14_Quantum_Matrix_Inversion.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Available frameworks:\n",
      "Qiskit\n",
      "D-Wave Ocean\n"
     ]
    }
   ],
   "source": [
    "%run -i \"assignment_helper.py\"\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 1** (1 point). We want to solve the equation *Ax = b* with: $$A = \\begin{bmatrix}1 & 0 \\\\0 & -1 \\\\ \\end{bmatrix}$$ and $$b =\\begin{bmatrix} 0 \\\\ 1 \\\\ \\end{bmatrix}$$ with quantum matrix inversion. We will encode $A$ in the unitary matrix $$U=e^{iAt_0} \\space with \\space t_0=\\pi/2$$ and *b* in a register. With the ancilla (qubit 0), the eigenvalue registers (or the ancilla qubits of phase estimation, qubits 1 and 2), and the eigenstate, you will need a total of four qubits and one classical register for post-selection. Prepare the superposition in the eigenvalue register and the vector *b*. Place your solution in an object called `hhl`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "q0_0: ─────\n",
       "      ┌───┐\n",
       "q0_1: ┤ H ├\n",
       "      ├───┤\n",
       "q0_2: ┤ H ├\n",
       "      ├───┤\n",
       "q0_3: ┤ X ├\n",
       "      └───┘\n",
       "c0: 4/═════\n",
       "           </pre>"
      ],
      "text/plain": [
       "           \n",
       "q0_0: ─────\n",
       "      ┌───┐\n",
       "q0_1: ┤ H ├\n",
       "      ├───┤\n",
       "q0_2: ┤ H ├\n",
       "      ├───┤\n",
       "q0_3: ┤ X ├\n",
       "      └───┘\n",
       "c0: 4/═════\n",
       "           "
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "###\n",
    "### YOUR CODE HERE\n",
    "###\n",
    "\n",
    "qr = QuantumRegister(4)\n",
    "cr = ClassicalRegister(4)\n",
    "\n",
    "hhl = QuantumCircuit(qr, cr)\n",
    "\n",
    "hhl.h(qr[1])\n",
    "hhl.h(qr[2])\n",
    "hhl.x(qr[3])\n",
    "\n",
    "hhl.draw()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "grade": true,
     "grade_id": "exercise1",
     "locked": true,
     "points": "1",
     "solution": false
    }
   },
   "outputs": [],
   "source": [
    "amplitudes = get_amplitudes(hhl)\n",
    "assert np.allclose(amplitudes, np.array([0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j,\n",
    "       0. +0.j, 0.5+0.j, 0. +0.j, 0.5+0.j, 0. +0.j, 0.5+0.j, 0. +0.j, 0.5+0.j, 0. +0.j]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2** (2 points). Start the quantum phase estimation by applying $$C-U^{2^0} \\space and \\space C-U^{2^1}$$ Extend the circuit with the two appropriate gates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-31T15:46:05.496117Z",
     "start_time": "2019-01-31T15:46:05.463504Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<ipython-input-4-819d6348ebae>:5: DeprecationWarning: The QuantumCircuit.cu1 method is deprecated as of 0.16.0. It will be removed no earlier than 3 months after the release date. You should use the QuantumCircuit.cp method instead, which acts identically.\n",
      "  hhl.cu1(2*np.pi, qr[1], qr[3])\n"
     ]
    },
    {
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      "text/html": [
       "<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: &quot;Courier New&quot;,Courier,monospace\">                            \n",
       "q0_0: ──────────────────────\n",
       "      ┌───┐                 \n",
       "q0_1: ┤ H ├─■───────────────\n",
       "      ├───┤ │               \n",
       "q0_2: ┤ H ├─┼────────■──────\n",
       "      ├───┤ │U1(2π)  │U1(π) \n",
       "q0_3: ┤ X ├─■────────■──────\n",
       "      └───┘                 \n",
       "c0: 4/══════════════════════\n",
       "                            </pre>"
      ],
      "text/plain": [
       "                            \n",
       "q0_0: ──────────────────────\n",
       "      ┌───┐                 \n",
       "q0_1: ┤ H ├─■───────────────\n",
       "      ├───┤ │               \n",
       "q0_2: ┤ H ├─┼────────■──────\n",
       "      ├───┤ │U1(2π)  │U1(π) \n",
       "q0_3: ┤ X ├─■────────■──────\n",
       "      └───┘                 \n",
       "c0: 4/══════════════════════\n",
       "                            "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "###\n",
    "### YOUR CODE HERE\n",
    "###\n",
    "\n",
    "hhl.cu1(2*np.pi, qr[1], qr[3])\n",
    "hhl.cu1(np.pi, qr[2], qr[3])\n",
    "\n",
    "hhl.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "grade": true,
     "grade_id": "exercise2",
     "locked": true,
     "points": "2",
     "solution": false
    }
   },
