content/ex1/ex1-id.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Latihan 1 - Gerbang Toffoli\n",
    "\n",
    "## Latar belakang sejarah\n",
    "\n",
    "40 tahun yang lalu, 50 pemikir mengatur diri untuk berfoto di halaman MIT Edicott House. Beberapa berasal dari Konferensi Komputasi Fisika, yang diselenggarakan bersama oleh MIT dan IBM, mengira mereka akan membuat sejarah pada tahun 1981. Yang bisa dibilang merupakan tempat lahir komputasi fisika, dan terutama bidang komputasi kuantum yang sedang berkembang, yang merupakan topik yang serius yang layak untuk buku cetak atau mata kuliah.\n",
    "\n",
    "Pada konferensi ini dimana Fenyman mengucapkan kutipan yang sekarang menjadi terkenal: \"Alam tidak klasikal, sial, dan jika anda ingin membuat simulasi dari alam, anda lebih baik membuat dari mekanika kuantum, dan astaga itu merupakan masalah yang luar biasa, karena itu tidak terlihat sangat mudah\"[1] Pada awal bulan ini, kita merayakan 40 tahun dari ulang tahun konferensi penting ini. Anda dapat membaca lebih lanjut [disini](https://youtu.be/GR6ANm6Z0yk).\n",
    "\n",
    "![](resources/conference-photo.jpeg)\n",
    "\n",
    "Salah satu tema yang dibahas didalam konferensi yaitu komputasi reversibel, yang dimana Tommaso Toffoli dan Edward Fredkin di MIT yang telah dipikirkan dalam beberapa tahun terakhir.[2-3] Toffoli datang dengan versi reversibel gerbang AND/NAND (yang sekarang disebut sebagai gerbang Toffoli, atau gerbang *controled-controled NOT*). Karena gerbang NAND merupakan gerbang universal dalam komputasi klasik, gerbang Toffoli merupakan gerbang logika reversibel. Komputasi kuantum merupakan bentuk khusus dari komputasi reversibel; semua gerbang reversibel dapat di terapkan ke komputer kuantum, dan oleh karena itu gerbang Toffoli juga merupakan gerbang logika kuantum. Akan tetapi, gerbang Toffoli bukanlah gerbang universal untuk komputasi kuantum \n",
    "\n",
    "Di latihan ini, kita akan mengeksplorasi gerbang Toffoli dan gerbang universal untuk komputasi kuantum.\n",
    "\n",
    "### References\n",
    "1. Feynman, Richard P. \"Simulating physics with computers.\" Int. J. Theor. Phys 21.6/7 (1982).\n",
    "1. Toffoli, Tommaso. \"Reversible computing.\" International colloquium on automata, languages, and programming. Springer, Berlin, Heidelberg, 1980.\n",
    "1. Fredkin, Edward, and Tommaso Toffoli. \"Conservative logic.\" International Journal of theoretical physics 21.3 (1982): 219-253."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gerbang logika klasik\n",
    "\n",
    "Di komputasi klasikal sering digunakan model logika Boolean atau gerbang logikal klasik. Gerbang tersebut mewakili fungsi Boolean, fungsi yang hanya bersifat biner (0,1) masukan (*input*) dan keluaran(*output*). 1 aspek menarik dari logika Boolean, semua kemungkinan fungsi biner dapat dihasilkan dari kombinasi beberapa gerbang logika berbeda. Demikian set tersebut disebut set lengkap secara fungsional. 1 set yang terkenal yaitu AND dan NOT. 2 gerbang tersebut dapat mengekspresikan semua fungsi yang memungkinkan. Sama juga untuk OR dan NOT. Yang dimana ada di set yang lebih kecil, yang dimana NAND dan NOR adalah universal, akan tetapi fungsi AND, NOT dan OR sering dilihat sebagai dasar dari komputasi klasikal.\n",
    "\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "**Tujuan**\n",
