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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# D-Wave Examples\n",
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"\n",
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"This notebook shows how to formulate and solve QUBO problems using D-Wave's dimod software package. The problems and formulations are either from the paper *Ising formulations of many NP problems* by Andrew Lucas (2014), or from *A Tutorial on Formulating and Using QUBO Models*, by Fred Glover, Gary Kochenberger, and Yu Du (2019).\n",
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"\n",
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"## 1. Common functions"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"import dimod\n",
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"import time\n",
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"import numpy as np\n",
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"from dwave.system import DWaveSampler, AutoEmbeddingComposite\n",
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"\n",
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"def solve_qubo(Q,\n",
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" sampler=\"CPU\", # CPU or QPU\n",
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" k=10,\n",
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" chain_strength=None):\n",
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" \"\"\"\n",
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" Given an upper triangular matrix Q of size NxN, solves the quadratic unconstrained binary\n",
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" optimization (QUBO) problem given by\n",
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" \n",
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" minimize sum(x[i] * Q[i,j] * x[j]\n",
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" for i in range(N),\n",
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" for j in range(i+1, N))\n",
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" \n",
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" Uses dimod.SimulatedAnnealingSampler, which solves the problem k times through simulated\n",
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" annealing (on a regular CPU). This method returns the best solution found.\n",
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" \"\"\"\n",
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" assert isinstance(Q, np.ndarray)\n",
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" assert sampler in [\"CPU\", \"QPU\"]\n",
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" n = Q.shape[0]\n",
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" nz = len(Q[Q!=0])\n",
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" print(\"Solving QUBO problem (%d vars, %d nz) on %s...\" % (n, nz, sampler))\n",
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" \n",
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" start = time.time()\n",
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" if sampler == \"CPU\":\n",
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" sampler = dimod.SimulatedAnnealingSampler()\n",
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" response = sampler.sample_qubo(Q, num_reads=k)\n",
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" else:\n",
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" if chain_strength is None:\n",
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" chain_strength = int(10 * np.max(np.abs(Q)))\n",
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" sampler = AutoEmbeddingComposite(DWaveSampler(solver=dict(qpu=True)))\n",
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" response = sampler.sample_qubo(Q, num_reads=k, chain_strength=chain_strength)\n",
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" elapsed = time.time() - start\n",
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" \n",
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" print(\"Solved in %.2f seconds\" % elapsed)\n",
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" solution = min(response.data([\"sample\", \"energy\"]), key=lambda s: s.energy)\n",
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" return solution, response"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 2. Number partitioning problem\n",
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"\n",
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"Given a set $S=\\left\\{s_1,\\ldots,s_n\\right\\}$ of $n$ positive integers, find a partition $(S_0, S_1)$ of $S$ minimizing the difference between $\\sum_{s \\in S_0} s$ and $\\sum_{s \\in S_1} s$.\n",
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"In this formulation, the upper triangular matrix $Q$ is given by\n",
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"$$\n",
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" Q_{i,j} = \\begin{cases}\n",
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" s_i (s_j - c) & \\text{if } i=j \\\\\n",
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" 2 s_i s_j & \\text{if } i>j,\n",
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" \\end{cases}\n",
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"$$\n",
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"where $c = \\sum_{s \\in S} s$."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 2.1 Create random instance and Q matrix"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [],
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"source": [
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"from scipy.stats import randint\n",
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"\n",
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"# Pick random integer numbers between 1 and 100\n",
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"np.random.seed(42)\n",
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"S = randint(low=1, high=11).rvs(30)\n",
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"\n",
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"# Generate Q matrix\n",
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"c, n = sum(S), len(S)\n",
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"Q = np.zeros((n, n), dtype=int)\n",
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"for i in range(n):\n",
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" Q[i,i] = S[i] * (S[i] - c)\n",
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" for j in range(i + 1, n):\n",
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" Q[i,j] = 2 * S[i] * S[j]"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 2.2 Solve on the CPU"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Solving QUBO problem (30 vars, 465 nz) on CPU...\n",
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"Solved in 18.87 seconds\n",
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"sum(S0) 86\n",
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"sum(S1) 86\n"
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]
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}
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],
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"source": [
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"# Solve problem\n",
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"solution, response = solve_qubo(Q)\n",
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"\n",
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"# Display solution\n",
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"S0 = [S[i] for (i, xi) in solution.sample.items() if xi > 0.5]\n",
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"S1 = [S[i] for (i, xi) in solution.sample.items() if xi <= 0.5]\n",
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"print(\"sum(S0) %8d\" % sum(S0))\n",
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"print(\"sum(S1) %8d\" % sum(S1))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 2.3 Solve on the QPU"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Solving QUBO problem (30 vars, 465 nz) on QPU...\n",
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"Solved in 17.30 seconds\n",
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"sum(S0) 86\n",
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"sum(S1) 86\n"
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]
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}
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],
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"source": [
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"# Solve problem\n",
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"solution, qresponse = solve_qubo(Q, sampler=\"QPU\")\n",
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"\n",
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"# Display solution\n",
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"S0 = [S[i] for (i, xi) in solution.sample.items() if xi > 0.5]\n",
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"S1 = [S[i] for (i, xi) in solution.sample.items() if xi <= 0.5]\n",
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"print(\"sum(S0) %8d\" % sum(S0))\n",
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"print(\"sum(S1) %8d\" % sum(S1))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 2.4 Inspect QPU solution and embedding"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"'http://127.0.0.1:18000/?problemId=72672b81-b943-4dc6-ade7-24b0adce7c5d'"
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]
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},
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"execution_count": 11,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"import dwave.inspector\n",
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"dwave.inspector.show(qresponse)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.7"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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