Added sample notebook with noise (#1980)

Demonstrates use of Pauli noise in Python and display histogram widget
in the notebook.

---------

Co-authored-by: Dmitry Vasilevsky <dmitryv@microsoft.com>

samples/notebooks/noise.ipynb

@@ -0,0 +1,208 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Simulating Pauli noise\n",
+    "This notebook shows how to run simulations with Pauli noise, such as bit-flip or depolarizing noise.\n",
+    "\n",
+    "First, make sure prerequisites are available. Packages `qsharp` and `qsharp_widgets` must be already installed."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import qsharp\n",
+    "import qsharp_widgets"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Simulation with noise\n",
+    "\n",
+    "Define a simple program that creates a Bell state on two qubits and measures both qubits."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {
+    "vscode": {
+     "languageId": "qsharp"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "%%qsharp\n",
+    "\n",
+    "operation BellPair() : Result[] {\n",
+    "    use q = Qubit[2];\n",
+    "    H(q[0]);\n",
+    "    CNOT(q[0], q[1]);\n",
+    "    MResetEachZ(q)\n",
+    "}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Run 20 shots without noise and display results."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "results = qsharp.run(\"BellPair()\", 20)\n",
+    "results"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Note that measurements always agree within a shot as expected. Now run 20 shots of the same program with 10% [depolarizing noise](https://en.wikipedia.org/wiki/Quantum_depolarizing_channel). Depolarizing noise is applied to each gate and each measurement."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "results = qsharp.run(\"BellPair()\", 20, noise=qsharp.DepolarizingNoise(0.1))\n",
+    "results"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Note that measurements do not always agree within the shot."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Histograms\n",
+    "\n",
+    "Define a program to prepare a cat state on five qubits and measure each qubit."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {
+    "vscode": {
+     "languageId": "qsharp"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "%%qsharp\n",
+    "\n",
+    "operation Cat5() : Result[] {\n",
+    "    use q = Qubit[5];\n",
+    "    H(q[0]);\n",
+    "    ApplyCNOTChain(q);\n",
+    "    MResetEachZ(q)\n",
+    "}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "First, run this program without noise. Roughly half of the outcomes should be $\\ket{00000}$ and another half should be $\\ket{11111}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "result = qsharp.run(\"Cat5()\", 1000)\n",
+    "qsharp_widgets.Histogram(result)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now, run the same program with bit-flip noise of 1%, 5%, 10%, 25%."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "for p in [0.01, 0.05, 0.1, 0.25]:\n",
+    "    result = qsharp.run(\"Cat5()\", 1000, noise=qsharp.BitFlipNoise(p))\n",
+    "    display(f\"Noise probability = {p}\")\n",
+    "    display(qsharp_widgets.Histogram(result))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We can see that with 1% noise, cat state can still be clearly seen, but when noise approaches 25%, the cat state is indistinguishable from noise."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Arbitrary Pauli noise\n",
+    "\n",
+    "Standard bit-flip, phase-flip, and [depolarizing](https://en.wikipedia.org/wiki/Quantum_depolarizing_channel) noise are available, but arbitrary Pauli noise is also possible. The following example runs the same Cat5 program. First it applies noise with 20% probability (bit-flip half the time and phase-flip half the time). In a second experiment it applies Pauli-Y noise with 10% probability."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "result = qsharp.run(\"Cat5()\", 1000, noise=(0.1, 0.0, 0.1))\n",
+    "display(qsharp_widgets.Histogram(result))\n",
+    "result = qsharp.run(\"Cat5()\", 1000, noise=(0.0, 0.1, 0.0))\n",
+    "display(qsharp_widgets.Histogram(result))"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}