can estimate the qubit dephasing time (T2) due to charge noise by the following expression: $T_2 = \\frac{\\hbar}{A\\pi |\\epsilon_1|}$ where $A$ is on the order of $1E-4$\n",
+        " according to Phys. Rev. A 76, 042319 (2007). Since this is essentially just the inverse of the charge dispersion for $\\epsilon_1$, we can easily calculate T2 as a function of $E_J/E_C$ with the following:"
+      ],
+      "metadata": {
+        "id": "gdEHEA0wcYPL"
+      }
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "x = np.linspace(0,80,101)     # ratio of Ej/Ec, varying from 0 to 80\n",
+        "T2 = []                       # empty list for T2"
+      ],
+      "metadata": {
+        "id": "o1k7C88pbnh6"
+      },
+      "execution_count": 86,
+      "outputs": []
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "E_c = 1000.0\n",
+        "for i in x:\n",
+        "    H_half = Hcpb(nlevels=15, Ej=i*E_c, Ec=E_c, ng=0.5)\n",
+        "    H_zero = Hcpb(nlevels=15, Ej=i*E_c, Ec=E_c, ng=0.0)\n",
+        "    eps = abs(H_half.evalue_k(1) - H_zero.evalue_k(1))\n",
+        "    T2.append(1.0/(2.0*(1E-4)*(1E6)*eps) )\n",
+        "\n",
+        "plt.plot(x, T2)\n",
+        "plt.yscale(\"log\")\n",
+        "plt.xlabel(\"Ej/Ec\")\n",
+        "plt.ylabel(\"T2 (sec)\")"
+      ],
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/",
+          "height": 453
+        },
+        "id": "j52M1m_-bnkp",
+        "outputId": "d5099a5a-32ff-462d-b418-cc922cc4c242"
+      },
+      "execution_count": 87,
+      "outputs": [
+        {
+          "output_type": "execute_result",
+          "data": {
+            "text/plain": [
+              "Text(0, 0.5, 'T2 (sec)')"
+            ]
+          },
+          "metadata": {},
+          "execution_count": 87
+        },
+        {
+          "output_type": "display_data",
+          "data": {
+            "text/plain": [
+              "<Figure size 640x480 with 1 Axes>"
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+          },
+          "metadata": {}
+        }
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "Operator matrix elements visualized with text representation of each entry."
+      ],
+      "metadata": {
+        "id": "FF2_ML3uh19q"
+      }
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "tmon.plot_matrixelements('cos_phi_operator', evals_count=10, show3d=False, show_numbers=True);\n"
+      ],
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/",
+          "height": 417
+        },
+        "id": "_txRAJZlQMCL",
+        "outputId": "6a7278d1-9556-4f84-9333-59bdc9846213"
+      },
+      "execution_count": 117,
+      "outputs": [
+        {
+          "output_type": "display_data",
+          "data": {
+            "text/plain": [
+              "<Figure size 640x480 with 2 Axes>"
+            ],
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+          },
+          "metadata": {}
+        }
+      ]
+    },
+    {
+      "cell_type": "code",
+      "source": [],
+      "metadata": {
+        "id": "KolbsH0XigV1"
+      },
+      "execution_count": null,
+      "outputs": []
+    },
+    {
+      "cell_type": "code",
+      "source": [],
+      "metadata": {
+        "id": "GcmFLUX0QMEl"
+      },
+      "execution_count": null,
+      "outputs": []
+    },
+    {
+      "cell_type": "code",
+      "source": [],
+      "metadata": {
+        "id": "RSJolMAoHxSd"
+      },
+      "execution_count": 15,
+      "outputs": []
+    }
+  ]
+}
\ No newline at end of file

From cd2392bb9b2eadc6cff9b8106e93e815f598e1ba Mon Sep 17 00:00:00 2001
From: Jason Saroni <77505813+jsaroni@users.noreply.github.com>
Date: Thu, 29 May 2025 00:51:00 -0500
Subject: [PATCH 2/3] Add files via upload

An example of interfacing scqubits with Qiskit metal.