    "\n",
    "Membangun gerbang Toffoli dengan set gerbang basis (CX, RZ, SX dan X) dari IBM Quantum System.\n",
    "\n",
    "</div>\n",
    "\n",
    "<div class=\"alert alert-block alert-danger\">\n",
    "    \n",
    "Latihan ini bertujuan untuk mengajari anda untuk mengerti konsep dasar dari gerbang kuantum dan cara membangun sirkuit kuantum secara\n",
    "1. visual dengan *Circuit Composer widget*\n",
    "2. secara terprogram dengan *Qiskit*. \n",
    "\n",
    "Jika anda sudah familiar dengan gerbang kuantum dab *Qiskit*. Anda dapat meloncat ke bagian <a href=#problem>permasalahan</a>.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Menyingkirakan peringatan yang tidak perlu\n",
    "import warnings\n",
    "from matplotlib.cbook import MatplotlibDeprecationWarning\n",
    "warnings.filterwarnings('ignore', category=MatplotlibDeprecationWarning)\n",
    "\n",
    "# Mengimport pustaka/libraries standar Qiskit\n",
    "from qiskit import QuantumCircuit, execute, Aer, IBMQ, QuantumRegister, ClassicalRegister\n",
    "from qiskit.compiler import transpile, assemble\n",
    "from qiskit.tools.jupyter import *\n",
    "from qiskit.visualization import *\n",
    "from ibm_quantum_widgets import *\n",
    "\n",
    "# Sangat berguna jika menggunakan pi\n",
    "import math\n",
    "pi=math.pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Apa itu sirkuit kuantum\n",
    "\n",
    "Sirkuit kuantum merupakan model dari komputasi kuantum yang dimana komputasi adalah rangkaian dari gerbang kuantum. Gerbang kuantum sering diwakili dengan rotasi didalam *Bloch sphere*. Mari kita lihat beberapa gerbang kuantum populer\n",
    "\n",
    "### Gerbang X \n",
    "\n",
    "Gerbang X dapat diwakili dengan matriks Pauli-X: \n",
    "\n",
    "$X = \\begin{pmatrix}\n",
    "0 & 1 \\\\\n",
    "1 & 0 \\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Gerbang X sama dengan rotasi di sumbu X dari Bloch sphere sebesar $\\pi$ radian. Yang memetakan $|0\\rangle$ menjadi $|1\\rangle$ dan $|1\\rangle$ menjadi $|0\\rangle$. Yang setara dengan gerbang NOT dari komputer klasik dan kadang disebut sebagai *bit-flip*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_gate=QuantumCircuit(1) # Buat 1 sirkuit kuantum dengan 1 qubit\n",
    "x_gate.x(0)\n",
    "x_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(x_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gerbang SX\n",
    "\n",
    "Gerbang SX sama dengan rotasi di sumbu X dari Bloch sphere sebesar $\\pi/2$. Dipanggil sebagai gerbang SX yang menunjukan akar dua dari gerbang X . Menerapkan gerbang ini secara dua kali akan menghasilkan gerbang Pauli-X standar. Lawan dari SX adalah SX belati, yang memiliki rotasi sebesar $\\pi/2$ ke arah yang berlawanan.\n",
    "\n",
    "$SX = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1+i & 1-i \\\\\n",
    "1-i & 1+i \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sx_gate = QuantumCircuit(1)\n",
    "sx_gate.sx(0)  \n",
    "sx_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(sx_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gerbang RZ\n",
    "\n",
    "Gerbang RZ melakukan rotasi sebesar $\\phi$ mengelilingi sumbu Z (Dimana $\\phi$ adalah angka riil). Yang memiliki matriks sebagai berikut:\n",
    "\n",
    "$RZ = \\begin{pmatrix}\n",
    "1 & 0 \\\\\n",
