docs/source/tutorials/operator_spreading.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "b06fd49d",
   "metadata": {},
   "source": [
    "# Operator spreading"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11f2f811-3e10-433d-a677-666dc0b17071",
   "metadata": {},
   "source": [
    "## Overview"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc6be64c-4851-4476-90be-4766cbe73600",
   "metadata": {},
   "source": [
    "In this tutorial, we will introduce the operator spreading that serves as a diagnostic of the chaotic dynamics and information scrambling. We will examine operator spreading as a function of the circuit depth $L$ that can be regarded as the time $t$ in discrete quantum systems. The operator spreading coefficient considered here reads:\n",
    "$$\n",
    "C_{y}(j,t)=\\frac{1}{2} \\text{Tr}([O(t), \\sigma^{(j)}_{y}]^{\\dagger}[O(t),\\sigma^{(j)}_{y}]),\n",
    "$$\n",
    "where $\\sigma_{y}^{(j)}$ is the Pauli-y matrix at the $j$-th qubit. $O(0)$ is the Pauli-x matrix located at $N/2$, i.e. $\\sigma_{x}^{(N/2)}$. And $O(t)=U(t)^{\\dagger}O(0)U(t)$, where $U(t)$ is the unitary matrix of the circuit with $t$ layers."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d734f3ee-a0fb-4df9-adf5-a691cd4a9041",
   "metadata": {},
   "source": [
    "A physical picture for the operator spreading is that if the Heisenberg operator growth $O(t)$ does not reach the site $j$, $[O(t), \\sigma_{y}^{(j)}]=0$, while the equality will break down when sites $N/2, j$ become correlated inside the causal region."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d1246b8b-a25c-49f1-83a3-bb2fa80266e4",
   "metadata": {},
   "source": [
    "## Setup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "f2e763c0-8fc9-420c-947d-4a99e9d73f47",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import tensorflow as tf\n",
    "import tensorcircuit as tc\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "tc.set_backend(\"tensorflow\")\n",
    "tc.set_dtype(\"complex128\")\n",
    "dtype = np.complex128"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "73c63294-e270-4578-83c1-a6fe4904d266",
   "metadata": {},
   "source": [
    "### Parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "177a6048-02b0-4e10-b89e-fac0af8f764d",
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 6  # The number of qubits\n",
    "L = 2  # The number of circuit layers\n",
    "num_trial = 1  # The number of random circuit instances"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e658f4a0-78c6-4ab6-82e2-77a0da41620d",
   "metadata": {},
   "source": [
    "### The Unitary Matrix $U(t)$ of the Quantum Circuit"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "46dd164c-b3ce-414d-899f-f040677de65f",
   "metadata": {},
   "source": [
    "The circuit architecture is shown below, each block includes two Pauli rotation gates along the y-axis ($R_{y}$) followed by the $CZ$ gate. We can get the unitary matrix $U(t)$ of the circuit by setting the input state as an identity matrix."
   ]
  },
  {
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    }
   },
   "cell_type": "markdown",
   "id": "9dd090d9-304f-4f7f-b7c6-cb1ab62a79d2",
   "metadata": {},
   "source": [
    "![total.jpeg](attachment:5dc85aba-f14e-42f3-b7ec-85870ef08c32.jpeg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "7faad4ff-fb3e-40f0-b340-9c93fcfadfaf",
   "metadata": {},
   "outputs": [],
   "source": [
    "@tf.function()\n",
    "def get_unitary_of_circuit(v):\n",
    "    layers = len(v)\n",
    "    c = tc.Circuit(\n",
    "        N, inputs=np.eye(2**N, dtype=dtype)\n",
    "    )  # when we choose inputs=np.eye(2**N,...), the output wavefunction\n",
    "    # is the unitary matrix U(t) of the quantum circuit\n",
    "\n",
    "    # Ry+Cz\n",
    "    for layer in range(layers):\n",
    "        if layer % 2 == 0:\n",
    "            for i in range(0, N, 2):\n",
    "                c.ry(i, theta=v[layer, i])\n",
    "                c.ry(i + 1, theta=v[layer, i + 1])\n",
    "                c.cz(i, i + 1)\n",
    "        else:\n",
    "            for i in range(1, N, 2):\n",
    "                c.ry(i, theta=v[layer, i])\n",
    "                c.ry((i + 1) % N, theta=v[layer, (i + 1) % N])\n",
    "                c.cz(i, (i + 1) % N)\n",
    "    U = c.wavefunction()  # the output of wavefunction is a vector with dim=2^(2N)\n",
    "    U = tf.reshape(U, [2**N, 2**N])  # reshape vector to matrix\n",
    "    return U"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "956d3d13-b694-4c5a-8742-72faee8d8907",
   "metadata": {},
   "source": [
    "### Operator at $t=0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e99718b4-26de-4f74-8694-163513cf2249",