From 1ac1ebecdcfd16401227b2b735062c235b07310a Mon Sep 17 00:00:00 2001
From: Jason Saroni <77505813+jsaroni@users.noreply.github.com>
Date: Thu, 12 Jun 2025 02:12:02 -0400
Subject: [PATCH 3/3] Updating to show usage of Qiskit metal in showing
 orthogonality of wavefunctions and scqubits in showing matrix elements of the
 relevant operator.

---
 ...ansmon_qubit_qiskit_metal_interfaced.ipynb | 4695 ++++++++++++++++-
 1 file changed, 4489 insertions(+), 206 deletions(-)

diff --git a/examples/demo_transmon_qubit_qiskit_metal_interfaced.ipynb b/examples/demo_transmon_qubit_qiskit_metal_interfaced.ipynb
index 4304ccd..e9d53af 100644
--- a/examples/demo_transmon_qubit_qiskit_metal_interfaced.ipynb
+++ b/examples/demo_transmon_qubit_qiskit_metal_interfaced.ipynb
@@ -1,36 +1,22 @@
 {
-  "nbformat": 4,
-  "nbformat_minor": 0,
-  "metadata": {
-    "colab": {
-      "provenance": []
-    },
-    "kernelspec": {
-      "name": "python3",
-      "display_name": "Python 3"
-    },
-    "language_info": {
-      "name": "python"
-    }
-  },
   "cells": [
     {
       "cell_type": "markdown",
-      "source": [
-        "The following tutorial is based on Koch et al. (2007) PRA. It models the transmon qubit in the cooper-pair charge basis, assuming wrapped junction phase variable."
-      ],
       "metadata": {
         "id": "EuGcWuEZZCIK"
-      }
+      },
+      "source": [
+        "The following tutorial is based on Koch et al. (2007) PRA where the physics context is described. It models the transmon qubit in the cooper-pair charge basis, assuming wrapped junction phase variable."
+      ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "#Import key modules for this demo"
-      ],
       "metadata": {
         "id": "iE0Cjg0nWBSw"
-      }
+      },
+      "source": [
+        "#Import key modules for this demo"
+      ]
     },
     {
       "cell_type": "code",
@@ -50,10 +36,7 @@
     },
     {
       "cell_type": "code",
-      "source": [
-        "!pip install PySide2\n",
-        "!pip install qiskit-metal"
-      ],
+      "execution_count": 7,
       "metadata": {
         "colab": {
           "base_uri": "https://localhost:8080/"
@@ -61,110 +44,110 @@
         "id": "qoAOt7LRKlNZ",
         "outputId": "38bf78f4-a14d-4b00-b5d2-8eef0921762b"
       },
-      "execution_count": 7,
       "outputs": [
         {
-          "output_type": "stream",
           "name": "stdout",
+          "output_type": "stream",
           "text": [
             "Requirement already satisfied: PySide2 in /usr/local/lib/python3.11/dist-packages (5.13.2)\n",
             "Requirement already satisfied: shiboken2==5.13.2 in /usr/local/lib/python3.11/dist-packages (from PySide2) (5.13.2)\n"
           ]
         }
+      ],
+      "source": [
+        "!pip install PySide2\n",
+        "!pip install qiskit-metal"
       ]
     },
     {
       "cell_type": "code",
+      "execution_count": 11,
+      "metadata": {
+        "id": "0YOEc4VjO7QT"
+      },
+      "outputs": [],
       "source": [
         "import os\n",
         "\n",
         "# Set this environment variable to prevent qiskit-metal from trying to set up a Qt backend\n",
         "os.environ[\"QISKIT_METAL_HEADLESS\"] = \"1\""
-      ],
-      "metadata": {
-        "id": "0YOEc4VjO7QT"
-      },
-      "execution_count": 11,
-      "outputs": []
+      ]
     },
     {
       "cell_type": "code",
-      "source": [
-        "# Now import qiskit_metal\n",