    "0 & e ^{i \\phi } \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "rz_gate = QuantumCircuit(1)\n",
    "rz_gate.rz(pi/2, 0)\n",
    "rz_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(rz_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Karena rotasi di sumbu Z tidak memiliki perbedaan ketika diterapkan di keadaan $|0\\rangle$, oleh karna itu kita akan menggunakan keadaan yang dibuat dengan menerapkan gerbang SX dan kemudian menerapkan RZ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "rz_gate.sx(0)\n",
    "rz_gate.rz(pi/2, 0)\n",
    "rz_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(rz_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gerbang Hadamard\n",
    "Gerbang Hadamard mewakili rotasi sebesar $\\pi$ di antara sumbu X dan Z.\n",
    "Yang memetakan basis keadaan $|0\\rangle$ menjadi $\\frac{|0\\rangle + |1\\rangle}{\\sqrt{2}}$, Yang dimana pengukuran akan memiliki kemungkinan yang sama antara `1` atau `0`, yang membuat 'superposisi' dari keadaan. Keadaan ini juga ditulis $|+\\rangle$. Yang dilakukan gerbang Hadamard adalah untuk mentransformasi dari $|0\\rangle$ $|1\\rangle$ dan basis $|+\\rangle$ $|-\\rangle$. \n",
    "\n",
    "$H = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1 & 1 \\\\\n",
    "1 & -1 \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Berikan gerbang H ke qubit |0>\n",
    "h_gate = QuantumCircuit(1)\n",
    "h_gate.h(0)\n",
    "h_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Lihat hasil dari gerbang H\n",
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(h_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gerbang CX (Gerbang CNOT)\n",
    "\n",
    "Gerbang *controlled* NOT (atau CNOT atau CX) berlaku untuk 2 qubit. Yang mengerjakan operasi NOT (yang sama dengan menerapkan gerbang X) ke qubit kedua ketika qubit pertama adalah $|1\\rangle$ dan jika tidak, tidak ada operasi yang di kerjakan. \n",
    "\n",
    "Catatan: Angka bit di *string* dalam Qiskit adalah kanan ke kiri.\n",
    "\n",
    "$CX = \\begin{pmatrix}\n",
    "1 & 0 & 0 & 0  \\\\\n",
    "0 & 1 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 1 \\\\\n",
    "0 & 0 & 1 & 0 \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cx_gate = QuantumCircuit(2)\n",
    "cx_gate.cx(0,1)\n",
    "cx_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gerbang CCX (gerbang Toffoli)\n",
    "\n",
    "Gerbang CCX (gerbang *controlled controlled* X) yang juga dikenal sebagai gerbang Toffoli. Gerbang CCX adalah gerbang 3 bit, dengan 2 kontrol dan 1 target sebagai masukkan dan keluaran. Jika 2 bit pertama berada dikeadaan $|1\\rangle$, akan menerapkan Pauli-X (atau NOT) di bit ketiga. Jika tidak, maka tidak ada operasi yang dikerjakan. \n",
    "\n",
    "Catatan: Angka bit di *string* dalam Qiskit adalah kanan ke kiri.\n",
    "\n",
    "$CCX = \\begin{pmatrix}\n",
    "1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
    "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ccx_gate = QuantumCircuit(3)\n",
    "ccx_gate.ccx(0,1,2)\n",
    "ccx_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Membuat gerbang logika dengan bantuan gerbang kuantum\n",
    "\n",
    "### Gerbang NOT\n",
    "\n",
    "Gerbang NOT membalik nilai bit dan, sesuai yang telah disebutkan, dimana gerbang X dapat disebut sebagai gerbang NOT. Tabel kebenaran dari gerbang NOT sebagai berikut:\n",
    "\n",
