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"source": [
"# PRELAB --Intro to IBM Qiskit"
]
},
{
"cell_type": "markdown",
"id": "c081821d",
"metadata": {},
"source": [
"## Problem 1\n",
"### (a)\n",
"Show that\n",
"\n",
"$$\n",
"\\begin{align}\n",
"H|0\\rangle &= |+\\rangle\\\\\n",
"H|1\\rangle &= |-\\rangle\n",
"\\end{align}\n",
"$$\n",
"\n",
"where \n",
"\n",
"$$\n",
"H = \\frac{1}{\\sqrt{2}} \\left(\\begin{array}{cc} 1& 1\\\\1& -1 \\end{array} \\right)\n",
"$$\n",
"\n",
"is the Hadamard gate.\n",
"\n",
"Recall that $|\\pm\\rangle = (|0\\rangle\\pm |1\\rangle)/\\sqrt{2}$ is an eigenstate of the Pauli-$X$ operator."
]
},
{
"cell_type": "markdown",
"id": "e2a064fe",
"metadata": {},
"source": [
"### (b)\n",
"Show that (up to a global phase)\n",
"\n",
"$$\n",
"\\begin{align}\n",
"H &= e^{-i\\frac{\\pi}{2} \\sigma_{\\hat{i}}}\n",
"\\end{align}\n",
"$$\n",
"\n",
"where $ \\hat{i} = \\frac{1}{\\sqrt{2}}\\left( \\begin{array}{c} 1\\\\0\\\\1 \\end{array} \\right) $."
]
},
{
"cell_type": "markdown",
"id": "029af910",
"metadata": {},
"source": [
"## Problem 2\n",
"Show that the following circuit performs a SWAP operation:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "bb5988a7",
"metadata": {},
"outputs": [
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"data": {
"text/html": [
"<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: "Courier New",Courier,monospace\"> ┌───┐ \n",
"q_0: ──■──┤ X ├──■──\n",
" ┌─┴─┐└─┬─┘┌─┴─┐\n",
"q_1: ┤ X ├──■──┤ X ├\n",
" └───┘ └───┘</pre>"
],
"text/plain": [
" ┌───┐ \n",
"q_0: ──■──┤ X ├──■──\n",
" ┌─┴─┐└─┬─┘┌─┴─┐\n",
"q_1: ┤ X ├──■──┤ X ├\n",
" └───┘ └───┘"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from qiskit import QuantumCircuit\n",
"from qiskit.circuit import Gate\n",
"qc = QuantumCircuit(2)\n",
"qc.cx(0,1)\n",
"qc.cx(1,0)\n",
"qc.cx(0,1)\n",
"qc.draw()"
]
},
{
"cell_type": "markdown",
"id": "ad2a66e1",
"metadata": {},
"source": [
"## Problem 3\n",
"### (a)\n",
"Show that $ HXH = Z $ and $ HZH = X $."
]
},
{
"cell_type": "markdown",
"id": "7f11a99f",
"metadata": {},
"source": [
"### (b)\n",
"Using the identities above, which two-qubit gate is the following quantum circuit equivalent to?"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "2ed97488",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: "Courier New",Courier,monospace\"> \n",
"q_0: ──────■──────\n",
" ┌───┐ │ ┌───┐\n",
"q_1: ┤ H ├─■─┤ H ├\n",
" └───┘ └───┘</pre>"
],
"text/plain": [
" \n",
"q_0: ──────■──────\n",
" ┌───┐ │ ┌───┐\n",
"q_1: ┤ H ├─■─┤ H ├\n",
" └───┘ └───┘"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from qiskit import QuantumCircuit\n",
"from qiskit.circuit import Gate\n",
"qc = QuantumCircuit(2)\n",
"qc.h(1)\n",
"qc.cz(0,1)\n",
"qc.h(1)\n",
"qc.draw()"
]
},
{
"cell_type": "markdown",
"id": "6e283f04",
"metadata": {},
"source": [
"### (c)\n",
"In real-world experiments, qubits often _do not_ have all-to-all connectivities. For example, the quantum circuit below performs effectively a two-qubit gate between q_1 and q_2 by using q_0 as an auxiliary qubit. Which two-qubit gate is it and prove that this is indeed the case."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "f02c583f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: "Courier New",Courier,monospace\"> ┌───┐ ┌───┐\n",
"q_0: ─■─┤ H ├─■─┤ H ├\n",
" │ └───┘ │ └───┘\n",
"q_1: ─■───────┼──────\n",
" │ \n",
"q_2: ─────────■──────\n",
" </pre>"
],
"text/plain": [
" ┌───┐ ┌───┐\n",
"q_0: ─■─┤ H ├─■─┤ H ├\n",
" │ └───┘ │ └───┘\n",
"q_1: ─■───────┼──────\n",
" │ \n",
"q_2: ─────────■──────\n",
" "
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from qiskit import QuantumCircuit\n",
"from qiskit.circuit import Gate\n",
"qc = QuantumCircuit(3)\n",
"qc.cz(0,1)\n",
"qc.h(0)\n",
"qc.cz(0,2)\n",
"qc.h(0)\n",
"qc.draw()"
]
},
{
"cell_type": "markdown",
"id": "44c877ae",
"metadata": {},
"source": [
"## Problem 4\n",
"Quantum Fourier transform (QFT) is an essential primitive for many quantum algorithms. It is a unitary transformation that maps\n",
"\n",
"$$ |X\\rangle = \\sum_{j=0}^{N-1} x_j |j\\rangle $$\n",
"\n",
"to \n",
"\n",
"$$ |Y\\rangle = \\sum_{k=0}^{N-1} y_k |k\\rangle$$\n",
"\n",
", where $ y_k = \\frac{1}{\\sqrt{N}} \\sum_{j=0}^{N-1} x_j \\omega_N^{jk}$ and $ \\omega_N^{jk}=e^{2\\pi i \\frac{jk}{N}}$. $N=2^n$ represents the size of the Hilbert space spanned by $n$ qubits.\n",
"\n",
"Show that the Hadamard gate is a 1-qubit QFT."
]
},
{
"cell_type": "markdown",
"id": "e8977e2b",
"metadata": {},
"source": [
"## Problem 5\n",
"Typically, a CNOT gate performed on a control qubit q_c and a target qubit q_t results in a change in the state of q_t depending on the state of q_c. Show that in the $X$-basis, instead, the state of q_c changes depending on the state of q_t. This phenomenon is called phase kickback, a mechanism essential to the quantum phase estimation algorithm as you will see in lab."
]
}
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