data-fitting.ipynb

{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solving a Data Fitting Problem"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This data fitting quantum algorithm involves 3 sub-routines:\n",
    "1. a quantum algorithm for performing a pseudo-inverse\n",
    "2. an algorithm for estimating the fit quality\n",
    "3. an algorithm for learning the fit parameters $\\lambda$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Input: \n",
    "- quantum state $|y\\rangle$\n",
    "- upper bound $\\kappa$ square root of the conditional number $FF^{\\dag}$ and $F^{\\dag}F$\n",
    "- sparseness $s$ of $F$\n",
    "- error tolerance $\\epsilon$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from qiskit import ClassicalRegister, QuantumCircuit, execute, Aer\n",
    "from qiskit.visualization import plot_histogram"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sub-Routine 1: Pseudo-Inverse"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Also, we set additional specifications for $F$ and $y$. $F \\in \\mathbb{R}^{4\\times3}$. And $y \\in \\mathbb{R}^4$, not a linear combination of the columns of $F$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "See below we define an $F$ with rank 3 and a $y$ that isn't a linear combination of $F$'s columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "rank 3\n",
      "F= [[1.  1.  1. ]\n",
      " [0.  0.  1. ]\n",
      " [1.  0.  0. ]\n",
      " [0.5 1.  0. ]]\n",
      "y= [[0]\n",
      " [0]\n",
      " [1]\n",
      " [0]]\n"
     ]
    }
   ],
   "source": [
    "F = np.matrix([[1, 1, 1], [0, 0, 1], [1, 0, 0], [0.5, 1, 0]])\n",
    "\n",
    "\n",
    "print(\"rank\", np.linalg.matrix_rank(F))\n",
    "\n",
    "y = np.array([0, 0, 1, 0])\n",
    "y.resize((4, 1))\n",
    "print(\"F=\", F)\n",
    "print(\"y=\", y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The data fitting problem starts with the need to solving the equation.\n",
    "$$\\lambda = F^{+}y$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where $F^{+} = (F^{\\dag}F)^{-1}F^{\\dag}$. However, the inverse operation is computationally expensive, so we can construct the problem as follows:\n",
    "$$\n",
    "    (F^\\dag F )\\lambda = F^\\dag y\n",
    "$$\n",
    "Which can be solved efficiently using HHL, where $A = F^\\dag F$ and $b = F^\\dag y$. $A$ is $3\\times3$ however, so we need to pad it for its dimensions to be a power of $2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = F.getH() * F\n",
    "\n",
    "A = np.pad(A, ((0, 1), (0, 1)))\n",
    "A[-1][-1] = 1 \n",
    "\n",
    "b = F.getH() * y\n",
    "b = np.vstack((b, [0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sub-Routine 2: Estimating Fit Quality"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/tmp/ipykernel_162827/305850212.py:4: DeprecationWarning: The HHL class is deprecated as of Qiskit Terra 0.22.0 and will be removed\n",
      "        no sooner than 3 months after the release date.\n",
      "        It is replaced by the tutorial at https://qiskit.org/textbook/ch-applications/hhl_tutorial.html\"\n",
      "        \n",
      "  hhl = HHL(quantum_instance=backend)\n"
     ]
    }
   ],
   "source": [
    "from qiskit.algorithms.linear_solvers.hhl import HHL\n",
    "\n",
    "backend = Aer.get_backend('aer_simulator')\n",
    "hhl = HHL(quantum_instance=backend)\n",
    "accurate_solution = hhl.solve(A, b)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We must perform the swap test to distinguish between $|y\\rangle$ and $I(F)|\\lambda\\rangle$. The overlap of these two quantum states can be learned through repeated sampling of this swap test to get the approximate value $|\\langle y|I(F)|\\lambda\\rangle|^2$ of the mean of the distribution. The fit quality $E$ can be estimated to be less than $2(1-|\\langle y|I(F)|\\lambda\\rangle|)$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sub-Routine 3: Learning Lambda"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This routine involves repeatedly sampling the solution lambda output from HHL. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nShots = 10000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "c = ClassicalRegister(2, 'c')\n",
    "qc = QuantumCircuit(c)\n",
    "\n",
    "lambda_circ = accurate_solution.state\n",
    "lambda_circ = lambda_circ.compose(qc)\n",
    "lambda_circ.measure(range(2), c)\n",
    "\n",
    "\n",
    "backend_qasm = Aer.get_backend('qasm_simulator')\n",
    "res = execute(lambda_circ, backend_qasm, shots=nShots).result() \n",
    "\n",
    "counts = res.get_counts()\n",
    "plot_histogram(counts)    \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From this histogram we have the relative importance of each of the fit values and are able to obtain $|lambda\\rangle$ with confidence computed from sub-routine 2. It is apparent that because we padded $|b\\rangle$ with a single $0$ that there is no likelihood of observing $11\\rangle$."
   ]
  }
 ],
 "metadata": {
  "language_info": {
   "name": "python"
  },
  "orig_nbformat": 4
 },
 "nbformat": 4,
 "nbformat_minor": 2
}