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{
"cells": [
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"id": "rqYMIG5_jAmf",
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"outputId": "b707028f-ed28-4f8c-b2b5-7fd158f4a437"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Time in seconds since beginning of run: 1695143320.5080006\n",
"Tue Sep 19 17:08:40 2023\n"
]
}
],
"source": [
"# This cell is added by sphinx-gallery\n",
"# It can be customized to whatever you like\n",
"%matplotlib inline\n",
"# !pip install pennylane custatevec-cu11 pennylane-lightning-gpu\n",
"\n",
"import time\n",
"seconds = time.time()\n",
"print(\"Time in seconds since beginning of run:\", seconds)\n",
"local_time = time.ctime(seconds)\n",
"print(local_time)"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"id": "30MBcn8AjAmh",
"outputId": "e5632ea8-3c38-43f6-f9e6-f882f7e6bd8d"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"[[0.69 0.28 0.65]\n",
" [0.28 0.82 0.26]\n",
" [0.65 0.26 0.88]]\n"
]
}
],
"source": [
"import numpy as np\n",
"import networkx as nx\n",
"import matplotlib.pyplot as plt\n",
"\n",
"\n",
"def create_data_point(n):\n",
" \"\"\"\n",
" Returns a random undirected adjacency matrix of dimension (n,n).\n",
" The diagonal elements are interpreted as node attributes.\n",
" \"\"\"\n",
" mat = np.random.rand(n, n)\n",
" A = (mat + np.transpose(mat))/2\n",
" return np.round(A, decimals=2)\n",
"\n",
"A = create_data_point(3)\n",
"print(A)"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "fZNWu6j_jAmh"
},
"source": [
"Let\\'s also write a function to generate permuted versions of this\n",
"adjacency matrix.\n"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"id": "9AlMlrsVjAmi",
"outputId": "8610e272-241a-4ea2-a55b-53bfb7b59747"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"[[0.82 0.26 0.28]\n",
" [0.26 0.88 0.65]\n",
" [0.28 0.65 0.69]]\n"
]
}
],
"source": [
"def permute(A, permutation):\n",
" \"\"\"\n",
" Returns a copy of A with rows and columns swapped according to permutation.\n",
" For example, the permutation [1, 2, 0] swaps 0->1, 1->2, 2->0.\n",
" \"\"\"\n",
"\n",
" P = np.zeros((len(A), len(A)))\n",
" for i,j in enumerate(permutation):\n",
" P[i,j] = 1\n",
"\n",
" return P @ A @ np.transpose(P)\n",
"\n",
"A_perm = permute(A, [1, 2, 0])\n",
"print(A_perm)"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "Js9CqRTZjAmi"
},
"source": [
"If we create [networkx]{.title-ref} graphs from both adjacency matrices\n",
"and plot them, we see that they are identical as claimed.\n"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 487
},
"id": "Q7N1C_MKjAmi",
"outputId": "614e7de6-25c6-45a1-95d8-b5183baafc34"
},
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": [
"<Figure size 640x480 with 2 Axes>"
],
"image/png": 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\n"
},
"metadata": {}
}
],
"source": [
"fig, (ax1, ax2) = plt.subplots(1, 2)\n",
"\n",
"# interpret diagonal of matrix as node attributes\n",
"node_labels = {n: A[n,n] for n in range(len(A))}\n",
"np.fill_diagonal(A, np.zeros(len(A)))\n",
"\n",
"G1 = nx.Graph(A)\n",
"pos1=nx.spring_layout(G1)\n",
"nx.draw(G1, pos1, labels=node_labels, ax=ax1, node_size = 800, node_color = \"#ACE3FF\")\n",
"edge_labels = nx.get_edge_attributes(G1,'weight')\n",
"nx.draw_networkx_edge_labels(G1,pos1,edge_labels=edge_labels, ax=ax1)\n",
"\n",
"# interpret diagonal of permuted matrix as node attributes\n",
"node_labels = {n: A_perm[n,n] for n in range(len(A_perm))}\n",
"np.fill_diagonal(A_perm, np.zeros(len(A)))\n",
"\n",
"G2 = nx.Graph(A_perm)\n",
"pos2=nx.spring_layout(G2)\n",