-        "import qiskit_metal"
-      ],
+      "execution_count": 12,
       "metadata": {
         "id": "e-eHuv1PO7S9"
       },
-      "execution_count": 12,
-      "outputs": []
+      "outputs": [],
+      "source": [
+        "# Now import qiskit_metal\n",
+        "import qiskit_metal"
+      ]
     },
     {
       "cell_type": "code",
+      "execution_count": 15,
+      "metadata": {
+        "id": "YmvJTBQ3O7Ve"
+      },
+      "outputs": [],
       "source": [
         "from qiskit_metal.analyses.hamiltonian.transmon_charge_basis import Hcpb\n",
         "from qiskit_metal.analyses.hamiltonian.transmon_CPB_analytic import Hcpb_analytic\n",
         "import pandas as pd\n",
         "import matplotlib.pyplot as plt\n",
         "import numpy as np\n"
-      ],
-      "metadata": {
-        "id": "YmvJTBQ3O7Ve"
-      },
-      "execution_count": 15,
-      "outputs": []
+      ]
     },
     {
       "cell_type": "code",
+      "execution_count": 74,
+      "metadata": {
+        "id": "QpRuTrajapEO"
+      },
+      "outputs": [],
       "source": [
         "%matplotlib inline\n",
         "%config InlineBackend.figure_format = 'svg'\n",
         "\n",
         "import numpy as np\n",
         "import scqubits as scq"
-      ],
-      "metadata": {
-        "id": "QpRuTrajapEO"
-      },
-      "execution_count": 74,
-      "outputs": []
+      ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "#The Hamiltonian"
-      ],
       "metadata": {
         "id": "ZMh9LZ6JbZ-I"
-      }
+      },
+      "source": [
+        "#The Hamiltonian"
+      ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "The Hamiltonian of the transmon qubit in the can be written in the charge basis as:"
-      ],
       "metadata": {
         "id": "W1u0TH73WuXf"
-      }
+      },
+      "source": [
+        "The Hamiltonian of the transmon qubit in the can be written in the charge basis as:"
+      ]
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "qReRvMBJX2ah"
+      },
       "source": [
         "\\begin{align}\n",
         "        H = 4E_C(\\hat{n} - n_g)^2 - \\frac{1}{2}E_J\\sum_n\\left(\n",
         "          \\left| n \\right \\rangle \\left \\langle n+1 \\right | + \\text{h.c.}  \\right)\n",
         "\\end{align}"
-      ],
-      "metadata": {
-        "id": "qReRvMBJX2ah"
-      }
+      ]
     },
     {
       "cell_type": "code",
-      "source": [
-        "tmon = scq.Transmon.create()\n",
-        "print(tmon)"
-      ],
+      "execution_count": 76,
       "metadata": {
         "colab": {
           "base_uri": "https://localhost:8080/"
@@ -172,11 +155,10 @@
         "id": "lxBDWrwTcwy6",
         "outputId": "b2b64eff-53af-47ff-bcfd-d3ed0d38562a"
       },
-      "execution_count": 76,
       "outputs": [
         {
-          "output_type": "stream",
           "name": "stderr",
+          "output_type": "stream",
           "text": [
             "WARNING:py.warnings:Warning: Starting with v3.2, scqubits uses the optional package 'ipyuetify' for graphical user interfaces. To use this functionality, add the package via `conda install -c conda-forge ipyvuetify` or `pip install ipyvuetify`.\n",
             "For use with jupyter lab, additionally execute `jupyter labextension install jupyter-vuetify`.\n",
@@ -185,8 +167,8 @@
           ]
         },
         {
-          "output_type": "stream",
           "name": "stdout",
+          "output_type": "stream",