   "outputs": [],
   "source": [
    "amplitudes = get_amplitudes(hhl)\n",
    "assert np.allclose(amplitudes, np.array([ 0. +0.j,  0. +0.j,  0. +0.j,  0. +0.j,  0. +0.j,  0. +0.j,\n",
    "                                          0. +0.j,  0. +0.j,  0.5+0.j,  0. +0.j,  0.5+0.j,  0. +0.j,\n",
    "                                         -0.5+0.j,  0. +0.j, -0.5+0.j,  0. +0.j]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3** (1 point). Apply the quantum inverse Fourier transformation. Don't forget the swap just before the transformation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-31T15:46:05.500661Z",
     "start_time": "2019-01-31T15:46:05.497677Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: &quot;Courier New&quot;,Courier,monospace\">                                                    \n",
       "q0_0: ──────────────────────────────────────────────\n",
       "      ┌───┐                                    ┌───┐\n",
       "q0_1: ┤ H ├─■────────────────X───────■─────────┤ H ├\n",
       "      ├───┤ │                │ ┌───┐ │U1(-π/2) └───┘\n",
       "q0_2: ┤ H ├─┼────────■───────X─┤ H ├─■──────────────\n",
       "      ├───┤ │U1(2π)  │U1(π)    └───┘                \n",
       "q0_3: ┤ X ├─■────────■──────────────────────────────\n",
       "      └───┘                                         \n",
       "c0: 4/══════════════════════════════════════════════\n",
       "                                                    </pre>"
      ],
      "text/plain": [
       "                                                    \n",
       "q0_0: ──────────────────────────────────────────────\n",
       "      ┌───┐                                    ┌───┐\n",
       "q0_1: ┤ H ├─■────────────────X───────■─────────┤ H ├\n",
       "      ├───┤ │                │ ┌───┐ │U1(-π/2) └───┘\n",
       "q0_2: ┤ H ├─┼────────■───────X─┤ H ├─■──────────────\n",
       "      ├───┤ │U1(2π)  │U1(π)    └───┘                \n",
       "q0_3: ┤ X ├─■────────■──────────────────────────────\n",
       "      └───┘                                         \n",
       "c0: 4/══════════════════════════════════════════════\n",
       "                                                    "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "###\n",
    "### YOUR CODE HERE\n",
    "###\n",
    "\n",
    "hhl.swap(qr[1], qr[2])\n",
    "hhl.h(qr[2])\n",
    "hhl.cu1(-np.pi / 2, qr[1], qr[2])\n",
    "hhl.h(qr[1])\n",
    "\n",
    "hhl.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "grade": true,
     "grade_id": "exercise3",
     "locked": true,
     "points": "1",
     "solution": false
    }
   },
   "outputs": [],
   "source": [
    "amplitudes = get_amplitudes(hhl)\n",
    "assert np.allclose(amplitudes, np.array([0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j,\n",
    "                                         0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 4** (1 point). After this step, swap the eigenvalue registers again (this time to perform the inversion) and apply the controlled rotation. Use the same angle as in the lecture notebook."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-01-31T15:46:05.508179Z",
     "start_time": "2019-01-31T15:46:05.502279Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<ipython-input-8-c2e2da35df10>:9: DeprecationWarning: The QuantumCircuit.cu3 method is deprecated as of 0.16.0. It will be removed no earlier than 3 months after the release date. You should use the QuantumCircuit.cu method instead, where cu3(ϴ,φ,λ) = cu(ϴ,φ,λ,0).\n",
      "  hhl.cu3(theta_0, 0, 0, qr[1], qr[0])  # CRY0\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: &quot;Courier New&quot;,Courier,monospace\">                                                       ┌──────────────┐»\n",
       "q0_0: ─────────────────────────────────────────────────┤ U3(1π/8,0,0) ├»\n",
       "      ┌───┐                                    ┌───┐   └──────┬───────┘»\n",
       "q0_1: ┤ H ├─■────────────────X───────■─────────┤ H ├─X────────■────────»\n",
       "      ├───┤ │                │ ┌───┐ │U1(-π/2) └───┘ │                 »\n",
       "q0_2: ┤ H ├─┼────────■───────X─┤ H ├─■───────────────X─────────────────»\n",
       "      ├───┤ │U1(2π)  │U1(π)    └───┘                                   »\n",
       "q0_3: ┤ X ├─■────────■─────────────────────────────────────────────────»\n",
       "      └───┘                                                            »\n",