    "| Masukkan | Keluaran |\n",
    "| --- | --- | \n",
    "| 1 | 0 |\n",
    "| 0 | 1 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "not_gate=QuantumCircuit(1,1) # Membuat sirkuit kuantum dengan 1 qubit dan 1 bit klasik\n",
    "not_gate.x(0)\n",
    "not_gate.measure(0,0)\n",
    "not_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gerbang AND\n",
    "\n",
    "Keluaran dari gerbang AND benar jika dan hanya jika kedua masukkan adalah benar. Tabel kebenaran dari gerbang AND terlihat sebagai berikut :\n",
    "\n",
    "| A (Masukkan) | B (Masukkan) | Keluaran |\n",
    "| --- | --- | --- |\n",
    "| 0 | 0 | 0 | \n",
    "| 0 | 1 | 0 |\n",
    "| 1 | 0 | 0 |\n",
    "| 1 | 1 | 1 |\n",
    "\n",
    "Dengan gerbang Toffoli, kita mendapatkan hasil dari gerbang AND dengan menafsirkan 2 bit kontrol sebagai bit masukan dan target bit sebagai bit keluaran."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "and_gate=QuantumCircuit(3,1) # Membuat sirkuit kuantum dengan 3 qubit dan 1 bit klasik\n",
    "and_gate.ccx(0,1,2)\n",
    "and_gate.measure(2,0)\n",
    "and_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gerbang OR\n",
    "\n",
    "Gerbang OR mengembalikan benar jika salah satu input adalah benar.\n",
    "Tabel kebenaran gerbang OR sebagai berikut:\n",
    "\n",
    "| A (Masukkan) | B (Masukkan) | Keluaran |\n",
    "| --- | --- | --- |\n",
    "| 0 | 0 | 0 | \n",
    "| 0 | 1 | 1 |\n",
    "| 1 | 0 | 1 |\n",
    "| 1 | 1 | 1 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "or_gate=QuantumCircuit(3,1) # Membuat sirkuit kuantum dengan 3 qubit dan 1 bit klasik\n",
    "or_gate.cx(1,2)\n",
    "or_gate.cx(0,2)\n",
    "or_gate.ccx(0,1,2)\n",
    "or_gate.measure(2,0)\n",
    "or_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Menggunakan widget ssirkuit komposer\n",
    "\n",
    "Anda mungkin familiar dengan *IBM quantum cricuit composer*, yang dimana anda dapat menghasilkan sirkuit dengan antar muka grafis. Dengan sirkuit komposer widget yang baru, dengan fitur yang sama dapat digunakan di *jupyter notebook*. Anda dapat belajar cara menggunakan widget [disini](https://quantum-computing.ibm.com/lab/docs/iql/composer-widget)\n",
    "\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "**Latihan 1a:** Membuat gerbang NOR (merupakan negasi or) dengan sirkuit komposer\n",
    "    \n",
    "Latihan ini mendorong anda untuk bermain dengan komposer. Yang dimana tidak dinilai. Anda dapat melewatkan ini jika anda mau. Mengeksekusi sel berikut akan menampilkan komposer, yang anda dapat menambahkan gerbang dengan menarik ke tempat yang benar.    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from ibm_quantum_widgets import CircuitComposer\n",
    "editor = CircuitComposer()\n",
    "editor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Menggunakan widget sirkuit komposer untuk berkerja dengan sirkuit yang ada\n",
    "\n",
    "Kita dapat menggunakan sirkuit komposer untuk membuka sirkuit yang sebelumnya dibuat. Dengan kode dibawah kita akan membuka sirkuit diatas, yang dimana mewakili gerbang OR. Anda dapat untuk cek,Jika anda membangun OR secara benar.\n",
    "\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "Anda dapat mengubah sirkuit yang terbuka dengan tarik dan lepas. Coba menghapus pengukuran di akhir.\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "from ibm_quantum_widgets import CircuitComposer\n",