           "text": [
             "Transmon------------| [Transmon_9]\n",
             "                    | EJ: 15.0\n",
@@ -199,23 +181,32 @@
             "\n"
           ]
         }
+      ],
+      "source": [
+        "tmon = scq.Transmon.create()\n",
+        "print(tmon)"
       ]
     },
     {
       "cell_type": "code",
+      "execution_count": 79,
+      "metadata": {
+        "id": "AX8Vj7EZcw2R"
+      },
+      "outputs": [],
       "source": [
         "x = np.linspace(-2.0,2.0,101) # this represents the charging energy (ng)\n",
         "H_norm = Hcpb(nlevels=2, Ej=1000.0, Ec=1000.0, ng=0.5) # Hamiltonian definition\n",
         "norm = H_norm.fij(0,1) # normalization constant"
-      ],
-      "metadata": {
-        "id": "AX8Vj7EZcw2R"
-      },
-      "execution_count": 79,
-      "outputs": []
+      ]
     },
     {
       "cell_type": "code",
+      "execution_count": 80,
+      "metadata": {
+        "id": "5qaCI49Zapnw"
+      },
+      "outputs": [],
       "source": [
         "E0, E1, E2 = [], [], []\n",
         "\n",
@@ -225,32 +216,28 @@
         "    E0.append(H.evalue_k(0)/norm)\n",
         "    E1.append(H.evalue_k(1)/norm)\n",
         "    E2.append(H.evalue_k(2)/norm)\n"
-      ],
-      "metadata": {
-        "id": "5qaCI49Zapnw"
-      },
-      "execution_count": 80,
-      "outputs": []
+      ]
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "STp-uA8laevC"
+      },
       "source": [
         "#Verifying Orthonormality of the Wavefunctions\n",
         "\n"
-      ],
-      "metadata": {
-        "id": "STp-uA8laevC"
-      }
+      ]
     },
     {
-      "cell_type": "code",
+      "cell_type": "markdown",
+      "metadata": {},
       "source": [
-        "Psi0, theta0 = H.psi_k(0)\n",
-        "Psi1, theta1 = H.psi_k(1)\n",
-        "print(np.dot(Psi0,Psi1.conj()))\n",
-        "print(np.dot(Psi0, Psi0.conj()))\n",
-        "print(np.dot(Psi1, Psi1.conj()))"
-      ],
+        "Qiskit metal is used to show orthonormality of wavefunctions."
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": 81,
       "metadata": {
         "colab": {
           "base_uri": "https://localhost:8080/"
@@ -258,113 +245,108 @@
         "id": "YFkasdgOad-7",
         "outputId": "c52e5b87-5ad4-4315-9b98-5c1ca8a152c7"
       },
-      "execution_count": 81,
       "outputs": [
         {
-          "output_type": "stream",
           "name": "stdout",
+          "output_type": "stream",
           "text": [
             "(8.326672684688674e-17-7.392060725770952e-18j)\n",
             "(0.9999999999999996+0j)\n",
             "(0.9999999999999998+0j)\n"
           ]
         }
+      ],
+      "source": [
+        "Psi0, theta0 = H.psi_k(0)\n",
+        "Psi1, theta1 = H.psi_k(1)\n",
+        "print(np.dot(Psi0,Psi1.conj()))\n",
+        "print(np.dot(Psi0, Psi0.conj()))\n",
+        "print(np.dot(Psi1, Psi1.conj()))"
       ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "#Charge dispersion"
-      ],
       "metadata": {
         "id": "UocxZpEnhFIO"
-      }
+      },
+      "source": [
+        "#Charge dispersion"
+      ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "The peak-to-peak value of the charge dispersion for the mth energy level is given by the expression: $\\epsilon_m = E_m(n_g=0.5)-E_m(n_g=0.0)$. We can plot $\\epsilon_m/E_{01}$ as a function of $E_J/E_C$ for the first few energy levels and reproduce the figure published in Phys. Rev. A 76, 042319 (2007) (Figure 4(a)). We can start by defining a value of charging energy and creating empty lists for $\\epsilon_0$ through $\\epsilon_4$"
