{ "cells": [ { "cell_type": "markdown", "id": "07c02b16", "metadata": {}, "source": [ "# The Quantum Approximate Optimization Algorithm\n", "\n", "\n", "In this notebook we are going to see a simple application of the QAOA to a MAXCUT problem on a quantum computer of 5 qubits using Qiskit." ] }, { "cell_type": "markdown", "id": "44e43434", "metadata": {}, "source": [ "## The MAXCUT problem\n", "\n", "We are going to apply the algorithm to a MAXCUT problem, but what does it mean? Suppose to have a graph $G= (V,E)$ and you want to color the vertices of this graph using two colors in such a way that the maximum possible number of edges has its two vertices of different colors.\n", "\n", "We can formally define the problem as an optimization problem where the function to optimize is \n", "\n", "$$\n", "C(\\vec{x}) = \\sum_{(j,k) \\in E} x_{j}(1-x_{k})\n", "$$\n", "\n", "where $\\vec{x} \\in \\{ 0,1\\}^{|V|}$ is a bit string with its length equal to the number of vertices of the graph. To further complicate the problem, you can consider a weighted MAXCUT problem, by including a matrix of weights $w_{j,k}$ in the cost function. In our case we will continue with the standard MAXCUT problem. \n", "\n", "This problem is known to be a NP-hard problem, that means that no polynomial-time algorithm for MAXCUT in general graphs is known. So in a polynomial time our hope will be restricted to an approximate solution.\n", "\n", "#### Relationship with physics?\n", "\n", "In statistical physics and disordered systems, the MAXCUT problem is equivalent to minimizing the Hamiltonian of a spin glass model, most simply the Ising model. For the Ising model on a graph G and only nearest-neighbor interactions, the Hamiltonian is\n", "\n", "$$\n", "H(s) = - \\sum_{(j,k) \\in E} J_{j,k} s_j s_k\n", "$$\n", "\n", "where each vertex $j$ of the graph is a spin site that can take a spin value $s_j = \\pm 1$" ] }, { "cell_type": "code", "execution_count": 1, "id": "f1de61aa", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt \n", "\n", "\n", "# We import the tools to handle general Graphs\n", "import networkx as nx" ] }, { "cell_type": "markdown", "id": "fdfdb083", "metadata": {}, "source": [ "The MAXCUT problem on which we will focus is the following:" ] }, { "cell_type": "code", "execution_count": 2, "id": "6359b70f", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Generating the butterfly graph with 5 nodes \n", "n = 5\n", "V = np.arange(0,n,1)\n", "E =[(0,1,1.0),(0,2,1.0),(1,2,1.0),(3,2,1.0),(3,4,1.0),(4,2,1.0)] \n", "\n", "G = nx.Graph()\n", "G.add_nodes_from(V)\n", "G.add_weighted_edges_from(E)\n", "\n", "# Generate plot of the Graph\n", "\n", "colors = ['dodgerblue' for node in G.nodes()]\n", "default_axes = plt.axes(frameon=False)\n", "pos = nx.spring_layout(G)\n", "\n", "nx.draw_networkx(G, node_color=colors, node_size=300, alpha=1, ax=default_axes, pos=pos)" ] }, { "cell_type": "markdown", "id": "6717361c", "metadata": {}, "source": [ "## Map the problem to a quantum computer\n", "\n", "As we saw thorughout the course, one of the first steps in quantum computing is to map the problem we want to solve into something a quantum device is able to handle. In this case the mapping is very simple: we can map the cost function into a Pauli Hamiltonian of the form \n", "\n", "$$\n", "H = \\sum_{(j,k)} \\frac{1}{2} ( 1-Z_{j}Z_{k}) \n", "$$\n", "\n", "where $Z_{j}Z_{k}$ is an operator that acts as an identity on all the qubits different from $j,k$ and as a Pauli-$Z$ on them. With this Hamiltonian, the two 'colors' of our qubits will be the two states $|0 \\rangle $ and $|1\\rangle$. At the end of the algorithm we are going to measure the qubits in the $Z$- basis, the bit string produced as a results will represent the solution suggested by the algorithm." ] }, { "cell_type": "code", "execution_count": 5, "id": "c912e072", "metadata": {}, "outputs": [], "source": [ "# importing Qiskit\n", "from qiskit_aer import Aer\n", "from qiskit import QuantumRegister, ClassicalRegister, QuantumCircuit" ] }, { "cell_type": "markdown", "id": "9c86da54", "metadata": {}, "source": [ "We want to use our quantum computer to produce a state that measured returns a good approximation of the solution of the MAXCUT problem. In order to do this, we make an ansatz on the structure of the circuit: \n", "\n", "$$ |\\psi_p(\\vec{\\gamma},\\vec{\\beta})\\rangle = e^{ -i\\beta_p B } e^{ -i\\gamma_p H } \\ldots e^{ -i\\beta_1 B } e^{ -i\\gamma_1 H } |+\\rangle_{|V|} $$\n", "\n", "where $H$ is the Hamiltonian indicated abve and $B= \\sum_{j \\in V} X_j $, while the vectors of angles $\\vec{\\gamma},\\vec{\\beta}$ have to be optimized in order to maximize the expectation value \n", "\n", "$$\n", "F_p(\\vec{\\gamma},\\vec{\\beta}) = \\langle \\psi_p(\\vec{\\gamma},\\vec{\\beta})|H|\\psi_p(\\vec{\\gamma},\\vec{\\beta})\\rangle \n", "$$\n", "\n", "Note that the ansatz is not fixed, the higher value of $p$ we consider, the deeper will be the ansatz. In the following, we are going to consider $p=1$. With this choice of $p$, we have a circuit that first prepare all the qubits in the $ |+\\rangle$ state by acting with an Hadamard gate on each of them, then applies a parametrized rotation that depends on the Hamiltonian and then a parameterized rotation on $B$.\n", "\n", "We note that, since the Hamiltonian is composed of a sum of operators that commutes with each others, the operator $ e^{ -i\\gamma H}$ can be decomposed exactly into a product of rotations depending on the $ZZ$ operator, which can be easily implemented on a quantum computer. The same can be said also for rotation depending on $B$, so the final form of the ansatz will be\n", "\n", "$$\n", "|\\psi_1(\\gamma,\\beta)\\rangle = \\prod_{(j,k) \\in E} e^{-\\frac{i}{2}\\gamma (1- Z_j Z_k)} \\prod_{l \\in V} e^{ -i\\beta X_l } |+\\rangle_{|V|}\n", "$$" ] }, { "cell_type": "code", "execution_count": 6, "id": "b104cb8b", "metadata": {}, "outputs": [], "source": [ "def QAOA_circuit(γ,β,G): \n", " \n", " # Returns an appropriate quantum circuit mapping p blocks (length of γ,β) \n", " #of evolution according to the cost Hamiltonian induced by graph G\n", " \n", " # Create the |+> state on every qubit \n", " QAOA=QuantumCircuit(len(G.nodes()),len(G.nodes()))\n", " QAOA.h(range(len(G.nodes())))\n", " QAOA.barrier()\n", " \n", " for i in range(len(γ)): #exp(-i*γ*H_c) repeated with different angles γ for each block\n", " for edge in G.edges():\n", " k = edge[0]\n", " l = edge[1]\n", " QAOA.cp(2*γ[i], k, l)\n", " QAOA.p(-γ[i], k)\n", " QAOA.p(-γ[i], l)\n", " \n", " # then apply the single qubit X - rotations with angle β to all qubits (exp(-i*β*H_b))\n", " QAOA.barrier()\n", " QAOA.rx(-2*β[i], range(len(G.nodes())))\n", "\n", " # Finally measure the result in the computational basis\n", " QAOA.barrier()\n", " QAOA.measure(range(len(G.nodes())),range(len(G.nodes())))\n", " \n", " return QAOA" ] }, { "cell_type": "markdown", "id": "706bea2e", "metadata": {}, "source": [ "## Finding the optimal angles \n", "\n", "Now we have mapped the problem and obtained the ansatz circuit. However, this depends on some parameters that has not been set yet. \n", "\n", "In finding the optimal set of angles emerges the hybrid nature of the QAOA: in fact, depending on what type of problem we are facing, there are different way to choose the best angles for our circuit. As an example, if the ansatz is not too deep, a simple grid search could be used to find them, or we can use some classic optimization algorithms, using optimizers like ADAM or Adagrad, like they do in [Pennylane](https://pennylane.ai/qml/demos/tutorial_qaoa_maxcut.html). Again, we could use Qiskit already implemented [optimizers](https://qiskit.org/documentation/apidoc/qiskit.aqua.components.optimizers.html) such as COBYLA or the Conjugate Gradient to find the solution, using the quantum computer to iteratively measure the cost function and it's gradient, such as we did in the VQE (this type of ansatz satisfies the conditions for the parameter shift rule to be applied).\n", "\n", "In our case, the structure of the graph and of the ansatz is such that we can perform a grid search and find the optimal values of the angles, that are the following:\n", "\n", "$$\n", "\\beta = 0.2 \\quad , \\quad \\gamma = 1.9\n", "$$" ] }, { "cell_type": "code", "execution_count": 7, "id": "431ce3f3", "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/clemensgiuliani/epfl/teaching/qc24/venv_qiskit/lib/python3.12/site-packages/qiskit/visualization/circuit/matplotlib.py:269: UserWarning: Style JSON file 'iqx.json' not found in any of these locations: /Users/clemensgiuliani/epfl/teaching/qc24/venv_qiskit/lib/python3.12/site-packages/qiskit/visualization/circuit/styles/iqx.json, iqx.json. Will use default style.\n", " self._style, def_font_ratio = load_style(self._style)\n" ] }, { "data": { "image/png": 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"text/plain": [ "
" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#Create the circuit\n", "\n", "QAOA = QAOA_circuit([1.9],[0.2],G)\n", "QAOA.draw('mpl', style={'name': 'iqx'})" ] }, { "cell_type": "markdown", "id": "38a1b389", "metadata": {}, "source": [ "## Run the algorithm on simulator\n", "\n", "Now that we have all we need to run the algorithm, let's do it on a simulator" ] }, { "cell_type": "code", "execution_count": 9, "id": "b9a1b8d1", "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "{'10110': 339, '00000': 127, '01000': 250, '11110': 262, '11011': 457, '01111': 242, '01010': 346, '10100': 346, '00111': 224, '00010': 258, '10001': 381, '01001': 357, '11000': 246, '10101': 381, '00101': 378, '10010': 364, '01101': 359, '10111': 236, '11100': 245, '11111': 133, '11101': 220, '00110': 356, '01100': 345, '00100': 461, '00011': 264, '11010': 395, '00001': 261, '01011': 390, '11001': 382, '10000': 266, '01110': 337, '10011': 392}\n" ] } ], "source": [ "# run on local simulator\n", "backend = Aer.get_backend(\"qasm_simulator\")\n", "shots = 10000\n", "\n", "QAOA_results = backend.run(QAOA, shots=shots).result()\n", "\n", "answer = QAOA_results.get_counts()\n", "\n", "f, ax = plt.subplots(figsize=(14,5))\n", "\n", "plt.suptitle(\"Bit string from the QAOA\")\n", "plt.bar(answer.keys(), answer.values(), color='dodgerblue')\n", "plt.xticks(rotation=45)\n", "\n", "plt.show()\n", "\n", "print(answer)" ] }, { "cell_type": "markdown", "id": "39c2ab7e", "metadata": {}, "source": [ "## Evaluate the solutions\n", "\n", "As we can see, there are a lot of possibile answers, since the solution of this problem is not unique. We want now to evaluate how good are the solution found by the algorithm. We need a function that, given a bitstring, evaluate the cost related to the graph we are considering. We can do it by using the classical cost function declared at the beginning of the notebook." ] }, { "cell_type": "code", "execution_count": 10, "id": "527391b8", "metadata": {}, "outputs": [], "source": [ "def cost_function_C(x,G):\n", " \n", " E = G.edges()\n", " if( len(x) != len(G.nodes())):\n", " return np.nan\n", " \n", " C = 0;\n", " for index in E:\n", " e1 = index[0]\n", " e2 = index[1]\n", " \n", " C = C + x[e1]*(1-x[e2]) + x[e2]*(1-x[e1])\n", " \n", " return C" ] }, { "cell_type": "code", "execution_count": 11, "id": "4259cb2e", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", " --- SIMULATION RESULTS ---\n", "\n", "The sampled mean value is M1_sampled = 3.30 \n", "\n", "The approximate solution is x* = 10110 with C(x*) = 4 \n", "\n", "The cost function is distributed as: \n", "\n" ] }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Now evaluate the solutions\n", "\n", "# Solution evaluation and a dictionary to contain the different values of them\n", "avr_C = 0\n", "max_C = [0,0]\n", "hist = {}\n", "\n", "for k in range(len(G.edges())+1):\n", " hist[str(k)] = hist.get(str(k),0)\n", "\n", "for sample in list(answer.keys()):\n", "\n", " # use sampled bit string x to compute C(x)\n", " x = [int(num) for num in list(sample)]\n", " tmp_eng = cost_function_C(x,G)\n", " \n", " # compute the expectation value and energy distribution\n", " avr_C = avr_C + answer[sample]*tmp_eng\n", " hist[str(round(tmp_eng))] = hist.get(str(round(tmp_eng)),0) + answer[sample]\n", " \n", " # save best bit string\n", " if( max_C[1] < tmp_eng):\n", " max_C[0] = sample\n", " max_C[1] = tmp_eng\n", " \n", "M1_sampled = avr_C/shots\n", "\n", "print('\\n --- SIMULATION RESULTS ---\\n')\n", "print('The sampled mean value is M1_sampled = %.02f \\n' % (M1_sampled))\n", "print('The approximate solution is x* = %s with C(x*) = %d \\n' % (max_C[0],max_C[1]))\n", "print('The cost function is distributed as: \\n')\n", "\n", "\n", "f, ax = plt.subplots(figsize=(10,5))\n", "\n", "plt.suptitle(\"Evaluations of solutions provided by the QAOA\")\n", "plt.bar(hist.keys(),hist.values(), color='dodgerblue')\n", "plt.xticks(rotation=45)\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "d3ae364e", "metadata": {}, "source": [ "As you can see, there is an high probability of obtaining a solution that has a cost equal to 4, that we can verify to be the optimal solution. Let's color the graph by using the bitstring we obtained from the algorithm, using `orange` when the bit is `1` and `dodgerblue` when the bit is `0`:" ] }, { "cell_type": "code", "execution_count": 12, "id": "3ac76be3", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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