05_Gate-Model_Quantum_Computing.ipynb

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      "Available frameworks:\n",
      "Qiskit\n",
      "D-Wave Ocean\n"
     ]
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   "source": [
    "%run -i \"assignment_helper.py\"\n",
    "%matplotlib inline"
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    "# Defining circuits\n",
    "\n",
    "**Exercise 1** (2 points). Quantum computers typically initialize their qubit registers in the |0> state. This means that if there is any particular state we would like to work with, first we have to figure out how to create that state with a circuit. Some states are easier to prepare than others. If you are just given a random vector, say. $$\\begin{bmatrix}0.36\\\\  0.8704\\end{bmatrix}$$ it is not easy to figure out how to prepare it. In fact, the very purpose of quantum computing is to prepare a probability distribution of interest, that is, a state. So in some ways, generic state preparation is as hard as or equivalent to quantum computation. On the other hand, some states are easy to prepare; for instance, the state: $$\\frac{-|0\\rangle + |1\\rangle}{\\sqrt{2}}$$ Create a circuit in your preferred framework that prepares this state. The object should be called `circuit`."
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    "###\n",
    "### YOUR CODE HERE\n",
    "###\n",
    "\n",
    "circuit = QuantumCircuit()\n",
    "qr = QuantumRegister(1)\n",
    "circuit.add_register(qr)\n",
    "\n",
    "circuit.h(qr[0])\n",
    "circuit.z(qr[0])\n",
    "circuit.x(qr[0])\n",
    "\n",
    "\n",
    "circuit.draw()"
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       "array([-0.70710678+8.65956056e-17j,  0.70710678+0.00000000e+00j])"
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   "source": [
    "get_amplitudes(circuit)"
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    "amplitudes = get_amplitudes(circuit)\n",
    "assert np.allclose(amplitudes, np.array([-1/np.sqrt(2), 1/np.sqrt(2)]))"
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   "source": [
    "**Exercise 2** (2 points). We know that entanglement is an important resource for quantum computing, but so far we entangled only a pair of qubits. Create a circuit to prepare the state: $$\\frac{|000\\rangle + |111\\rangle}{\\sqrt{2}}$$which is a three-qubit entangled state. The circuit should be in an object called `circuit` with three quantum and three classical registers. If you use PyQuil, declare the classical registers in a variable called `ro`."
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       "       ┌───┐          \n",
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    "###\n",
    "### YOUR CODE HERE\n",
    "###\n",
    "\n",
    "circuit = QuantumCircuit()\n",
    "\n",
    "qr = QuantumRegister(3)\n",
    "cr = ClassicalRegister(3)\n",
    "\n",
    "circuit.add_register(qr)\n",
    "circuit.add_register(cr)\n",
    "\n",
    "### Gates\n",
    "circuit.h(qr[0])\n",
    "circuit.cx(qr[0], qr[1])\n",
    "circuit.cx(qr[1], qr[2])\n",
    "\n",
    "\n",
    "circuit.draw()"
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    "amplitudes = get_amplitudes(circuit)\n",
    "assert np.allclose(amplitudes, np.array([1/np.sqrt(2), 0, 0, 0, 0, 0, 0, 1/np.sqrt(2)]))"
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    "**Exercise 3** (1 point). This state is entangled and exhibits very strong correlations between the qubits. To see this, add a measurement on each qubit. We'll run the circuit a hundred times and study the statistics."
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      "{'000': 47, '111': 53}\n"
     ]
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       "                       0  1  2 </pre>"
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       "                       0  1  2 "
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    "###\n",
    "### YOUR CODE HERE\n",
    "###\n",
    "\n",
    "circuit.measure(0,0)\n",
    "circuit.measure(1,1)\n",
    "circuit.measure(2,2)\n",
    "\n",
    "print(get_counts(circuit))\n",
    "\n",
    "circuit.draw()"
   ]
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    "counts = get_counts(circuit)\n",
    "assert abs(counts['000']/100-.5) < 0.1\n",
    "assert abs(counts['111']/100-.5) < 0.1\n",
    "assert  counts['000'] + counts['111'] == 100"
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   "source": [
    "The measurement statistics show that if any of the qubits measured gives 0, so do the other two. If a qubit is measured 1, then the other two will be 1 as well."
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