    "editor2 = CircuitComposer(circuit=or_gate)\n",
    "editor2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Di contoh dibawah kita menyimpan sirkuit terakhir dari komposer didalam variabel qc2 dan menerapkan X untuk output dan mengukur lagi.\n",
    "\n",
    "Gunakan sirkuit dari editor terakhir dan terapkan X ke output dan mengukur lagi."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "qc2 = editor2.circuit\n",
    "\n",
    "qc2.x(2)\n",
    "qc2.measure((2), (0))\n",
    "\n",
    "qc2.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sekarang kita membuat NOR yang merupakan negasi dari OR, yang wajib identik dengan sirkuit yang anda bangun di latihan pertama. \n",
    "\n",
    "## Gabungan gerbang kuantum dan biayanya\n",
    "\n",
    "Komputer kuantum secara normal tidak memiliki implementasi fisik dari semua gerbang. Akan tetapi mereka menggunakan bagian kecil dari gerbang dasar, yang dimana membentuk set gerbang universal, yang sama dengan masalah klasikal, instruksi set yang dimana dapat digunakan untuk implementasikan kesemua kemungkinan operasi.\n",
    "\n",
    "Untuk alasan ini, sirkuit kuantum wajib diubah menjadi gerbang basis sebelum dieksekusi. Yang dimana diubah secara otomatis oleh *Qiskit transpiler* ketika sirkuit kuantum dikirim ke sistem kuantum IBM. Tapi untuk tujuanpembelajaran, anda diekspetasi untuk membangun sirkuit dengan menggunakan gerbang basis. Gerbang basis di dalam IBM Quantum System adalah CX, ID, RZ, SX dan X. Anda dapat melihat [sistem `ibmq_mumbai`](https://quantum-computing.ibm.com/services?skip=0&systems=all&system=ibmq_mumbai) untuk menjadi contoh.\n",
    "\n",
    "Sekarang lihat sirkuit dibawah ini:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "qc = QuantumCircuit(2)\n",
    "qc.sxdg(0)\n",
    "qc.t(1)\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mari kita bagai mana dekomposisi dari sirkuit kuantum komputer diatas, yang menggunakan gerbang basis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "qc = QuantumCircuit(2)\n",
    "qc.sx(0)\n",
    "qc.sx(0)\n",
    "qc.sx(0)\n",
    "qc.rz(pi/4,1)\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Seperti yang anda dapat lihat, kita menggunakan gerbang basis, akan tetapi lebih banyak gerbang yang digunakan. Semakin banyak gerbang yang dimiliki sirkuit, semakin kompleks untuk dieksekusi. Jadi, ketika kita ingin menghitung biaya dari sirkuit, kita wajib mempertimbangkan jumlah gerbang yang digunakan. Akan tetapi, tidak semua gerbang memiliki biaya yang sama, oleh karena itu ketika kita menghitung biaya sirkuit, kita mengunakan rumus sebagai berikut:\n",
    "\n",
    "$$\n",
    "Biaya = 10 Jumlah_{CNOT} + Jumlah_{other}\n",
    "$$\n",
    "\n",
    "Dimana $Jumlah_{CNOT}$ adalah jumlah gerbanh CNOT dan $Jumlah_{other}$ adalah jumlah gerbang lain.\n",
    "\n",
    "### Gerbang Hadamard\n",
    "\n",
    "Seperti yang sudah dikatakan semua operasi dapat diekspresikan dengan gerbang basis. Seperti contoh yang kita bangun gerbang Hadamard dengan gerbang basis. Kita tidak memiliki gerbang basis yang dapat melakukan rotasi diantara sumbu X dan sumbu Z, oleh karna itu kita akan mengunakan rotasi di sumbu X dan Z untuk mendapatkan hasil yang sama. \n",
    "\n",
    "Bisakah anda tebak rotasi apa yang kita perlukan?"