"nx.draw(G2, pos2, labels=node_labels, ax=ax2, node_size = 800, node_color = \"#ACE3FF\")\n",
"edge_labels = nx.get_edge_attributes(G2,'weight')\n",
"nx.draw_networkx_edge_labels(G2,pos2,edge_labels=edge_labels, ax=ax2)\n",
"\n",
"ax1.set_xlim([1.2*x for x in ax1.get_xlim()])\n",
"ax2.set_xlim([1.2*x for x in ax2.get_xlim()])\n",
"plt.tight_layout()\n",
"plt.show()"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"id": "eUhWZ9_3jAmi"
},
"outputs": [],
"source": [
"import pennylane as qml\n",
"\n",
"def perm_equivariant_embedding(A, betas, gammas):\n",
" \"\"\"\n",
" Ansatz to embedd a graph with node and edge weights into a quantum state.\n",
"\n",
" The adjacency matrix A contains the edge weights on the off-diagonal,\n",
" as well as the node attributes on the diagonal.\n",
"\n",
" The embedding contains trainable weights 'betas' and 'gammas'.\n",
" \"\"\"\n",
" n_nodes = len(A)\n",
" n_layers = len(betas) # infer the number of layers from the parameters\n",
"\n",
" # initialise in the plus state\n",
" for i in range(n_nodes):\n",
" qml.Hadamard(i)\n",
"\n",
" for l in range(n_layers):\n",
"\n",
" for i in range(n_nodes):\n",
" for j in range(i):\n",
" \t# factor of 2 due to definition of gate\n",
" qml.IsingZZ(2*gammas[l]*A[i,j], wires=[i,j])\n",
"\n",
" for i in range(n_nodes):\n",
" qml.RX(A[i,i]*betas[l], wires=i)"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "glOVsqRvjAmi"
},
"source": [
"We can use this ansatz in a circuit.\n"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 57
},
"id": "0oMs3Tu0jAmi",
"outputId": "37e1342f-fe50-4aa8-e6c0-dbc7a3a4b5dc"
},
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": [
"<Figure size 58400x2600 with 1 Axes>"
],
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2BUrM5749BnlZAAAAAElFTkSuQmCC\n"
},
"metadata": {}
}
],
"source": [
"n_qubits = 25\n",
"n_layers = 2\n",
"\n",
"dev = qml.device(\"default.qubit\", wires=n_qubits)\n",
"\n",
"@qml.qnode(dev)\n",
"def eqc(adjacency_matrix, observable, trainable_betas, trainable_gammas):\n",
" \"\"\"Circuit that uses the permutation equivariant embedding\"\"\"\n",
"\n",
" perm_equivariant_embedding(adjacency_matrix, trainable_betas, trainable_gammas)\n",
" return qml.expval(observable)\n",
"\n",
"\n",
"A = create_data_point(n_qubits)\n",
"betas = np.random.rand(n_layers)\n",
"gammas = np.random.rand(n_layers)\n",
"observable = qml.PauliX(0) @ qml.PauliX(1) @ qml.PauliX(3)\n",
"\n",
"qml.draw_mpl(eqc, decimals=2)(A, observable, betas, gammas)\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "TIHRQVnDjAmi"
},
"source": [
"Validating the equivariance\n",
"===========================\n",
"\n",
"Let\\'s now check if the circuit is really equivariant!\n",
"\n",
"This is the expectation value we get using the original adjacency matrix\n",
"as an input:\n"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"id": "ZCFN2RkSjAmj",
"outputId": "63eb3bb2-3fd8-4129-e245-ef4b7a0a6f92"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Model output for A: 0.09092655485833454\n"
]
}
],
"source": [
"result_A = eqc(A, observable, betas, gammas)\n",
"print(\"Model output for A:\", result_A)"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "ncmkAwH-jAmj"
},
"source": [
"If we permute the adjacency matrix, this is what we get:\n"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"id": "KjwEdmGXjAmj",
"outputId": "faa6e1fe-8652-40cf-f3c2-2d4cb419fc29"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Model output for permutation of A: 0.6991826852663847\n"
]
}
],
"source": [
"perm = [2, 3, 0, 1, 4]\n",
"A_perm = permute(A, perm)\n",
"result_Aperm = eqc(A_perm, observable, betas, gammas)\n",