-      ],
       "metadata": {
         "id": "VunqiRmbcAUp"
-      }
+      },
+      "source": [
+        "The peak-to-peak value of the charge dispersion for the mth energy level is given by the expression: $\\epsilon_m = E_m(n_g=0.5)-E_m(n_g=0.0)$. We can plot $\\epsilon_m/E_{01}$ as a function of $E_J/E_C$ for the first few energy levels and reproduce the figure published in Phys. Rev. A 76, 042319 (2007) (Figure 4(a)). We can start by defining a value of charging energy and creating empty lists for $\\epsilon_0$ through $\\epsilon_4$"
+      ]
     },
     {
       "cell_type": "code",
+      "execution_count": 83,
+      "metadata": {
+        "id": "dypU9-Yab0s0"
+      },
+      "outputs": [],
       "source": [
         "E_c=100.0 # charging energy\n",
         "epsilon0, epsilon1, epsilon2, epsilon3 = [], [], [], []    # charge dispersion for m=0 through m=4\n",
         "x = np.linspace(1,140,101)"
-      ],
-      "metadata": {
-        "id": "dypU9-Yab0s0"
-      },
-      "execution_count": 83,
-      "outputs": []
+      ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "#Anharmonicity"
-      ],
       "metadata": {
         "id": "ga7GZViWbPcS"
-      }
+      },
+      "source": [
+        "#Anharmonicity"
+      ]
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "4T74-QzHj9J-"
+      },
       "source": [
         "We know that for the transmon qubit, having the Josephson Energy much larger than the charging energy ($E_J>>E_C$) results in a decrease in anharmonicity. The latter is critical for a functional qubit in which the energy difference between the lowest two states ($E_{01}$) is sufficiently different than the energy difference between the second and third states, ($E_{12}$). The absolute anharmonicity is defined as $\\alpha = E_{12}-E_{01}$, while the relative anharmonicity is defined as\n",
         ".\n",
         "\n",
         "We can easily make a plot of the anharmonicity as a function of $E_J/E_C$ using the Hcpb class. Let’s have the ratio of $E_J/E_C$ (which we’ll call x) vary from 0 to 80, and then we’ll create empty lists for $\\alpha$ and $\\alpha_r$\n",
         "\n"
-      ],
-      "metadata": {
-        "id": "4T74-QzHj9J-"
-      }
+      ]
     },
     {
       "cell_type": "code",
+      "execution_count": 82,
+      "metadata": {
+        "id": "-bKdEjF-WtST"
+      },
+      "outputs": [],
       "source": [
         "x = np.linspace(0,80,101)   #EJ/EC\n",
         "alpha = []\n",
         "alpha_r = []"
-      ],
-      "metadata": {
-        "id": "-bKdEjF-WtST"
-      },
-      "execution_count": 82,
-      "outputs": []
+      ]
     },
     {
       "cell_type": "code",
+      "execution_count": 84,
+      "metadata": {
+        "id": "i6fXf09UWtXc"
+      },
+      "outputs": [],
       "source": [
         "for i in x:\n",
         "    H_anharm = Hcpb(nlevels=15, Ej=i*E_c, Ec=E_c, ng=0.5)\n",
         "    alpha.append(H_anharm.anharm())\n",
         "    alpha_r.append(H_anharm.anharm()/H_anharm.fij(0,1))"
-      ],
-      "metadata": {
-        "id": "i6fXf09UWtXc"
-      },
-      "execution_count": 84,
-      "outputs": []
+      ]
     },
     {
       "cell_type": "code",
-      "source": [
-        "plt.figure(1)\n",
-        "plt.subplot(131)\n",
-        "plt.plot(x,alpha)\n",