   ]
  },
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   "execution_count": null,
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   "source": [
    "q=QuantumRegister(1)\n",
    "c=ClassicalRegister(1)\n",
    "qc=QuantumCircuit(q,c)\n",
    "qc.rz(pi/2, 0)\n",
    "qc.sx(0)\n",
    "qc.rz(pi/2, 0)\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Seperti yang mungkin anda ingat, sirkuit yang punya diatas, ketika kita memvisualisasikan rotasi dari gerbang RZ. Diatas kita melihat bahwa RZ pertama tidak melakukan apa apa, ketika kita berada di keadaan $|0\\rangle$ atau $|1\\rangle$. Jadi mungkin kita merasa tidak penting. Akan tetapi, jika kita berada di keadaan $|+\\rangle$ dan $|-\\rangle$ rotasi pertama akan memiliki efek. Kita memiliki masalah yang berbeda, setelah mengaplikasikan gerbang SX kita berada di keadaan $|0\\rangle$ atau $|1\\rangle$ dan RZ kedua tidak memiliki efek sama sekali.\n",
    "\n",
    "### Rotasi terkontrol\n",
    "\n",
    "Kita telah melihat NOT terkontrol diatas, sekarang perlihatkan contoh membangun rotasi terkontrol di sumbu Y. Rotasi $\\theta$ dapat memiliki nilai selain $\\pi$, ini hanya contoh. "
   ]
  },
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   "execution_count": null,
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   "source": [
    "qc = QuantumCircuit(2)\n",
    "theta = pi # Theta dapat bernilai apapun (pi dipilih untuk mempermudah visualisasi)\n",
    "qc.ry(theta/2,1)\n",
    "qc.cx(0,1)\n",
    "qc.ry(-theta/2,1)\n",
    "qc.cx(0,1)\n",
    "qc.draw(output='mpl')"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ketika 1 melewati sirkuit ini, 1 dapat dilihat jika qubit pertama adalah 0 maka 2 rotasi saling membatalkan satu sama yang lain dan tidak terjadi rotasi. \n",
    "\n",
    "Akan tetapi, jika qubit pertama bernilai 1 maka kita akan mendapatkan keadaan dimana dilakukan rotasi sebesar $\\theta / 2$ dua kali yang membentuk rotasi awal $\\theta$. Ini berkerja di sumbu X dan sumbu Y karena kedua sumbu bersifat ortogonal. \n",
    "\n",
    "<div class=\"alert alert-block alert-danger\">\n",
    "Untuk rotasi disumbu lain anda mungkin mengunakan trik yang lain.\n",
    "</div>\n",
    "\n",
    "### *Controlled controlled rotation*\n",
    "\n",
    "Diatas kita melihat contoh bagiamana melakukan rotasi di sumbu $Y$.\n",
    "Sekarang kita berasumsi kita memiliki rotasi terkontrol (di sumbu yang kita inginkan) dan ingin membuat rotasi terkontrol ganda, yang dimana rotasi diaplikasikan jika kedua qubit kontrol bernilai 1 yang sama dengan gerbang CCX."
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "qc = QuantumCircuit(3)\n",
    "theta = pi # Theta dapat bernilai apapun (pi dipilih untuk mempermudah visualisasi)\n",
    "qc.cp(theta/2,1,2)\n",
    "qc.cx(0,1)\n",
    "qc.cp(-theta/2,1,2)\n",
    "qc.cx(0,1)\n",
    "qc.cp(theta/2,0,2)\n",
    "qc.draw()"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "Didalam sirkuit ini jika qubit pertama dan kedua bernilai 0, maka tidak ada yang tejadi. Jika hanya qubit kedua bernilai 1,kita akan menerapkan rotasi sebesar $\\pi/2$ dan setelah itu diterapkan rotasi $-\\pi/2$ yang dimana saling membatalkan. Jika hanya qubit pertama bernilai 1, maka qubit kedua akan bernilai 1 setelah CX pertama akan berotasi sebesar$-\\pi/2$ yang akan di terapakan dan setela itu rotasi sebesar $\\pi/2$ akan diterpkan dan kedua rotasi akan saling membatalkan\n",
    "\n",
    "Jika kedua qubit bernilai 1 (qubit pertama dan kedua) maka akan terjadi rotasi sebesar $\\pi/2$ kemudian qubit kedua akan menjadi 0 sehingga rotasi kedua tidak terjadi dan akan dikembalikan menjadi 1. Setelah rotasi yang lain sebesar $\\pi/2$ akan diterapkan sejak qubit pertama bernilai 1. maka kita memiliki 2 kali rotasi sebesar $\\pi/2$ yang membentuk rotasi sebesar $\\pi$."