   "metadata": {},
   "outputs": [],
   "source": [
    "# j means the operator locates at the j-th qubit\n",
    "# a means the type of operator, a=1 (2,3) is the Pauli-x (y,z) matrix\n",
    "def get_operator(j, a):\n",
    "    I = tc.gates._i_matrix\n",
    "    if a == 1:\n",
    "        σ = tc.gates._x_matrix  # Pauli-x matrix\n",
    "    elif a == 2:\n",
    "        σ = tc.gates._y_matrix  # Pauli-y matrix\n",
    "    elif a == 3:\n",
    "        σ = tc.gates._z_matrix  # Pauli-z matrix\n",
    "    h = []\n",
    "    for i in range(N):\n",
    "        if i == j:\n",
    "            h.append(σ)\n",
    "        else:\n",
    "            h.append(I)\n",
    "\n",
    "    # Pauli-a matrix locates at j-th qubit\n",
    "    operator = h[0]\n",
    "    for i in range(N - 1):\n",
    "        operator = np.kron(operator, h[i + 1])\n",
    "    return tf.cast(operator, dtype=dtype)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9e4ec8f8-ae7e-44f9-8936-dc48c8729b44",
   "metadata": {},
   "source": [
    "### Operator Spreading"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4f75c3e8-8c00-4796-bc85-5b89891eaba5",
   "metadata": {},
   "source": [
    "The operator spreading $C_{y}(j,t)=\\frac{1}{2} \\text{Tr}([O(t), \\sigma^{(j)}_{y}]^{\\dagger}[O(t),\\sigma^{(j)}_{y}])$ can be written as:\n",
    "$$\n",
    "C_{y}(j,t) = 1- \\frac{\\rm{Re}(\\rm{Tr}(\\sigma^{(j)}_{y} O(t)^{\\dagger} \\sigma^{(j)}_{y} O(t)))}{2^N}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "b5d61f0a-09cc-4bce-8130-f6457bb29429",
   "metadata": {},
   "outputs": [],
   "source": [
    "@tf.function()\n",
    "def get_spreading(v, σ_y):\n",
    "    U = get_unitary_of_circuit(v)\n",
    "    U_dagger = tf.transpose(U, conjugate=True)\n",
    "    σ_x = get_operator(int(N / 2), 1)  # O(0)\n",
    "    σ_x_t = U_dagger @ σ_x @ U  # O(t)\n",
    "    σ_x_t_dagger = tf.transpose(σ_x_t, conjugate=True)\n",
    "    C = tf.linalg.trace(σ_y @ σ_x_t_dagger @ σ_y @ σ_x_t)\n",
    "    C = 1 - tf.math.real(C) / (2**N)\n",
    "    return C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "090a9101-58a4-4111-9fde-8f0ad557c26d",
   "metadata": {},
   "outputs": [],
   "source": [
    "def main(layers=1):\n",
    "    # use vmap to get operator spreading coefficients of different random circuit instances\n",
    "    # num_trial: the number of random circuit instances\n",
    "    # layers: the number of circuit layers\n",
    "    get_spreading_vmap = tc.backend.vmap(get_spreading, vectorized_argnums=0)\n",
    "\n",
    "    C = get_spreading_vmap(\n",
    "        tf.random.uniform(\n",
    "            [num_trial, layers, N], minval=0.0, maxval=2 * np.pi, dtype=tf.float64\n",
    "        ),\n",
    "        [[get_operator(j, 2) for j in range(N)] for _ in range(num_trial)],\n",
    "    )\n",
    "\n",
    "    return tf.reduce_mean(C, 0)[0]  # averaged over different random circuit instances"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "b9a97ffe-560b-4730-b826-038d9a57722b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Operator spreading')"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "C = main(L)  # operator spreading averaged over different random circuit instances\n",
    "\n",
    "plt.bar([i for i in range(N)], C)\n",
    "plt.xticks(fontsize=16)\n",
    "plt.yticks(fontsize=16)\n",
    "plt.ylabel(\"Operator spreading\", fontsize=16)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c094ab16-0329-4024-9311-216391a305cf",
   "metadata": {},
   "source": [
    "### Results"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "575be3d8-6c29-45ab-b991-34ce61a293e7",
   "metadata": {},
   "source": [
    "The operator spreading for a system of $n = 6$ qubits as a function of $1 ≤ j ≤ n = 6$ for a different number of\n",
    "layers $L$, averaged over 20 random circuit instances. When $L=2$, the Heisenberg operator growth of $O(t)$ does not reach the site $0,1$, $C=0$; when $L=10$, the equality will break down."
   ]
  },
  {
   "attachments": {
    "22584b62-a923-42ee-96a6-9030cbdba506.jpeg": {
     "image/jpeg": 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    }
   },
   "cell_type": "markdown",
   "id": "9a3080e7-a403-42dd-980e-5b13909e00da",
   "metadata": {},
   "source": [
    "![spreading.jpeg](attachment:22584b62-a923-42ee-96a6-9030cbdba506.jpeg)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ab329ec-ecd4-4391-927e-e62ae7935f07",
   "metadata": {},
   "source": [
    "## Reference \n",
    "\n",
    "[1] https://arxiv.org/pdf/2201.01452.pdf."
   ]
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