       "c0: 4/═════════════════════════════════════════════════════════════════»\n",
       "                                                                       »\n",
       "«      ┌──────────────┐\n",
       "«q0_0: ┤ U3(π/16,0,0) ├\n",
       "«      └──────┬───────┘\n",
       "«q0_1: ───────┼────────\n",
       "«             │        \n",
       "«q0_2: ───────■────────\n",
       "«                      \n",
       "«q0_3: ────────────────\n",
       "«                      \n",
       "«c0: 4/════════════════\n",
       "«                      </pre>"
      ],
      "text/plain": [
       "                                                       ┌──────────────┐»\n",
       "q0_0: ─────────────────────────────────────────────────┤ U3(1π/8,0,0) ├»\n",
       "      ┌───┐                                    ┌───┐   └──────┬───────┘»\n",
       "q0_1: ┤ H ├─■────────────────X───────■─────────┤ H ├─X────────■────────»\n",
       "      ├───┤ │                │ ┌───┐ │U1(-π/2) └───┘ │                 »\n",
       "q0_2: ┤ H ├─┼────────■───────X─┤ H ├─■───────────────X─────────────────»\n",
       "      ├───┤ │U1(2π)  │U1(π)    └───┘                                   »\n",
       "q0_3: ┤ X ├─■────────■─────────────────────────────────────────────────»\n",
       "      └───┘                                                            »\n",
       "c0: 4/═════════════════════════════════════════════════════════════════»\n",
       "                                                                       »\n",
       "«      ┌──────────────┐\n",
       "«q0_0: ┤ U3(π/16,0,0) ├\n",
       "«      └──────┬───────┘\n",
       "«q0_1: ───────┼────────\n",
       "«             │        \n",
       "«q0_2: ───────■────────\n",
       "«                      \n",
       "«q0_3: ────────────────\n",
       "«                      \n",
       "«c0: 4/════════════════\n",
       "«                      "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "###\n",
    "### YOUR CODE HERE\n",
    "###\n",
    "\n",
    "theta_0 = 0.392699\n",
    "theta_1 = 0.19634955\n",
    "\n",
    "hhl.swap(qr[1], qr[2])\n",
    "hhl.cu3(theta_0, 0, 0, qr[1], qr[0])  # CRY0\n",
    "hhl.cu3(theta_1, 0, 0, qr[2], qr[0]);  # CRY1\n",
    "\n",
    "hhl.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "grade": true,
     "grade_id": "exercise4",
     "locked": true,
     "points": "1",
     "solution": false
    }
   },
   "outputs": [],
   "source": [
    "amplitudes = get_amplitudes(hhl)\n",
    "assert np.allclose(amplitudes, np.array([0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,\n",
    "                                         0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,\n",
    "                                         0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,\n",
    "                                         0.99518473+0.j, 0.09801714+0.j, 0.        +0.j, 0.        +0.j]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 5** (2 points). Uncompute the eigenvalue register"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: &quot;Courier New&quot;,Courier,monospace\">                                                       ┌──────────────┐»\n",
       "q0_0: ─────────────────────────────────────────────────┤ U3(1π/8,0,0) ├»\n",
       "      ┌───┐                                    ┌───┐   └──────┬───────┘»\n",
       "q0_1: ┤ H ├─■────────────────X───────■─────────┤ H ├─X────────■────────»\n",
       "      ├───┤ │                │ ┌───┐ │U1(-π/2) └───┘ │                 »\n",
       "q0_2: ┤ H ├─┼────────■───────X─┤ H ├─■───────────────X─────────────────»\n",
       "      ├───┤ │U1(2π)  │U1(π)    └───┘                                   »\n",
       "q0_3: ┤ X ├─■────────■─────────────────────────────────────────────────»\n",
       "      └───┘                                                            »\n",
       "c0: 4/═════════════════════════════════════════════════════════════════»\n",
       "                                                                       »\n",
       "«      ┌──────────────┐                                    \n",
       "«q0_0: ┤ U3(π/16,0,0) ├────────────────────────────────────\n",
       "«      └──────┬───────┘   ┌───┐                  ┌───┐     \n",
       "«q0_1: ───────┼─────────X─┤ H ├─■──────────────X─┤ H ├─────\n",
       "«             │         │ └───┘ │U1(π/2) ┌───┐ │ └───┘┌───┐\n",
       "«q0_2: ───────■─────────X───────■────────┤ H ├─X───■──┤ H ├\n",