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Permasalahan\n",
    "\n",
    "<div id='problem'></div>\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "Kita telah melihat bagaimana cara membuat gerbang Hadamard dari set basis, dan sekarang kita ingin membangun gerbang Toffoli. Kenapa gerbang Toffoli? Seperti yang di jelaskan diatas, gerbang Toffoli merupakan gerbang universal untuk komputasi klasik yang sama dengan gerbang NAND, yang dimana bersifat reversibel. Selanjutnya akan membangun gerbang universal simple untuk komputasi kuantum jika digabung dengan gerbang Hadamard. \n",
    "\n",
    "Kita sudah melihat beberapa contoh bagaimana mengekspresikan gerbang yang kompleks dengan gerbang basis, sekarang kita ingin mengunakan ilmu yang didapatkan untuk membangun gerbang Toffoli dengan gerbang basis. Untuk dapat memecahkan menyelesaikan latihan ini, contoh diatas untuk cara membangun dan memakai rotasi terkontrol, akan menjadi berguna. Tantangan terbesar adalah membangun rotasi terkontrol yang diinginkan.\n",
    "    \n",
    "Anda dapat mengunakan kode dibawah untuk menggunakan widget komposer untuk membangun sirkuit anda.\n",
    "    \n",
    "</div>\n",
    "\n",
    "\n",
    "<div class=\"alert alert-block alert-danger\">\n",
    "\n",
    "Sebagai pengingat gerbang basis untuk IBM Quantum System adalah CX, RZ, SX dan X, jadi tidak diizinkan ada gerbang lain. \n",
    "    \n",
    "$$\n",
    "Biaya = 10 Jumlah_{CNOT} + Jumlah_{other}\n",
    "$$\n",
    "    \n",
    "</div>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qiskit import QuantumRegister, ClassicalRegister, QuantumCircuit\n",
    "from qiskit import IBMQ, Aer, execute\n",
    "from ibm_quantum_widgets import CircuitComposer\n",
    "editorEx = CircuitComposer() \n",
    "editorEx\n",
    "##### Bangun sirkuit anda dengan widget komposer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Anda juga dapat membangun sirkuit secara terprogram dengan menggunakan Qiskit\n",
    "\n",
    "circuit = QuantumCircuit(3)\n",
    "\n",
    "# Tulis kode anda diantara baris ini - MULAI\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Tulis kode anda diantara baris ini - AKHIR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Eksekusi sirkuit dengan qasm_simulator (simulator QASM)\n",
    "qc = editorEx.circuit \n",
    "#qc = circuit # Tutup komentar baris ini jika anda ingin mengumpulkan sirkuit dengan Qiskit\n",
    "\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Cek jawaban anda dengan kode dibawah ini\n",
    "from qc_grader import grade_ex1\n",
    "grade_ex1(qc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Kumpul jawaban anda. Anda dapat mengumpulkan jawaban anda sepanjang waktu\n",
    "from qc_grader import submit_ex1\n",
    "submit_ex1(qc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Additional information\n",
    "\n",
    "**Created by:** Marcel Pfaffhauser, Brahmani Thota, Junye Huang \n",
    "\n",
    "**Translate by:** Felix Montalfu\n",
    " \n",
    "**Version:** 1.0.0"
   ]
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