"print(\"Model output for permutation of A: \", result_Aperm)"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "ZOIqQmysjAmj"
},
"source": [
"Why are the two values different? Well, we constructed an *equivariant*\n",
"ansatz, not an *invariant* one! Remember, an *invariant* ansatz means\n",
"that embedding a permutation of the adjacency matrix leads to the same\n",
"state as an embedding of the original matrix. An *equivariant* ansatz\n",
"embeds the permuted adjacency matrix into a state where the qubits are\n",
"permuted as well.\n",
"\n",
"As a result, the final state before measurement is only the same if we\n",
"permute the qubits in the same manner that we permute the input\n",
"adjacency matrix. We could insert a permutation operator\n",
"`qml.Permute(perm)` to achieve this, or we simply permute the wires of\n",
"the observables!\n"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {
"id": "MgYVIMP6jAmj"
},
"outputs": [],
"source": [
"observable_perm = qml.PauliX(perm[0]) @ qml.PauliX(perm[1]) @ qml.PauliX(perm[3])"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "pb21hBrLjAmj"
},
"source": [
"Now everything should work out!\n"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"id": "tpzF6L3TjAmj",
"outputId": "7b928ecb-b9a0-4c4b-d71b-cda4f9c05b34"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Model output for permutation of A, and with permuted observable: 0.693935787549807\n"
]
}
],
"source": [
"result_Aperm = eqc(A_perm, observable_perm, betas, gammas)\n",
"print(\"Model output for permutation of A, and with permuted observable: \", result_Aperm)"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "0wblJk9mjAmj"
},
"source": [
"Et voilà!\n",
"\n",
"Conclusion\n",
"==========\n",
"\n",
"Equivariant graph embeddings can be combined with other equivariant\n",
"parts of a quantum machine learning pipeline (like measurements and the\n",
"cost function). [Skolik et al.\n",
"(2022)](https://arxiv.org/pdf/2205.06109.pdf), for example, use such a\n",
"pipeline as part of a reinforcement learning scheme that finds heuristic\n",
"solutions for the traveling salesman problem. Their simulations compare\n",
"a fully equivariant model to circuits that break permutation\n",
"equivariance and show that it performs better, confirming that if we\n",
"know about structure in our data, we should try to use this knowledge in\n",
"machine learning.\n",
"\n",
"References\n",
"==========\n",
"\n",
"1. Andrea Skolik, Michele Cattelan, Sheir Yarkoni,Thomas Baeck and\n",
" Vedran Dunjko (2022). Equivariant quantum circuits for learning on\n",
" weighted graphs.\n",
" [arXiv:2205.06109](https://arxiv.org/abs/2205.06109)\n",
"2. Quynh T. Nguyen, Louis Schatzki, Paolo Braccia, Michael Ragone,\n",
" Patrick J. Coles, Frédéric Sauvage, Martín Larocca and Marco Cerezo\n",
" (2022). Theory for Equivariant Quantum Neural Networks.\n",
" [arXiv:2210.08566](https://arxiv.org/abs/2210.08566)\n",
"\n",
"About the author\n",
"================\n"
]
},
{
"cell_type": "code",
"source": [
"seconds = time.time()\n",
"print(\"Time in seconds since end of run:\", seconds)\n",
"local_time = time.ctime(seconds)\n",
"print(local_time)"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"id": "HFTIJBPqph2f",
"outputId": "f32329c0-fe49-4ef3-9961-56e1d7b84492"
},
"execution_count": 34,
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Time in seconds since end of run: 1695144085.5710328\n",
"Tue Sep 19 17:21:25 2023\n"
]
}
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"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": "3.9.17"
},
"colab": {
"provenance": [],
"gpuType": "V100"
},
"accelerator": "GPU"
},
"nbformat": 4,
"nbformat_minor": 0
}