       "«                                        └───┘     │  └───┘\n",
       "«q0_3: ────────────────────────────────────────────■───────\n",
       "«                                                          \n",
       "«c0: 4/════════════════════════════════════════════════════\n",
       "«                                                          </pre>"
      ],
      "text/plain": [
       "                                                       ┌──────────────┐»\n",
       "q0_0: ─────────────────────────────────────────────────┤ U3(1π/8,0,0) ├»\n",
       "      ┌───┐                                    ┌───┐   └──────┬───────┘»\n",
       "q0_1: ┤ H ├─■────────────────X───────■─────────┤ H ├─X────────■────────»\n",
       "      ├───┤ │                │ ┌───┐ │U1(-π/2) └───┘ │                 »\n",
       "q0_2: ┤ H ├─┼────────■───────X─┤ H ├─■───────────────X─────────────────»\n",
       "      ├───┤ │U1(2π)  │U1(π)    └───┘                                   »\n",
       "q0_3: ┤ X ├─■────────■─────────────────────────────────────────────────»\n",
       "      └───┘                                                            »\n",
       "c0: 4/═════════════════════════════════════════════════════════════════»\n",
       "                                                                       »\n",
       "«      ┌──────────────┐                                    \n",
       "«q0_0: ┤ U3(π/16,0,0) ├────────────────────────────────────\n",
       "«      └──────┬───────┘   ┌───┐                  ┌───┐     \n",
       "«q0_1: ───────┼─────────X─┤ H ├─■──────────────X─┤ H ├─────\n",
       "«             │         │ └───┘ │U1(π/2) ┌───┐ │ └───┘┌───┐\n",
       "«q0_2: ───────■─────────X───────■────────┤ H ├─X───■──┤ H ├\n",
       "«                                        └───┘     │  └───┘\n",
       "«q0_3: ────────────────────────────────────────────■───────\n",
       "«                                                          \n",
       "«c0: 4/════════════════════════════════════════════════════\n",
       "«                                                          "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "###\n",
    "### YOUR CODE HERE\n",
    "###\n",
    "hhl.swap(qr[1], qr[2])\n",
    "hhl.h(qr[1])\n",
    "hhl.cu1(np.pi / 2, qr[1], qr[2])\n",
    "hhl.h(qr[2])\n",
    "hhl.swap(qr[1], qr[2])\n",
    "hhl.cz(qr[2], qr[3])\n",
    "hhl.h(qr[1])\n",
    "hhl.h(qr[2])\n",
    "\n",
    "hhl.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j,\n",
       "        0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j,\n",
       "        0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j,\n",
       "        0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j,\n",
       "        1.99673462e-16+1.22464680e-16j,  0.00000000e+00+0.00000000e+00j,\n",
       "       -9.00836673e-17+1.06030098e-16j, -1.79187517e-17+2.10906933e-17j,\n",
       "        9.95184726e-01-3.04687447e-16j,  9.80171449e-02-3.00090957e-17j,\n",
       "        8.78935449e-17+7.23348051e-17j,  2.66622119e-17+2.19425204e-17j])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "amplitudes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
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   "source": [
    "amplitudes = get_amplitudes(hhl)\n",
    "assert np.allclose(amplitudes, np.array([0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,\n",
    "                                         0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,\n",
    "                                         0.99518473+0.j, 0.09801714+0.j, 0.        +0.j, 0.        +0.j,\n",
    "                                         0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j]))"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "At this point, if we measure 1 in the ancilla (qubit 0), the state will be proportional to: $$A^{-1}b = \\sum_j \\beta_j\\lambda_j^{-1}\\left|u_j\\right\\rangle=0 \\frac{1}{1} |0\\rangle + 1 \\frac{1}{-1} |1\\rangle=-|1\\rangle$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'1000': 100}\n"
     ]
    }
   ],
   "source": [
    "hhl.measure(qr, cr)\n",
    "\n",
    "backend = qiskit.BasicAer.get_backend('qasm_simulator') # the device to run on\n",
    "result = execute(hhl, backend, shots=100).result()\n",
    "counts  = result.get_counts(hhl)\n",
    "print(counts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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