-        "plt.xlabel(\"Ej/Ec\")\n",
-        "plt.ylabel(\"alpha\")\n",
-        "plt.subplot(133)\n",
-        "plt.plot(x,alpha_r)\n",
-        "plt.ylim(-0.2, 1.0)\n",
-        "plt.xlabel(\"Ej/Ec\")\n",
-        "plt.ylabel(\"alpha_r\")"
-      ],
+      "execution_count": 85,
       "metadata": {
         "colab": {
           "base_uri": "https://localhost:8080/",
@@ -373,76 +355,1139 @@
         "id": "iA_iKSoqbnfe",
         "outputId": "9598e78b-85f9-4545-b33f-e4860d86fcc6"
       },
-      "execution_count": 85,
       "outputs": [
         {
-          "output_type": "execute_result",
           "data": {
             "text/plain": [
               "Text(0, 0.5, 'alpha_r')"
             ]
           },
+          "execution_count": 85,
           "metadata": {},
-          "execution_count": 85
+          "output_type": "execute_result"
         },
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           },
-          "metadata": {}
+          "metadata": {},
+          "output_type": "display_data"
         }
+      ],
+      "source": [
+        "plt.figure(1)\n",
+        "plt.subplot(131)\n",
+        "plt.plot(x,alpha)\n",
+        "plt.xlabel(\"Ej/Ec\")\n",
+        "plt.ylabel(\"alpha\")\n",
+        "plt.subplot(133)\n",
+        "plt.plot(x,alpha_r)\n",
+        "plt.ylim(-0.2, 1.0)\n",
+        "plt.xlabel(\"Ej/Ec\")\n",
+        "plt.ylabel(\"alpha_r\")"
       ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "#Dephasing Time (T2)"
-      ],
       "metadata": {
         "id": "oRYWjpV7cUxt"
-      }
+      },
+      "source": [
+        "#Dephasing Time (T2)"
+      ]
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "gdEHEA0wcYPL"
+      },
       "source": [
         "We can estimate the qubit dephasing time (T2) due to charge noise by the following expression: $T_2 = \\frac{\\hbar}{A\\pi |\\epsilon_1|}$ where $A$ is on the order of $1E-4$\n",
         " according to Phys. Rev. A 76, 042319 (2007). Since this is essentially just the inverse of the charge dispersion for $\\epsilon_1$, we can easily calculate T2 as a function of $E_J/E_C$ with the following:"
-      ],
-      "metadata": {
-        "id": "gdEHEA0wcYPL"
-      }
+      ]
     },
     {
       "cell_type": "code",
-      "source": [
-        "x = np.linspace(0,80,101)     # ratio of Ej/Ec, varying from 0 to 80\n",
-        "T2 = []                       # empty list for T2"
-      ],
+      "execution_count": 86,
       "metadata": {
         "id": "o1k7C88pbnh6"
       },
-      "execution_count": 86,
-      "outputs": []
+      "outputs": [],
+      "source": [
+        "x = np.linspace(0,80,101)     # ratio of Ej/Ec, varying from 0 to 80\n",
+        "T2 = []                       # empty list for T2"
+      ]
     },
     {
       "cell_type": "code",
-      "source": [
-        "E_c = 1000.0\n",
-        "for i in x:\n",
-        "    H_half = Hcpb(nlevels=15, Ej=i*E_c, Ec=E_c, ng=0.5)\n",
-        "    H_zero = Hcpb(nlevels=15, Ej=i*E_c, Ec=E_c, ng=0.0)\n",
-        "    eps = abs(H_half.evalue_k(1) - H_zero.evalue_k(1))\n",
-        "    T2.append(1.0/(2.0*(1E-4)*(1E6)*eps) )\n",
-        "\n",
-        "plt.plot(x, T2)\n",
-        "plt.yscale(\"log\")\n",
-        "plt.xlabel(\"Ej/Ec\")\n",
-        "plt.ylabel(\"T2 (sec)\")"
-      ],
+      "execution_count": 87,
       "metadata": {
         "colab": {
           "base_uri": "https://localhost:8080/",
@@ -451,44 +1496,1340 @@
