Applying a Hadamard Gate to a Qubit

Applying+a+Hadamard+Gate+to+a+Qubit.ipynb

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   "source": [
    "# Applying a Hadamard Gate to a Qubit\n",
    "_by James Cropcho_"
   ]
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   "source": [
    "import numpy as np\n",
    "BASIS_VECTOR_REPRESENTATIONS = (0, 1) # not fantastic naming ―JC"
   ]
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   "source": [
    "First, I created my Hadamard gate:"
   ]
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      "[[ 0.70710678  0.70710678]\n",
      " [ 0.70710678 -0.70710678]]\n"
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   "source": [
    "hadamard_gate = 2**-0.5 * np.matrix('1 1; 1 -1')\n",
    "print(hadamard_gate)"
   ]
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   "source": [
    "Then, I created my qubit, instantiated at $\\left|0\\right\\rangle$:"
   ]
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   "execution_count": 307,
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     "text": [
      "[[ 1.]\n",
      " [ 0.]]\n"
     ]
    }
   ],
   "source": [
    "qubit = np.matrix('1.; 0.')\n",
    "print(qubit)"
   ]
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   "cell_type": "markdown",
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   "source": [
    "Then, I applied the Hadamard gate to the qubit as a unitary transformation, in the hope that I would transform its state such that the amplitudes of $\\left|0\\right\\rangle$ and $\\left|1\\right\\rangle$ would each now be $\\frac{1}{\\sqrt2} \\approx 0.7071$. _I have overwritten the qubit's previous state to simulate the No-Cloning Theorem._"
   ]
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      "[[ 0.70710678]\n",
      " [ 0.70710678]]\n"
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   "source": [
    "qubit = hadamard_gate * qubit\n",
    "print(qubit)"
   ]
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   "source": [
    "Success! Because a qubit's state (i.e. a vector of amplitudes) must be in 2-norm, I then squared each component's amplitude to get the probability of measuring those values upon collapse of superposition:"
   ]
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   "cell_type": "code",
   "execution_count": 309,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.5]\n",
      " [ 0.5]]\n"
     ]
    }
   ],
   "source": [
    "print(np.square(qubit))"
   ]
  },
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   "source": [
    "50/50 for measuring either $0$ or $1$, just as one would expect. However, we are _not_ yet measuring, so once again, the qubit's new, present state:"
   ]
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     "text": [
      "[[ 0.70710678]\n",
      " [ 0.70710678]]\n"
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   "source": [
    "print(qubit)"
   ]
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   "cell_type": "markdown",
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   "source": [
    "Though the Hadamard gate is a fully deterministic operator, it transforms a qubit whose result will be certain upon measurment (e.g. $\\left|0\\right\\rangle$) into a \"quasi-probabilistic distribution\" superposition qubit state. The value one measures is not known until measurement. And so it was with my qubit at this time.\n",
    "\n",
    "Under classical probability theory, applying a deterministic operation to a probability may only ever yield another probabilistic result. However, under quantum mechanics amplitudes may also be negative and complex (unlike probabilities). This can have completely counterintuitive implications. Applying a Hadamard gate _again_ to this qubit results in a qubit state whose result will once more be certain upon measurment.\n",
    "\n",
    "Here, I applied my Hadamard gate for the second time to my qubit:"
   ]
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  {
   "cell_type": "code",
   "execution_count": 311,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.]\n",
      " [ 0.]]\n"
     ]
    }
   ],
   "source": [
    "qubit = hadamard_gate * qubit\n",
    "print(qubit)"
   ]
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   "cell_type": "markdown",
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   "source": [
    "Finally, I measure for the first time, and the system returns its first output, which like any measurement is some specific \"classical\" value! Below:"
   ]
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   "execution_count": 312,
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     "execution_count": 312,
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   "source": [
    "def measure(qubit):\n",
    "    amplitudes = np.stack(qubit, axis=1).tolist()[0]\n",
    "    probabilities = [amplitude**2 for amplitude in amplitudes]\n",
    "    \n",
    "    measurement = np.random.choice(BASIS_VECTOR_REPRESENTATIONS, p=probabilities)\n",
    "    del qubit # our measurement destroys the qubit's state\n",
    "\n",
    "    return measurement\n",
    "\n",
    "measure(qubit)"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The qubit's \"post-collapse\" measured state is $0$ (_not_ $\\left|0\\right\\rangle$) i.e. a non-error measurement of the qubit at line 311 will certainly be $0$.\n",
    "\n",
    "Thank you for exploring the Hadamard gate with me."
   ]
  }
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