{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "f44a230e-bd91-40a1-b63c-32905ff369b6",
   "metadata": {},
   "source": [
    "# Barren Plateaus"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c7e551d9-a7dc-4fe7-84c5-294b1ae267d6",
   "metadata": {},
   "source": [
    "## Overview"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "34f45c29-9f8f-4723-a96e-2ce283926c75",
   "metadata": {},
   "source": [
    "Barren plateaus are the greatest difficulties in the gradient-based optimization for a large family of random parameterized quantum circuits (PQC). The gradients vanish almost everywhere. In this example, we will show barren plateaus in quantum neural networks (QNNs)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e99265d2-3fac-4fb3-998c-463eddf18f65",
   "metadata": {},
   "source": [
    "## Setup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "c67b2f15-03be-4be2-85c6-6c3fa8ce5a35",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import tensorflow as tf\n",
    "import tensorcircuit as tc\n",
    "\n",
    "tc.set_backend(\"tensorflow\")\n",
    "tc.set_dtype(\"complex64\")\n",
    "\n",
    "Rx = tc.gates.rx\n",
    "Ry = tc.gates.ry\n",
    "Rz = tc.gates.rz"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e9616f63-f4dc-4272-8dc0-fc676a059651",
   "metadata": {},
   "source": [
    "## Parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "bbe10483-6182-47ae-96ee-ee5cab773278",
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 4  # The number of qubits\n",
    "nlayers = 1  # The number of circuit layers\n",
    "ncircuits = 3  # The number of circuits with different initial parameters\n",
    "ntrials = 2  # The number of random circuits with different structures"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fa8f0080-416c-48fa-9273-cda153c035d0",
   "metadata": {},
   "source": [
    "## Generating QNN"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "9652bb49-9d1c-453c-bd8f-66afd88d2b11",
   "metadata": {},
   "outputs": [],
   "source": [
    "def op_expectation(params, seed, n, nlayers):\n",
    "    paramsc = tc.backend.cast(params, dtype=\"float32\")  # parameters of gates\n",
    "    seedc = tc.backend.cast(seed, dtype=\"float32\")  # parameters of circuit structure\n",
    "\n",
    "    c = tc.Circuit(n)\n",
    "    for i in range(n):\n",
    "        c.ry(i, theta=np.pi / 4)\n",
    "    for l in range(nlayers):\n",
    "        for i in range(n):\n",
    "            # choose one gate from Rx, Ry, and Rz gates with equal prob=1/3; status is the seed.\n",
    "            c.unitary_kraus(\n",
    "                [Rx(paramsc[i, l]), Ry(paramsc[i, l]), Rz(paramsc[i, l])],\n",
    "                i,\n",
    "                prob=[1 / 3, 1 / 3, 1 / 3],\n",
    "                status=seedc[i, l],\n",
    "            )\n",
    "        for i in range(n - 1):\n",
    "            c.cz(i, i + 1)\n",
    "\n",
    "    return tc.backend.real(\n",
    "        c.expectation((tc.gates.z(), [0]), (tc.gates.z(), [1]))\n",
    "    )  # expectations of <Z_0Z_1>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "d8c49368-a0bd-4021-964d-942ec922bc37",
   "metadata": {},
   "outputs": [],
   "source": [
    "# use vmap and vvag to get the expectations of ZZ observable and gradients of different random circuit instances\n",
    "op_expectation_vmap_vvag = tc.backend.jit(\n",
    "    tc.backend.vmap(\n",
    "        tc.backend.vvag(op_expectation, argnums=0, vectorized_argnums=0),\n",
    "        vectorized_argnums=1,\n",
    "    )\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e6e4d667-e6d5-41f2-9ac1-39e151889bce",
   "metadata": {},
   "source": [
    "## Batch Variance Computation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c61124e8-0769-4118-a208-0a8984aa8e31",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The variance of the gradients is: 0.19805922\n"
     ]
    }
   ],
   "source": [
    "seed = tc.array_to_tensor(\n",
    "    np.random.uniform(low=0.0, high=1.0, size=[ntrials, n, nlayers]), dtype=\"float32\"\n",
    ")\n",
    "params = tc.array_to_tensor(\n",
    "    np.random.uniform(low=0.0, high=2 * np.pi, size=[ncircuits, n, nlayers]),\n",
    "    dtype=\"float32\",\n",
    ")\n",
    "\n",
    "e, grad = op_expectation_vmap_vvag(\n",
    "    params, seed, n, nlayers\n",
    ")  # the expectations of ZZ observable and gradients of different random circuits\n",
    "\n",
    "grad_var = tf.math.reduce_std(tf.math.reduce_std(grad, axis=0), axis=0)[\n",
    "    0, 0\n",
    "]  # the gradient variance of the first parameter\n",
    "print(\"The variance of the gradients is:\", grad_var.numpy())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1a47801a-0ff8-422e-807c-2382e06971ce",
   "metadata": {},
   "source": [
    "## Results"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78a5f073-aad1-40d6-ab24-68b4a7099f2b",
   "metadata": {},
   "source": [
    "The gradient variances in QNNs ($nlayers=50$, $ntrials=20$, $ncircuits=20$). The landscape become exponentially barren with increasing qubit number. "
   ]
  },
  {
   "attachments": {
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   },
   "cell_type": "markdown",
   "id": "7d1c8f77-42aa-4d5f-8862-c6d0e8dd0efa",
   "metadata": {},
   "source": [
    "![barren.jpg](attachment:028e2e26-1f06-452d-93f9-62f73c77e0f8.jpg)"
   ]
  }
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   "display_name": "Python 3 (ipykernel)",
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   "name": "python3"
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}