         "id": "j52M1m_-bnkp",
         "outputId": "d5099a5a-32ff-462d-b418-cc922cc4c242"
       },
-      "execution_count": 87,
       "outputs": [
         {
-          "output_type": "execute_result",
           "data": {
             "text/plain": [
               "Text(0, 0.5, 'T2 (sec)')"
             ]
           },
+          "execution_count": 87,
           "metadata": {},
-          "execution_count": 87
+          "output_type": "execute_result"
         },
         {
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+            ]
           },
-          "metadata": {}
+          "metadata": {},
+          "output_type": "display_data"
         }
+      ],
+      "source": [
+        "E_c = 1000.0\n",
+        "for i in x:\n",
+        "    H_half = Hcpb(nlevels=15, Ej=i*E_c, Ec=E_c, ng=0.5)\n",
+        "    H_zero = Hcpb(nlevels=15, Ej=i*E_c, Ec=E_c, ng=0.0)\n",
+        "    eps = abs(H_half.evalue_k(1) - H_zero.evalue_k(1))\n",
+        "    T2.append(1.0/(2.0*(1E-4)*(1E6)*eps) )\n",
+        "\n",
+        "plt.plot(x, T2)\n",
+        "plt.yscale(\"log\")\n",
+        "plt.xlabel(\"Ej/Ec\")\n",
+        "plt.ylabel(\"T2 (sec)\")"
       ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "Operator matrix elements visualized with text representation of each entry."
-      ],
       "metadata": {
         "id": "FF2_ML3uh19q"
-      }
+      },
+      "source": [
+        "Operator matrix elements visualized with text representation of each entry."
+      ]
     },
     {
       "cell_type": "code",
-      "source": [
-        "tmon.plot_matrixelements('cos_phi_operator', evals_count=10, show3d=False, show_numbers=True);\n"
-      ],
+      "execution_count": 117,
       "metadata": {
         "colab": {
           "base_uri": "https://localhost:8080/",
@@ -497,46 +2838,1988 @@
         "id": "_txRAJZlQMCL",
         "outputId": "6a7278d1-9556-4f84-9333-59bdc9846213"
       },
-      "execution_count": 117,
       "outputs": [
         {
-          "output_type": "display_data",
           "data": {
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+            ]
           },
-          "metadata": {}
+          "metadata": {},
+          "output_type": "display_data"
         }
+      ],
+      "source": [
+        "tmon.plot_matrixelements('cos_phi_operator', evals_count=10, show3d=False, show_numbers=True);\n"
       ]
     },
     {
       "cell_type": "code",
-      "source": [],
+      "execution_count": null,
       "metadata": {
         "id": "KolbsH0XigV1"
       },
-      "execution_count": null,
-      "outputs": []
+      "outputs": [],
+      "source": []
     },
     {
       "cell_type": "code",
-      "source": [],
+      "execution_count": null,
       "metadata": {
         "id": "GcmFLUX0QMEl"
       },
-      "execution_count": null,
-      "outputs": []
+      "outputs": [],
+      "source": []
     },
     {
       "cell_type": "code",
-      "source": [],
+      "execution_count": 15,
       "metadata": {
         "id": "RSJolMAoHxSd"
       },
-      "execution_count": 15,
-      "outputs": []
+      "outputs": [],
+      "source": []
     }
-  ]
-}
\ No newline at end of file
+  ],
+  "metadata": {
+    "colab": {
+      "provenance": []
+    },
+    "kernelspec": {
+      "display_name": "Python 3",
+      "name": "python3"
+    },
+    "language_info": {
+      "name": "python"
+    }
+  },
+  "nbformat": 4,
+  "nbformat_minor": 0
+}