514 lines (514 with data), 2.0 MB
{
"cells": [
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"id": "rqYMIG5_jAmf",
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"outputId": "8810bc32-aef6-4bb7-fffa-672a640987e3"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Time in seconds since beginning of run: 1695116803.0669148\n",
"Tue Sep 19 09:46:43 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": 19,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"id": "30MBcn8AjAmh",
"outputId": "8c40b105-2f40-44be-9eb2-49ef0103c239"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"[[0.86 0.54 0.36]\n",
" [0.54 0.01 0.3 ]\n",
" [0.36 0.3 0.5 ]]\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": 20,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"id": "9AlMlrsVjAmi",
"outputId": "6e4ea3df-5987-4594-f73c-92fde145838c"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"[[0.01 0.3 0.54]\n",
" [0.3 0.5 0.36]\n",
" [0.54 0.36 0.86]]\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": 21,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 487
},
"id": "Q7N1C_MKjAmi",
"outputId": "c63aab0a-b572-483c-b7c8-6713fd9baef1"
},
"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": 22,
"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": 23,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 57
},
"id": "0oMs3Tu0jAmi",
"outputId": "37f6e0dd-59b0-405c-9e6b-14bfe45fac2c"
},
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": [
"<Figure size 58400x2600 with 1 Axes>"
],
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},
"metadata": {}
}
],
"source": [
"n_qubits = 25\n",
"n_layers = 2\n",
"\n",
"dev = qml.device(\"lightning.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": null,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"id": "ZCFN2RkSjAmj",
"outputId": "ccc73008-2180-44e1-ab48-4bd549b8e0ed"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Model output for A: -0.00016385117549580974\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": 25,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"id": "KjwEdmGXjAmj",
"outputId": "9a2b760e-6e12-47f4-f448-52fa566a5cca"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Model output for permutation of A: 0.13585734143229955\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": 26,
"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": 27,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 0
},
"id": "tpzF6L3TjAmj",
"outputId": "616b9fd9-1653-4b8c-fe19-3c7bfddc3c5a"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Model output for permutation of A, and with permuted observable: -0.1251430323586184\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": "e84b11b7-2827-4ad3-831e-01e785c80bcf"
},
"execution_count": 28,
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Time in seconds since end of run: 1695116943.2209816\n",
"Tue Sep 19 09:49:03 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
}